Model comparison via simplicial complexes and persistent homology
Sean T. Vittadello333School of BioSciences, The University of Melbourne,
Parkville VIC 3010 Australia,111Corresponding author:
<EMAIL_ADDRESS>and Michael P. H. Stumpf1,444School of
Mathematics and Statistics, The University of Melbourne, Parkville VIC 3010
Australia.

In many scientific and technological contexts we have only a poor
understanding of the structure and details of appropriate mathematical models.
We often, therefore, need to compare different models. With available data we
can use formal statistical model selection to compare and contrast the ability
of different mathematical models to describe such data. There is, however, a
lack of rigorous methods to compare different models _a priori_. Here we
develop and illustrate two such approaches that allow us to compare model
structures in a systematic way by representing models in terms of simplicial
complexes. Using well-developed concepts from simplicial algebraic topology,
we define a distance between models based on their simplicial representations.
Employing persistent homology with a flat filtration provides for alternative
representations of the models as persistence intervals, which represent the
structure of the models, from which we can also obtain the distances between
models. We then expand on this measure of model distance to study the concept
of model equivalence in order to determine the conceptual similarity of
models. We apply our methodology for model comparison to demonstrate an
equivalence between a positional-information model and a Turing-pattern model
from developmental biology, constituting a novel observation for two classes
of models that were previously regarded as unrelated.

††footnotetext: Key words and phrases: Turing pattern, positional information,
algebraic topology, algebraic biology, persistent homology, persistence
barcode, model distance, model equivalence

##  1 Introduction

Scientific models are representations of the physical world that isolate
features of interest through various levels of abstraction [1]. The complexity
of biological phenomena, exemplified in systems biology, necessitates the
development of models to further the understanding of these systems [2, 3, 4,
5].

There are many approaches to modelling biological systems, including continuum
models [6, 7, 8], rule-based models [9, 10, 11], network models [12, 13, 14],
and mechanistic models [5, 15], which are not necessarily mutually exclusive
classifications. While a model should represent the corresponding phenomenon
as faithfully as possible, it is also a requirement that the model is
manageable. It is neither practicable to develop a complete model of a
biological system, due to _combinatorial complexity_ [10, 3, 16], nor is it
necessary, since many components of a given system have relatively small
effects on the system behaviour [4, 17].

A biological system may be represented by various alternative models, which
can differ with respect to the specific features isolated by each model and
the interconnections between these features. Contrarily, similar models may
represent ostensibly divergent biological systems, where the difference
between the models may be only conceptual or, indeed, simply different
parameterisations [18]. The ability to rigorously and efficiently compare
models is therefore imperative for understanding complex biological systems,
and can elucidate the relationship between apparently distinct phenomena [19,
20, 21, 22, 23]. Model comparison is, however, a nontrivial task for multiple
reasons: the various representational forms of models; the different levels of
abstraction and granularity between models; and the need for a systematic
formalism that provides elucidation of conceptual similarities between models.
In particular, we require a way of obtaining distances between models: with a
meaningful distance between models we can start to cluster ‘similar’ models
and look for shared characteristics; we can explore how ‘different’ proposed
alternative models for a given process really are; and we can use them to
distill design principles from large sets of models that have been proposed to
model a given biological process or problem.

There are very few attempts to establish distances between models [24, 25],
and many of the candidates for a distance between models have obvious
shortcomings: graph-based distances on model structures, for example, fail
because many dynamical systems cannot be described in terms of simple graphs
[18]; other well-defined distances used in functional analysis are too limited
in scope, as are distances among e.g. stoichiometric matrices, which only
apply for certain types of models; semantic distances between model
formulations in e.g. CellML or SMBL, can depend too strongly on
inconsistencies in modelling terminology, and also can fail to distinguish
between structural and semantic differences; distances among model outputs
such as information-theoretical distances, distances used in control theory,
or correlation, (dis)similarity measures require assessment of model outputs
and generally do not allow a direct structural comparison between models;
finally, coding distances or the minimum description length would be promising
candidates for distances between models. We could consider these, but they,
too, have shortcomings, especially for different modelling frameworks (say
ordinary differential equations or stochastic Petri-nets used to model the
same scenario [26]).

Therefore, in this article we present a novel formalism for model comparison
that is flexible, systematic, quantitative, and visual. Since models consist
of a finite collection of components and their interconnections, we represent
a given model as a combinatorial object where the components of the model are
represented as labelled vertices. Further, since the interconnections between
model components may have dimension higher than the dyadic connections in
combinatorial graphs, we employ simplicial complexes to represent the models
which allow for interconnections of any finite dimension. Representing models
as simplicial complexes ensures that any model can be represented within our
framework, irrespective of form, granularity, and level of abstraction. We
have to define model components – for this we may draw on domain expertise
relevant to the scientific modelling problem at hand – and model comparison is
then performed by comparison of the simplicial representations of the models
given these components. Our framework is both flexible, incorporating
different models and modelling approaches, and rigorous.

Given the model components, the simplicial representations provide a graphical
form for the models; however we can also obtain visual representations of the
models as collections of persistence intervals by applying persistent homology
to the simplicial complexes. Persistent homology identifies the global
topological structure of a space that persists across multiple scales [27,
28], and is often employed in topological data analysis to study high-
dimensional data sets [29]. Persistent homology is robust with respect to
small perturbations in the input data, and represents the qualitative
structure of the input data in a compact manner [29]. While we use tools that
are also employed in topological data analysis, our aims here are different:
we are trying to represent models in a framework that allows for meaningful
comparisons, rather than looking for topological structure in high-dimensional
data. Our application of persistent homology is novel since our simplicial
complexes correspond to models rather than data sets, and we utilise the
persistence intervals associated with a simplicial complex as an alternative
representation of the model and hence the corresponding biological system. The
persistence-interval representations tend to facilitate the visual comparison
between various models, which may be more difficult with the simplicial
representations and particularly with the models in their original forms.
Indeed, since three- and higher-dimensional simplices cannot be drawn in two
dimensions, only projected, representations as persistence barcodes provide
immediate information for all dimensions in an often clearer visual form.

We provide two general methodologies for model comparison. The first is
comparison by distance, which measures the difference between models in terms
of the differences between the corresponding labelled simplicial
representations, and can be calculated either directly using the simplicial
complexes or indirectly with the corresponding persistence intervals. The
second is comparison by equivalence, where we identify any equivalences
between the components of the simplicial representations of different models,
and then employ operations on the complexes to transform one into the other.
Where this is possible we can learn about shared underlying characteristics
between models.

As a particular application of our methodology we compare the two main classes
of models for developmental-pattern formation, namely Turing-pattern and
positional-information models. While there is an extensive literature for
models of both Turing-pattern and positional-information systems, the
relationship between these models, and between the corresponding biological
systems, has long been unclear [30]. One main outcome of our model-comparison
methodology is the demonstration that the Turing-pattern activator-inhibitor
model is equivalent to the positional-information annihilation model from a
significant conceptual perspective, where the fundamental difference between
the two models is the location of the source of the gradient-forming
morphogen. This novel observation suggests that the location of the morphogen
source may influence the particular mechanism, namely Turing-pattern or
positional information, for gradient formation.

The remainder of this article is organised as follows. In Section 2 we
introduce our methodology for model comparison, which consists of five
subsections: Subsection 2.1 provides a brief discussion of the required
background in algebraic topology; in Subsection 2.2 we define our notion of a
simplicial representation of a model; in Subsection 2.3 we develop the
persistent homology of the simplicial representations of models; Subsection
2.4 provides the definition of distance between two simplicial representations
of models; and, in Subsection 2.5 we establish our notion of model
equivalence. We then apply our model-comparison methodology to two examples in
Section 3, firstly a comparison of bisubstrate enzyme reactions, and secondly
a comparison of Turing-pattern and positional-information models. Finally, in
Section 4 we summarise the utility of our methodology for comparing models.

##  2 Methodology

###  2.1 Background

We begin with a brief, informal, and self-contained discussion of the
background in simplicial algebraic topology that is relevant for our work. A
more detailed overview of simplicial complexes and homology is provided in the
Electronic Supplementary Material document. There is also a growing literature
on related issues in topological data analysis [29], but there the aims and
the details of implementation differ considerably from what we set out to
achieve here.

A _$p$ -simplex_ is a generalisation of a filled triangle in the plane to an
arbitrary dimension $p$, whereby a point is a 0-simplex, a line segment is a
1-simplex, a filled triangle is a 2-simplex, a filled tetrahedron is a
3-simplex, and so forth. We can think of 0-simplices as vertices and
1-simplices as edges, so that a simple graph is therefore a set of 0-simplices
and 1-simplices. A _$k$ -face_ of a simplex is a $k$-dimensional subsimplex,
and the 0-faces of a simplex are said to _span_ the simplex. A face $\tau$ of
a simplex $\sigma$ is _proper_ if $\tau\neq\sigma$.

A _simplicial complex_ is a generalisation of a simple graph, allowing for
simplicies of dimension higher than one which represent higher-dimensional
interactions. Specifically, a simplicial complex $K$ consists of a set of
simplices such that: if $\sigma$ is a simplex in $K$ then every face of
$\sigma$ is also in $K$; and, the nonempty intersection of any two simplices
in $K$ is a simplex in $K$. A _simplicial subcomplex_ $L$ of a simplicial
complex $K$ is a collection $L\subseteq K$ that is also a simplicial complex,
and a _simplicial supercomplex_ $M$ of $K$ is a collection $M\supseteq K$ that
is also a simplicial complex. The _$k$ -skeleton_ of a simplicial complex $K$
is the subcomplex $K^{(k)}$ consisting of the simplices in $K$ with dimension
at most $k$. In particular, the 0-skeleton $K^{(0)}$ is the set of vertices
and the 1-skeleton $K^{(1)}$ is the _underlying graph_ of the simplicial
complex $K$.

A simplicial complex $K$ is a topological space, formed from the union of its
simplices, and simplicial homology characterises the complex using algebraic
techniques to compute the number of $k$-dimensional holes in the complex. So,
for example, 0-dimensional holes are connected components and 1-dimensional
holes are nonbounding cycles of edges. The homology of $K$ depends on the
underlying space of simplices and their intersections in $K$. Persistent
homology studies the topological features, namely the $k$-dimensional holes,
of a complex $K$ across multiple scales, based on a _filtration_ of a
simplicial complex which is an increasing sequence of subcomplexes. Each
$k$-dimensional hole in $K$ is created at some index in the filtration, and
either persists through to the full complex $K$ or is annihilated at some
intermediate index. Persistent homology therefore gives a multiset of
_persistence intervals_ ${\mathcal{P}}(K)$ describing the creation and
annihilation of all $k$-dimensional holes in $K$. We denote by
${\mathcal{P}}_{k}(K)$ the submultiset of ${\mathcal{P}}(K)$ consisting of the
intervals corresponding to $k$-dimensional holes. The persistence intervals
can be visualised as a _persistence barcode_ , displaying the intervals as
horizontal line segments.

###  2.2 Simplicial representations of models

By a _model_ we mean an abstraction of an observable phenomenon [1]. A
specific detail of a given model is a _component_ of the model, and all models
consist of a finite number of components and their interconnections, which
represent direct relationships between specific components. For example, a
Turing-pattern system of reaction-diffusion equations, which is a model of a
developmental-patterning process, includes components such as morphogens,
diffusion, boundary conditions, reactions, and a morphogen gradient, along
with the interconnections between particular components such as a morphogen
and its diffusion.

In comparing models, including when defining a distance between models, it is
important to ensure that model components are defined consistently across all
models. Beyond this, the definition of model components typically arises
naturally from the scientific-modelling context. It is essential to keep in
mind that model comparisons are always with regard to the chosen level of
conceptual detail used to represent the models. The intention of a simplicial
representation of a model is to identify the concepts underlying the model,
and their interconnections, thereby removing the formality of the mathematical
representations to allow for the comparison of models within the same
framework.

We begin by defining the set of model components:

######  Definition 2.1 (Model components, generated model).

Let ${\mathcal{C}}$ be the set of all components that appear in the collection
of models under consideration for comparison. We say that each such model is
_generated_ by a subset of components from ${\mathcal{C}}$.

######  Notation 2.2.

Denote by ${\mathbb{N}}^{\ast}$ the set of positive integers, and by
$\overline{{\mathbb{N}}^{\ast}}$ the set of extended positive integers
${\mathbb{N}}^{\ast}\cup\\{+\infty\\}$. For $n\in{\mathbb{N}}^{\ast}$ we
denote the subset $\\{\,m\in{\mathbb{N}}^{\ast}\mid m\leq n\,\\}$ of
${\mathbb{N}}^{\ast}$ as $[n]$.

We represent model components as 0-simplices, and label the 0-simplices in a
flexible manner to allow for more efficient or detailed labelling as required:

######  Notation 2.3 (Representations of model components).

Let $\operatorname{Ord}\colon{\mathcal{C}}\to\big{[}|{\mathcal{C}}|\big{]}$ be
a bijection that specifies an arbitrary order for the categorical data
elements in ${\mathcal{C}}$. We represent each model component from
${\mathcal{C}}$ as a 0-simplex which is labelled variously as $i$, $v_{i}$ in
recognition that 0-simplices are vertices, or the name of the model component
given by $\operatorname{Ord}^{-1}(i)$, where $i$ is the position in the total
order specified by $\operatorname{Ord}$.

The ordering function $\operatorname{Ord}$ provides for computationally-
efficient labelling of simplices. For example, a 2-simplex can be labelled as
$\\{1,2,3\\}$, where the 0-simplices are $\\{1\\}$, $\\{2\\}$, and $\\{3\\}$,
and the 1-simplices are $\\{1,2\\}$, $\\{1,3\\}$, and $\\{2,3\\}$. Note that
we have no need for the orientation of the simplicial complexes that is
provided by the total ordering of the vertices by $\operatorname{Ord}$, since
we only consider simplicial homology over ${\mathbb{Z}}/2{\mathbb{Z}}$.

To a given model we can associate various related simplicial complexes
depending on the required levels of abstraction and component detail. Each
associated simplicial complex consists of the relevant labelled 0-simplices
that represent the model components, along with the one- and higher-
dimensional simplices that represent the interconnections between the
components, where the dyadic interactions are 1-simplices, the triadic
interactions are 2-simplices, and so forth as required. The labelling of the
0-simplices induces a labelling of the higher-dimensional simplices through
their spanning 0-simplices. Since the models of interest are associated with
the same set of components ${\mathcal{C}}$, we can identify a particular
labelled simplex, representing specific components and interconnections,
within the different simplicial complexes associated with the models.

A general algorithm to associate a labelled simplicial complex $K$ with a
model is as follows:

  1. 1.

Identify the agents in the model.

  2. 2.

Identify the reactions and interactions of the agents in the model, which may
provide for various levels of conceptual detail.

  3. 3.

Identify any boundary conditions for the model.

  4. 4.

Identify, if desired, the parameters in the model, which may allow for various
levels of detail.

  5. 5.

Define the set ${\mathcal{C}}$ as consisting of the components identified in
steps 1–4.

  6. 6.

The simplicial representation $K$ has $\lvert{\mathcal{C}}\rvert$ vertices
with the labels from ${\mathcal{C}}$.

  7. 7.

The 1-simplices of $K$ correspond to direct connections between the relevant
concepts, which are immediate from the model. To determine whether there
should be a 1-simplex linking two vertices it may be easier to consider
whether the absence of the 1-simplex is consistent with the model structure.

  8. 8.

Higher-dimensional simplices may or may not be required, depending on the
particular model and the desired level of representation. Any higher-
dimensional simplices must represent mutual interconnections between all model
concepts involved.

With regard to steps 7–8 we note that, while all one- and higher-dimensional
simplices of $K$ are completely determined by the model, such interconnections
between the model components may not be obvious and may require a deep
understanding of the model.

As an example we consider the Lotka-Volterra model [31, Chapter 3, page 79],
which is described by

$\frac{\mathrm{d}x}{\mathrm{d}t}=x(a-by)\quad\text{and}\quad\frac{\mathrm{d}y}{\mathrm{d}t}=y(cx-d),$
(1)

where $x$ is the prey density, $y$ is the predator density, and $a$, $b$, $c$,
and $d$ are positive real constants. The underlying assumptions of the model
are: in the absence of predation the prey population grows without bound; the
reduction of the prey’s per capita growth rate due to predation is
proportional to both the prey and predator populations; in the absence of prey
the predator’s death rate decays exponentially; and, the prey’s contribution
to the predator’s growth rate is proportional to both the available prey and
the size of the predator population. In Figure 1 we provide a simplicial
representation of the Lotka-Volterra model.

Figure 1: Simplicial representation of the Lotka-Volterra model. The
representation contains higher-order simplices which capture the interactions
between the components of the mathematical model. The vertices labelled
‘Prey’, ‘Prey growth’, ‘Predation’, and ‘Oscillatory population’ span a
3-simplex; the vertices labelled ‘Predator’, ‘Predator growth’, ‘Predation’,
and ‘Oscillatory population’ span a 3-simplex; and the vertices labelled
‘Predator’, ‘Predator growth’, ‘Predator death’, and ‘Oscillatory population’
span a 3-simplex.

The simplicial representation accounts for the underlying concepts of the
model rather than the particular mathematical form of the model, as we want to
avoid the specific details associated with any particular mathematical
representation to allow for direct comparison with other models.

To arrive at the simplicial representation in Figure 1 we identify the
underlying concepts of Equation 1: the agents are ‘Prey’ and ‘Predator’,
corresponding to the population variables $x$ and $y$, respectively; the terms
$ax$ and $-bxy$ on the right side of the prey equation correspond to ‘Prey
growth’ and ‘Predation’, respectively; the terms $cxy$ and $-dy$ on the right
side of the predator equation correspond to ‘Predator growth’ and ‘Predator
death’, respectively; finally, the solutions of Equation 1 exhibit oscillatory
dynamics for both populations, which we identify as ‘Oscillatory population’
in each case. The vertices of the simplicial representation are labelled with
these concepts. To form the higher-dimensional simplices we first identify the
one-dimensional interconnections of these model concepts, based on Equation 1,
as indicated in Table 1.

Table 1: Conceptual interconnections for Equation 1. A ‘Y’ indicates that there is an interconnection between the two concepts. |  Prey |  Prey |  |  |  |  |
---|---|---|---|---|---|---|---
growth |  Predation |  Oscillatory |  |  |  |  |
population |  Predator |  Predator |  |  |  |  |
growth |  Predator |  |  |  |  |  |
death |  |  |  |  |  |  |
Prey |  |  Y |  Y |  Y |  |  |
Prey growth |  Y |  |  Y |  Y |  |  |
Predation |  Y |  Y |  |  Y |  |  |
Oscillatory population |  Y |  Y |  Y |  |  Y |  Y |  Y
Predator |  |  |  Y |  Y |  |  Y |  Y
Predator growth |  |  |  Y |  Y |  Y |  |  Y
Predator death |  |  |  |  Y |  Y |  Y |

For example, ‘Prey’ is directly connected in a conceptual manner with ‘Prey
growth’, ‘Predation’, and ‘Oscillatory population’, where each pair of these
concepts forms a 1-simplex, each subset of three of these four concepts forms
a 2-simplex, and all four of these concepts form a 3-simplex. Note that
‘Predation’ and ‘Predator death’ have no direct connection in the simplicial
representation since the effect of predator death on predation is indirect
through predator growth.

Consideration of the conceptual basis of models allows for constructive
comparison between models irrespective of their mathematical form. Since we
generally need to compare models with different forms, we construct simplicial
representations that contain the essential components of the models and their
interconnections — components here naturally depend on the scientific problem
studied. For instance, we may be interested in comparing a continuum model and
a discrete model of the same system in terms of the components of the system
that are identified in each model, in order to understand how the models are
related. The simplicial representation in Figure 1 is such a conceptual
representation of the Lotka-Volterra model, whereby the simplicial
representation provides the essential concepts and interactions underlying the
model: the vertices correspond to the underlying concepts of the model, the
edges link concepts that are directly interconnected, and the 2- and
3-simplices indicate higher-dimensional interconnections between related
concepts.

###  2.3 Homology of simplicial representations

Given a simplicial representation of a model we employ persistent homology to
provide a unique and visual characterisation of the model. Note that
simplicial homology of the full simplicial complex, while simpler than
persistent homology, is not applicable here since it does not in general
provide a unique characterisation of a model. Indeed, simplicial homology
calculates the number of $k$-dimensional holes of the full simplicial complex,
for nonnegative integers $k$, and the numbers of $k$-dimensional holes can be
equal for distinctly-labelled complexes. A simple example is given by two
simplicial complexes that are each just a 0-simplex, where the 0-simplex has a
different label in each complex. Then the two labelled complexes are distinct,
however they have the same numbers of $k$-dimensional holes since they both
consist of a single component, or 0-dimensional hole, and have no higher-
dimensional holes. We therefore require the filtration in persistent homology
in order to uniquely characterise distinct simplicial complexes. This is
different from the situation typically encountered in topological data
analysis.

To compare a collection of simplicial representations of models using
persistent homology we require filtrations for each complex that are
compatible, so that the same topological structures in different complexes
give the same persistence intervals. For this we utilise a reference complex
such that the simplicial complexes associated to all of the models for
comparison are subcomplexes.

######  Notation 2.4 (Reference complex).

Let ${\mathcal{C}}$ be the set of all components that appear in the collection
of models under consideration for comparison, and let $m$ be the maximum
dimension of the simplicial complexes associated to the models. Denote by
${\mathcal{R}}$ the $(|{\mathcal{C}}|-1)$-simplex spanned by the labelled
0-simplices $\\{v_{1},v_{2},\ldots,v_{|{\mathcal{C}}|}\\}$. The _reference
complex_ is the $m$-skeleton of ${\mathcal{R}}$, therefore
${\mathcal{R}}^{(m)}$. Note that each simplicial representation of a model
under consideration is a subcomplex of the reference complex.

Therefore, to obtain filtrations for the simplicial complexes that allow for
direct comparison of the persistence barcodes we first obtain a filtration for
the reference complex ${\mathcal{R}}^{(m)}$. The filtration of each simplicial
representation is then an induced filtration of the filtration for
${\mathcal{R}}^{(m)}$. In this way we can directly compare the persistence
intervals between different models, since identical persistence intervals will
arise from identical topological features associated with the same subset of
model components.

Note that we could always use the simplex ${\mathcal{R}}$ as the reference
complex, however it is computationally efficient to use the $m$-skeleton
${\mathcal{R}}^{(m)}$ where possible. If we then need to compare an additional
model generated by a subset of ${\mathcal{C}}$ whose simplicial representation
has dimension higher than $m$, we can simply extend the filtration of the
reference simplex to a filtration of a higher-dimensional skeleton of
${\mathcal{R}}$ (see Proposition 2.13).

We want the persistent homology of a particular simplicial complex to be
unique amongst all of the simplicial complexes of the models of interest, in
order to reflect all differences between the complexes and hence models. We
therefore employ a _flat filtration_ [32, Chapter 11, Page 83] whereby only
one simplex is added at each filtration index, and each simplex is added after
all of its proper faces.

######  Definition 2.5 (Flat filtration).

Let ${\mathcal{C}}$ be the set of all components that appear in the collection
of models under consideration for comparison, and let ${\mathcal{R}}^{(m)}$ be
the associated reference complex. Further, let
$w\colon{\mathcal{R}}^{(m)}\to\big{[}|{\mathcal{R}}^{(m)}|\big{]}$ be a
bijective weight function such that if $\tau$, $\sigma\in{\mathcal{R}}^{(m)}$
and $\tau$ is a proper face of $\sigma$ then $w(\tau)<w(\sigma)$. The _flat
filtration_ of $\big{(}{\mathcal{R}}^{(m)},w\big{)}$ is the nested sequence of
subcomplexes $\\{R_{i}\\}_{i=1}^{|{\mathcal{R}}^{(m)}|}$, where
$R_{i}=w^{-1}\big{(}(-\infty,i]\big{)}$ for
$i\in\big{[}|{\mathcal{R}}^{(m)}|\big{]}$.

The flat filtration on ${\mathcal{R}}^{(m)}$ induces a filtration on each
subcomplex of ${\mathcal{R}}^{(m)}$:

######  Definition 2.6 (Induced filtration of a subcomplex of the reference
complex).

Let ${\mathcal{C}}$ be the set of all components that appear in the collection
of models under consideration for comparison, and let
$\big{(}{\mathcal{R}}^{(m)},w\big{)}$ be the associated reference complex with
a flat filtration $\\{R_{i}\\}_{i=1}^{|{\mathcal{R}}^{(m)}|}$. Further, let
$L$ be a simplicial representation of a model generated by a subset of
components from ${\mathcal{C}}$. Then the _induced filtration_ for the
subcomplex $L\subseteq{\mathcal{R}}^{(m)}$ is the nested sequence of
subcomplexes $\\{F_{i}\\}_{i\in I}$ where, for each $i\in I$,
$F_{i}=w|_{L}^{-1}\big{(}(-\infty,i]\big{)}=R_{i}\cap L\neq\emptyset$, and
$i\leq j$ for all $j$ such that $1\leq j\leq|{\mathcal{R}}^{(m)}|$ and
$R_{i}\cap L=R_{j}\cap L$.

Note that for the induced filtration we exclude filtration indices $i$ for
which $R_{i}\cap L=\emptyset$, and since we may have $R_{i}\cap L=R_{j}\cap L$
for some filtration indices $i\neq j$ we take the smallest such index as the
corresponding filtration index. Note further that the induced filtration adds
one simplex at each index of the filtration, which allows us to use it as a
basis for a distance between models (given the set of model components
${\mathcal{C}}$) as we show below (see Definition 2.16 and Theorem 2.18).

We now establish that the simplicial representations and associated
filtrations provide the desired descriptions of the models. The following
theorem shows that the simplicial complex associated with a given model has a
unique multiset of persistence intervals with respect to the induced
filtration. Therefore the persistence intervals, visualised as a persistence
barcode, provide a unique ‘fingerprint’ of the model.

######  Theorem 2.7.

Let ${\mathcal{C}}$ be the set of all components that appear in the collection
of models under consideration for comparison, and let ${\mathcal{R}}^{(m)}$ be
the associated reference complex. Let $K$ and $L$ be two labelled simplicial
complexes associated with two models generated by subsets of ${\mathcal{C}}$,
and let $K$ and $L$ have the induced filtrations from a flat filtration on
${\mathcal{R}}^{(m)}$. Then $K=L$ if and only if
${\mathcal{P}}_{k}(K)={\mathcal{P}}_{k}(L)$ for all $k\geq 0$.

###### Proof.

The forward implication is trivial so consider the backward implication, for
which we provide a contrapositive proof. Suppose that $K\neq L$, and let
$\sigma^{p}$ be a simplex of smallest dimension $p$ in the symmetric
difference $K\triangle L$. Without loss of generality we may assume that
$\sigma^{p}\in K$. We show that there exists $k\geq 0$ such that
${\mathcal{P}}_{k}(K)\neq{\mathcal{P}}_{k}(L)$.

If $p=0$ then ${\mathcal{P}}_{0}(K)$ contains an interval with left endpoint
equal to the filtration index $i$ at which $\sigma^{p}$ is added to the
filtered complex, corresponding to the creation of a 0-dimensional homology
class. Since $\sigma^{p}\notin L$ there is no simplex added at index $i$ in
the filtration of $L$, hence no interval with left endpoint $i$ in
${\mathcal{P}}_{0}(L)$.

For $p\geq 1$ the boundary of $\sigma^{p}$ is a $(p-1)$-cycle $\tau^{p-1}$,
which is a nonbounding cycle in the filtered complex prior to the addition of
$\sigma^{p}$. So either $\tau^{p-1}$ corresponds to a linearly independent
$(p-1)$-dimensional homology class, or the homology class for $\tau^{p-1}$ is
a linear combination of the $(p-1)$-dimensional homology classes. If
$\tau^{p-1}$ is in a linearly independent homology class then
${\mathcal{P}}_{p-1}(K)$ contains an interval with right endpoint equal to the
index $j$ of the filtration where the homology class is annihilated by the
addition of $\sigma^{p}$. Since $\sigma^{p}\notin L$ there is no interval in
${\mathcal{P}}_{p-1}(L)$ with the same right endpoint $j$. Alternatively,
suppose the homology class for $\tau^{p-1}$ is a linear combination of the
$(p-1)$-dimensional homology classes. If the addition of $\sigma^{p}$ at some
index $i$ in the filtration creates a $p$-cycle, and hence a new
$p$-dimensional homology class, containing $\sigma^{p}$ then there is a
corresponding interval in ${\mathcal{P}}_{p}(K)$ with left endpoint $i$ and no
interval in ${\mathcal{P}}_{p}(L)$ with left endpoint $i$. Otherwise, if the
addition of $\sigma^{p}$ at some index $i$ in the filtration does not create a
$p$-cycle then, since $\tau^{p-1}$ is now a bounding cycle, it follows that
one of the previously linearly independent $(p-1)$-dimensional homology
classes is now linearly dependent and therefore annihilated at index $i$.
Therefore, there is a corresponding interval in ${\mathcal{P}}_{p-1}(K)$ with
right endpoint $i$, and no interval in ${\mathcal{P}}_{p-1}(L)$ with right
endpoint $i$. ∎

We can rephrase Theorem 2.7 in terms of the full multisets of persistence
intervals:

######  Corollary 2.8.

Let ${\mathcal{C}}$ be the set of all components that appear in the collection
of models under consideration for comparison, and let ${\mathcal{R}}^{(m)}$ be
the associated reference complex. Let $K$ and $L$ be two labelled simplicial
complexes associated with two models generated by subsets of ${\mathcal{C}}$,
and let $K$ and $L$ have the induced filtrations from a flat filtration on the
reference simplex ${\mathcal{R}}^{(m)}$. Then $K=L$ if and only if
${\mathcal{P}}(K)={\mathcal{P}}(L)$.

###### Proof.

The forward implication follows from Theorem 2.7. The backward implication
will also follow from Theorem 2.7 if we establish that
${\mathcal{P}}(K)={\mathcal{P}}(L)$ if and only if
${\mathcal{P}}_{k}(K)={\mathcal{P}}_{k}(L)$ for all $k\geq 0$. Note that
${\mathcal{F}}(K)=\\{\,{\mathcal{P}}_{k}(K)\mid\text{${\mathcal{P}}_{k}(K)\neq\emptyset$
and $k\geq 0$}\,\\}$ is a partition of ${\mathcal{P}}(K)$, and similarly for
${\mathcal{F}}(L)$. So $(a,b)\in{\mathcal{P}}(K)$ implies
$(a,b)\in{\mathcal{P}}_{k}(K)$ for some $k\geq 0$, and if
$(a,b)\in{\mathcal{P}}(L)$ then $(a,b)\in{\mathcal{P}}_{j}(L)$ for some $j\geq
0$. Since both $K$ and $L$ are filtered with induced filtrations, whereby at
most one simplex is added at each filtration index, the dimension of the
simplex added at index $a$ uniquely determines the dimension of the homology
class created at index $a$ corresponding to the interval $(a,b)$. Therefore
$k=j$, and it follows that ${\mathcal{P}}(K)={\mathcal{P}}(L)$ is equivalent
to ${\mathcal{P}}_{k}(K)={\mathcal{P}}_{k}(L)$ for all $k\geq 0$. ∎

For a general filtration the multiplicity of each persistence interval may be
greater than one. We now show that with the induced filtration, however, the
multiplicity of each persistence interval is equal to one, so that the
multiset of persistence intervals is a set.

######  Proposition 2.9.

Let ${\mathcal{C}}$ be the set of all components that appear in the collection
of models under consideration for comparison, and let ${\mathcal{R}}^{(m)}$ be
the associated reference complex. Let $K$ be a labelled simplicial complex
associated with a model generated by a subset of ${\mathcal{C}}$, and let $K$
have the induced filtration from a flat filtration on the reference complex
${\mathcal{R}}^{(m)}$. Then every interval in ${\mathcal{P}}(K)$ has
multiplicity one.

###### Proof.

It suffices to show that no two intervals in ${\mathcal{P}}(K)$ have the same
left endpoint. So suppose to the contrary that the two intervals $I_{1}$ and
$I_{2}$ in the multiset ${\mathcal{P}}(K)$ have the same left endpoint.

Note that $I_{1}$ and $I_{2}$ do not correspond to homology classes in
different dimensions. Indeed, one $p$-cycle and one $q$-cycle with $p<q$
cannot be created at the same index as this would require the simultaneous
addition of a $(p-1)$-simplex and a $(q-1)$-simplex, respectively, which is
not possible with the induced filtration.

So $I_{1}$ and $I_{2}$ must correspond to linearly-independent homology
classes in the same dimension, and therefore to the creation of two linearly-
independent nonbounding $p$-cycles $\sigma_{1}$ and $\sigma_{2}$ at the same
filtration index $i$ due to the addition of a $p$-simplex $\tau$. Then
$\sigma_{1}+\sigma_{2}$ under ${\mathbb{Z}}/2{\mathbb{Z}}$ addition is a
nonbounding cycle at index $i-1$, so the addition of $\tau$ at index $i$
yields one new homology class, contradicting the creation of two homology
classes. Therefore, there is at most one interval in ${\mathcal{P}}(K)$ with a
particular left endpoint. ∎

While we have established that the multiset of persistence intervals is a set,
we continue to describe it as a multiset for clarity of reference.

Since a submodel consists of a subset of the components and interconnections
of a given model, we expect a similar relationship to hold between the
simplicial representations and the corresponding persistent homology, which we
now confirm.

######  Proposition 2.10.

Let ${\mathcal{C}}$ be the set of all components that appear in the collection
of models under consideration for comparison, and let ${\mathcal{R}}^{(m)}$ be
the associated reference complex. Let $\mathsf{M}$ and $\mathsf{N}$ be two
models for comparison, where $\mathsf{M}$ is a submodel of $\mathsf{N}$, and
let $K$ and $L$ be the associated labelled simplicial complexes of
$\mathsf{M}$ and $\mathsf{N}$, respectively, generated by subsets of
${\mathcal{C}}$. Let $K$ and $L$ have the induced filtrations from a flat
filtration on ${\mathcal{R}}^{(m)}$. Then $K$ is a subcomplex of $L$. Further,
for each $k\geq 0$, the submultiset of finite intervals from
${\mathcal{P}}_{k}(K)$ is a submultiset of ${\mathcal{P}}_{k}(L)$, and each
infinite interval in ${\mathcal{P}}_{k}(K)$ is either in
${\mathcal{P}}_{k}(L)$ or corresponds to a finite interval in
${\mathcal{P}}_{k}(L)$ with the same left endpoint.

###### Proof.

Since $\mathsf{M}$ is a submodel of $\mathsf{N}$, all of the model components
and interconnections in $\mathsf{M}$ are also in $\mathsf{N}$, so the
corresponding simplicial complexes satisfy $K\subseteq L$.

Now, fix $k\geq 0$ and let $(a,b)$ be a finite interval in
${\mathcal{P}}_{k}(K)$ corresponding to the creation and subsequent
annihilation of a $k$-homology class at indices $a$ and $b$, respectively, in
the filtration. Since $K\subseteq L$, the subcomplex of $K$ corresponding to
the $k$-homology class is also a subcomplex of $L$, so
$(a,b)\in{\mathcal{P}}_{k}(L)$.

Suppose now that the infinite interval $(a,\infty)$ is in
${\mathcal{P}}_{k}(K)$, corresponding to the creation of a $k$-homology class
at index $a$ in the filtration which is not annihilated and persists in the
full complex $K$. Suppose further, without loss of generality, that
$(a,\infty)\notin{\mathcal{P}}_{k}(L)$. Since $K\subseteq L$, the subcomplex
of $K$ corresponding to the $k$-homology class is also a subcomplex of $L$, so
the $k$-homology class is also created in the filtration of $L$. Since
$(a,\infty)\notin{\mathcal{P}}_{k}(L)$, $L$ contains the $(k+1)$-simplex for
which the $k$-homology class is the bounding cycle, so the $k$-homology class
is annihilated at some index $b$ in the filtration of $L$ due to the addition
of the $(k+1)$-simplex. Thus, $(a,b)\in{\mathcal{P}}_{k}(L)$. ∎

We visualise the persistent homology of a simplicial complex, summarised by
the multisets of persistence intervals, as a persistence barcode. The barcodes
have the distinct advantage of allowing for a visual comparison of models.

######  Definition 2.11 (Persistence barcode).

Let ${\mathcal{C}}$ be the set of all components that appear in the collection
of models under consideration for comparison, and let ${\mathcal{R}}^{(m)}$ be
the associated reference complex. Let $K$ be a labelled simplicial complex
associated with a model generated by a subset of ${\mathcal{C}}$, and let $K$
have the induced filtration from a flat filtration on the reference complex
${\mathcal{R}}^{(m)}$. For each dimension $p\geq 0$ the $p$th _persistence
barcode_ $\operatorname{BAR}_{p}(K)$ for $K$ is a graphical representation of
the multiset of $p$-dimensional persistence intervals ${\mathcal{P}}_{p}(K)$.
Specifically, let
$f\colon{\mathcal{P}}_{p}(K)\to\big{[}|{\mathcal{P}}_{p}(K)|\big{]}$ be a
bijection, then $\operatorname{BAR}_{p}(K)$ is given by the set of points
$\big{\\{}\,\big{(}x,f(a,b)\big{)}\mid\text{$(a,b)\in{\mathcal{P}}_{p}(K)$ and
$x\in[a,b)$}\,\big{\\}}$, which is unique up to the bijection specifying the
order with respect to the second coordinate.

While two different flat filtrations of the reference complex
${\mathcal{R}}^{(m)}$ can produce two different multisets of persistence
intervals for a subcomplex $K$, the numbers of finite and infinite intervals
in each submultiset ${\mathcal{P}}_{k}(K)$ are unchanged.

######  Proposition 2.12.

Let ${\mathcal{C}}$ be the set of all components that appear in the collection
of models under consideration for comparison, and let ${\mathcal{R}}^{(m)}$ be
the associated reference complex. Let $K$ be a labelled simplicial complex
associated with a model generated by a subset of components in
${\mathcal{C}}$, let ${\mathcal{F}}$ and ${\mathcal{F}}^{\prime}$ be two
induced filtrations of $K$ corresponding to two different flat filtrations of
the reference complex ${\mathcal{R}}^{(m)}$, and let ${\mathcal{P}}(K)$ and
${\mathcal{P}}^{\prime}(K)$ be the corresponding multisets of persistence
intervals. Then, for all $k\geq 0$,
$|{\mathcal{P}}_{k}(K)|=|{\mathcal{P}}^{\prime}_{k}(K)|$, and in particular
the numbers of finite and infinite intervals are the same in
${\mathcal{P}}_{k}(K)$ and ${\mathcal{P}}^{\prime}_{k}(K)$.

###### Proof.

Fix $k\geq 0$. We first establish a bijection
$f\colon{\mathcal{P}}_{k}(K)\to{\mathcal{P}}^{\prime}_{k}(K)$. Let
$I\in{\mathcal{P}}_{k}(K)$, with left endpoint $i$ corresponding to the
creation of a nonbounding $k$-cycle $\sigma^{k}$ at index $i$ of filtration
${\mathcal{F}}$. Since the $k$-cycle is also created at some index $j$ of
filtration ${\mathcal{F}}^{\prime}$, there exists
$I^{\prime}\in{\mathcal{P}}^{\prime}_{k}(K)$ with left endpoint $j$, so we
define $f(I)=I^{\prime}$. The mapping $f$ is well defined, since each
topological feature appears at a unique filtration index.

We need to show that $f$ is injective and surjective. For injectivity, suppose
that $I_{1}$, $I_{2}\in{\mathcal{P}}_{k}(K)$ with $f(I_{1})=f(I_{2})$. In
particular, the intervals $f(I_{1})$ and $f(I_{2})$ have the same left
endpoint corresponding to the creation of a nonbounding $k$-cycle in the
filtration ${\mathcal{F}}^{\prime}$, and since this nonbounding $k$-cycle is
created at the left endpoints of both $I_{1}$ and $I_{2}$ we have
$I_{1}=I_{2}$. For surjectivity, let
$I^{\prime}\in{\mathcal{P}}^{\prime}_{k}(K)$. Then the left endpoint of
$I^{\prime}$ corresponds to the creation of a nonbounding $k$-cycle in the
filtration ${\mathcal{F}}^{\prime}$, and this nonbounding $k$-cycle is also
created at some index $i$ of the filtration ${\mathcal{F}}^{\prime}$, so there
exists $I\in{\mathcal{P}}_{k}(K)$ with left endpoint $i$, so that
$f(I)=I^{\prime}$.

Finally, the two induced filtrations correspond to the same simplicial complex
$K$, so ${\mathcal{P}}_{k}(K)$ and ${\mathcal{P}}^{\prime}_{k}(K)$ have the
same number of infinite intervals, and therefore the same number of finite
intervals. ∎

We can extend the model comparison to include new models for which not all
components are in ${\mathcal{C}}$ by appending the new components to
${\mathcal{C}}$ to obtain a superset $\mkern
1.5mu\overline{\mkern-1.5mu{\mathcal{C}}\mkern-1.5mu}\mkern 1.5mu$, and then
extending the ordering function on ${\mathcal{C}}$ to $\mkern
1.5mu\overline{\mkern-1.5mu{\mathcal{C}}\mkern-1.5mu}\mkern 1.5mu$. This may
be necessary as our set of candidate models grows, or if we want to increase
the level of detail in which we describe models. This extension method
preserves the simplicial representations, and associated persistence
intervals, of the models generated by ${\mathcal{C}}$. Notationally, we use an
overline to denote concepts in the extended system.

######  Proposition 2.13.

Let ${\mathcal{C}}$ and $\mkern
1.5mu\overline{\mkern-1.5mu{\mathcal{C}}\mkern-1.5mu}\mkern 1.5mu$ be two sets
of model components with $\mkern
1.5mu\overline{\mkern-1.5mu{\mathcal{C}}\mkern-1.5mu}\mkern
1.5mu\supseteq{\mathcal{C}}$. Let $\mkern
1.5mu\overline{\mkern-1.5mu\operatorname{Ord}\mkern-1.5mu}\mkern
1.5mu\colon\mkern 1.5mu\overline{\mkern-1.5mu{\mathcal{C}}\mkern-1.5mu}\mkern
1.5mu\to\big{[}|\mkern
1.5mu\overline{\mkern-1.5mu{\mathcal{C}}\mkern-1.5mu}\mkern 1.5mu|\big{]}$ be
a bijection specifying an order for $\mkern
1.5mu\overline{\mkern-1.5mu{\mathcal{C}}\mkern-1.5mu}\mkern 1.5mu$ such that
$\mkern 1.5mu\overline{\mkern-1.5mu\operatorname{Ord}\mkern-1.5mu}\mkern
1.5mu|_{{\mathcal{C}}}=\operatorname{Ord}$. Further, let ${\mathcal{R}}^{(m)}$
and $\mkern 1.5mu\overline{\mkern-1.5mu{\mathcal{R}}\mkern-1.5mu}\mkern
1.5mu^{(m)}$ be the reference complexes associated with ${\mathcal{C}}$ and
$\mkern 1.5mu\overline{\mkern-1.5mu{\mathcal{C}}\mkern-1.5mu}\mkern 1.5mu$,
respectively. Then we have the following:

  1. 1.

The reference complexes satisfy ${\mathcal{R}}^{(m)}\subseteq\mkern
1.5mu\overline{\mkern-1.5mu{\mathcal{R}}\mkern-1.5mu}\mkern 1.5mu^{(m)}$.

  2. 2.

Each flat filtration of ${\mathcal{R}}^{(m)}$ can be extended to a flat
filtration of $\mkern
1.5mu\overline{\mkern-1.5mu{\mathcal{R}}\mkern-1.5mu}\mkern 1.5mu^{(m)}$.

  3. 3.

Suppose ${\mathcal{R}}^{(m)}$ has a flat filtration, which is extended to a
flat filtration on $\mkern
1.5mu\overline{\mkern-1.5mu{\mathcal{R}}\mkern-1.5mu}\mkern 1.5mu^{(m)}$. For
a simplicial representation $K$ of a model generated by a subset of components
in ${\mathcal{C}}$ we have ${\mathcal{P}}_{k}(K)=\mkern
1.5mu\overline{\mkern-1.5mu{\mathcal{P}}\mkern-1.5mu}\mkern 1.5mu_{k}(K)$ for
all $k\geq 0$. In other words, the persistent homology for $K$ corresponding
to the induced filtration from the flat filtration on ${\mathcal{R}}^{(m)}$ is
the same as the persistent homology for $K$ corresponding to the induced
filtration from the extended flat filtration on $\mkern
1.5mu\overline{\mkern-1.5mu{\mathcal{R}}\mkern-1.5mu}\mkern 1.5mu^{(m)}$.

###### Proof.

For 1, note that ${\mathcal{R}}^{(m)}$ is spanned by a subset of the
0-simplicies that span $\mkern
1.5mu\overline{\mkern-1.5mu{\mathcal{R}}\mkern-1.5mu}\mkern 1.5mu^{(m)}$, so
${\mathcal{R}}^{(m)}$ is a subcomplex of $\mkern
1.5mu\overline{\mkern-1.5mu{\mathcal{R}}\mkern-1.5mu}\mkern 1.5mu^{(m)}$.

For 2, let $w\colon{\mathcal{R}}^{(m)}\to\big{[}|{\mathcal{R}}^{(m)}|\big{]}$
be a bijective weight function such that if $\tau$ is a proper face of
$\sigma$ then $w(\tau)<w(\sigma)$, yielding a flat filtration
$\\{R_{i}\\}_{i=1}^{|{\mathcal{R}}^{(m)}|}$ where each
$R_{i}=w^{-1}\big{(}(-\infty,i]\big{)}$. Extend $w$ to a bijective weight
function $\mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern
1.5mu\colon\mkern 1.5mu\overline{\mkern-1.5mu{\mathcal{R}}\mkern-1.5mu}\mkern
1.5mu^{(m)}\to\big{[}|\mkern
1.5mu\overline{\mkern-1.5mu{\mathcal{R}}\mkern-1.5mu}\mkern
1.5mu^{(m)}|\big{]}$ such that if $\tau$ is a proper face of $\sigma$ then
$\mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu(\tau)<\mkern
1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu(\sigma)$. Then the flat
filtration $\\{\mkern 1.5mu\overline{\mkern-1.5muR\mkern-1.5mu}\mkern
1.5mu_{i}\\}_{i=1}^{|\mkern
1.5mu\overline{\mkern-1.5mu{\mathcal{R}}\mkern-1.5mu}\mkern 1.5mu^{(m)}|}$ of
$\mkern 1.5mu\overline{\mkern-1.5mu{\mathcal{R}}\mkern-1.5mu}\mkern
1.5mu^{(m)}$, where $\mkern 1.5mu\overline{\mkern-1.5muR\mkern-1.5mu}\mkern
1.5mu_{i}=\mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern
1.5mu^{-1}\big{(}(-\infty,i]\big{)}$, extends the flat filtration of
${\mathcal{R}}^{(m)}$, that is, $R_{i}=\mkern
1.5mu\overline{\mkern-1.5muR\mkern-1.5mu}\mkern 1.5mu_{i}$ for
$i\in\big{[}|{\mathcal{R}}^{(m)}|\big{]}$.

For 3, let $K$ be a simplicial representation of a model generated by a subset
of components in ${\mathcal{C}}$. Since the flat filtration of
${\mathcal{R}}^{(m)}$ and the extended flat filtration of $\mkern
1.5mu\overline{\mkern-1.5mu{\mathcal{R}}\mkern-1.5mu}\mkern 1.5mu^{(m)}$ are
equal over the first $|{\mathcal{R}}^{(m)}|$ filtration indices, the
corresponding induced filtrations are equal, and it follows that
${\mathcal{P}}_{k}(K)=\mkern
1.5mu\overline{\mkern-1.5mu{\mathcal{P}}\mkern-1.5mu}\mkern 1.5mu_{k}(K)$ for
all $k\geq 0$. ∎

###  2.4 Model comparison by distance

To compare models of interest, given a set of model components
${\mathcal{C}}$, in a quantitative manner we introduce a measure of distance
between the simplicial representations. We provide a direct measure between
the simplicial complexes, along with a measure based on the corresponding
multisets of persistence intervals, and then show that the two measures yield
the same distance.

######  Definition 2.14 (Distance between simplicial complexes).

Let ${\mathcal{C}}$ be the set of all components that appear in the collection
of models under consideration for comparison. Let ${\mathcal{S}}$ be a
collection of labelled simplicial complexes corresponding to models generated
by subsets from ${\mathcal{C}}$. Define the function

$d_{{\mathcal{C}}}\colon{\mathcal{S}}\times{\mathcal{S}}\to{\mathbb{R}}\quad\text{such
that}\quad d_{{\mathcal{C}}}(K,L)=|K\triangle L|,$ (2)

so that $d_{{\mathcal{C}}}$ gives the cardinality of the symmetric difference
of the two labelled simplicial complexes.

Some important remarks regarding the distance function between simplicial
complexes:

  * •

The symmetric difference of two labelled simplicial complexes accounts for the
labelling of the simplices, so that two labelled simplices from different
simplicial complexes are considered identical when they are spanned by the
same set of labelled 0-simplices.

  * •

The function $d_{{\mathcal{C}}}$ calculates the number of labelled simplices
that must be added to or subtracted from one of the simplicial complexes to
obtain the other simplicial complex. In particular, this measure of distance
applies for any two simplicial representations, whether or not they have
labelled simplices in common.

  * •

The function $d_{{\mathcal{C}}}$ is invariant with respect to a reordering of
the model components in ${\mathcal{C}}$, and therefore a relabelling of the
vertices of the reference complex, since the original and relabelled
simplicial complexes are isomorphic.

  * •

We label the distance function $d_{{\mathcal{C}}}$ with the subscript
${\mathcal{C}}$ to emphasise that the distance between two simplicial
representations is dependent on the level of conceptual detail employed
according to the chosen set of model components ${\mathcal{C}}$.

  * •

Since we add simplices under the field ${\mathbb{Z}}/2{\mathbb{Z}}$ we have
that $K\triangle L=K+L$.

######  Notation 2.15.

For a multiset $S$ of intervals
$(a,b)\in{\mathbb{N}}^{\ast}\times\overline{{\mathbb{N}}^{\ast}}$, we denote
the projection onto the first coordinates as
$\operatorname{proj}_{1}(S)=\\{\,a\in{\mathbb{N}}^{\ast}\mid\text{$(a,b)\in S$
for some $b\in\overline{{\mathbb{N}}^{\ast}}$}\,\\}$, and the projection onto
the second coordinates as
$\operatorname{proj}_{2}(S)=\\{\,b\in\overline{{\mathbb{N}}^{\ast}}\mid\text{$(a,b)\in
S$ for some $a\in{\mathbb{N}}^{\ast}$}\,\\}$.

######  Definition 2.16 (Distance between multisets of persistence intervals).

Let ${\mathcal{C}}$ be the set of all components that appear in the collection
of models under consideration for comparison, and let ${\mathcal{R}}^{(m)}$ be
the associated reference complex with a flat filtration. Let ${\mathcal{M}}$
be a collection of multisets of persistence intervals ${\mathcal{P}}(K)$
corresponding to labelled simplicial complexes $K$ with labels from
${\mathcal{C}}$ and induced filtrations from the flat filtration. Further, let

$\Theta\colon{\mathcal{M}}\to
2^{\big{[}|{\mathcal{R}}^{(m)}|\big{]}}\quad\text{be such
that}\quad\Theta\big{(}{\mathcal{P}}(K)\big{)}=\operatorname{proj}_{1}\big{(}{\mathcal{P}}(K)\big{)}\cup\big{(}\operatorname{proj}_{2}\big{(}{\mathcal{P}}(K)\big{)}\setminus\\{+\infty\\}\big{)},$
(3)

which is the set of all left endpoints and finite right-endpoints of the
persistence intervals. Define the function

$\hat{d}_{{\mathcal{C}}}\colon{\mathcal{M}}\times{\mathcal{M}}\to{\mathbb{R}}\quad\text{by}\quad\hat{d}_{{\mathcal{C}}}\big{(}{\mathcal{P}}(K),{\mathcal{P}}(L)\big{)}=\big{|}\Theta\big{(}{\mathcal{P}}(K)\big{)}\triangle\Theta\big{(}{\mathcal{P}}(L)\big{)}\big{|}.$
(4)

We now show that $d_{{\mathcal{C}}}$ is a distance function on
${\mathcal{S}}$, that $\hat{d}_{{\mathcal{C}}}$ is a distance function on
${\mathcal{M}}$, and that the metric spaces $({\mathcal{S}},d)$ and
$({\mathcal{M}},\hat{d}_{{\mathcal{C}}})$ are isometric, so we can use either
metric to determine the distances between simplicial complexes. To prove these
claims we use the triangle inequality for the symmetric difference: for sets
$X$, $Y$, and $Z$, the cardinality of the symmetric difference satisfies
subadditivity, that is, $|X\triangle Z|\leq|X\triangle Y|+|Y\triangle Z|$.
This relation follows by observing that $X\triangle Z\subseteq(X\triangle
Y)\cup(Y\triangle Z)$.

We first need a lemma to show that the function $\hat{d}_{{\mathcal{C}}}$ is
invariant with respect to a change in the flat filtration on the reference
complex.

######  Lemma 2.17.

Let ${\mathcal{C}}$ be the set of all components that appear in the collection
of models under consideration for comparison, and let ${\mathcal{R}}^{(m)}$ be
the associated reference complex. The function $\hat{d}_{{\mathcal{C}}}$ is
invariant with respect to a change in the flat filtration of the reference
complex ${\mathcal{R}}^{(m)}$.

###### Proof.

Let ${\mathcal{P}}(K)$ and ${\mathcal{P}}(L)$ be the multisets of persistence
intervals for $K$ and $L$, respectively, corresponding to the induced
filtrations from the first flat filtration with associated weight function
$w$, and let ${\mathcal{P}}^{\prime}(K)$ and ${\mathcal{P}}^{\prime}(L)$ be
the multisets of persistence intervals for $K$ and $L$, respectively,
corresponding to the induced filtrations from the second flat filtration with
associated weight function $w^{\prime}$.

There exists a permutation $\pi$ on $\big{[}|{\mathcal{R}}^{(m)}|\big{]}$ such
that $\pi\circ w=w^{\prime}$, from which it follows that there is a bijection
$f\colon\Theta\big{(}{\mathcal{P}}(K)\big{)}\cup\Theta\big{(}{\mathcal{P}}(L)\big{)}\to\Theta\big{(}{\mathcal{P}}^{\prime}(K)\big{)}\cup\Theta\big{(}{\mathcal{P}}^{\prime}(L)\big{)}$
where the restriction of $f$ to $\Theta\big{(}{\mathcal{P}}(K)\big{)}$ is a
bijection onto $\Theta\big{(}{\mathcal{P}}^{\prime}(K)\big{)}$ and the
restriction of $f$ to $\Theta\big{(}{\mathcal{P}}(L)\big{)}$ is a bijection
onto $\Theta\big{(}{\mathcal{P}}^{\prime}(L)\big{)}$. It follows that there is
a bijection between
$\Theta\big{(}{\mathcal{P}}(K)\big{)}\triangle\Theta\big{(}{\mathcal{P}}(L)\big{)}$
and
$\Theta\big{(}{\mathcal{P}}^{\prime}(K)\big{)}\triangle\Theta\big{(}{\mathcal{P}}^{\prime}(L)\big{)}$,
so
$\hat{d}_{{\mathcal{C}}}\big{(}{\mathcal{P}}(K),{\mathcal{P}}(L)\big{)}=\hat{d}_{{\mathcal{C}}}\big{(}{\mathcal{P}}^{\prime}(K),{\mathcal{P}}^{\prime}(L)\big{)}$.
∎

######  Theorem 2.18.

Let ${\mathcal{C}}$ be the set of all components of models under
consideration, and let ${\mathcal{R}}^{(m)}$ be the associated reference
complex with a flat filtration. Let ${\mathcal{S}}$ be a collection of
labelled simplicial complexes with labels from ${\mathcal{C}}$, and let
${\mathcal{M}}$ be the collection of multisets of persistence intervals
${\mathcal{P}}(K)$, for $K\in{\mathcal{S}}$ with the induced filtrations. Then
the function $d_{{\mathcal{C}}}$ is a metric on ${\mathcal{S}}$, the function
$\hat{d}_{{\mathcal{C}}}$ is a metric on ${\mathcal{M}}$, and the metric
spaces $({\mathcal{S}},d)$ and $({\mathcal{M}},\hat{d}_{{\mathcal{C}}})$ are
isometric.

###### Proof.

To see that $d_{{\mathcal{C}}}$ is a metric on ${\mathcal{S}}$, suppose that
$K$, $L\in{\mathcal{S}}$. Then $d_{{\mathcal{C}}}(K,L)=0$ if and only if
$|K\triangle L|=0$ if and only if $K=L$, so the identity of indiscernibles
holds. Symmetry follows from the symmetric difference. It remains to show
subadditivity, so let $M\in{\mathcal{S}}$. The triangle inequality for the
symmetric difference gives $d_{{\mathcal{C}}}(K,L)=|K\triangle
L|\leq|K\triangle M|+|M\triangle
L|=d_{{\mathcal{C}}}(K,M)+d_{{\mathcal{C}}}(M,L)$, as required.

Next we show that $\hat{d}_{{\mathcal{C}}}$ is a metric on ${\mathcal{M}}$,
noting that we can consider any particular induced filtration since
$\hat{d}_{{\mathcal{C}}}$ is invariant with respect to a change in the flat
filtration of the reference complex by Lemma 2.17. So let ${\mathcal{P}}(K)$,
${\mathcal{P}}(L)\in{\mathcal{M}}$ be two multisets of persistence intervals
corresponding to the simplicial complexes $K$ and $L$. For the identity of
indiscernibles,
$\hat{d}_{{\mathcal{C}}}\big{(}{\mathcal{P}}(K),{\mathcal{P}}(L)\big{)}=0$ if
and only if
$\Theta\big{(}{\mathcal{P}}(K)\big{)}=\Theta\big{(}{\mathcal{P}}(L)\big{)}$,
so that $K$ and $L$ contain the same simplicies, if and only if $K=L$ if and
only if ${\mathcal{P}}(K)={\mathcal{P}}(L)$ by Corollary 2.8. Symmetry follows
from the symmetric difference, and subadditivity follows from the triangle
inequality for the symmetric difference.

Finally, to show that the two metric spaces
$({\mathcal{S}},d_{{\mathcal{C}}})$ and
$({\mathcal{M}},\hat{d}_{{\mathcal{C}}})$ are isometric we need to establish a
surjective isometry $f\colon{\mathcal{S}}\to{\mathcal{M}}$. Define $f$ such
that ${\mathcal{K}}\mapsto{\mathcal{P}}(K)$, which is a surjection. Recall
that $f$ is an isometry if and only if
$\hat{d}_{{\mathcal{C}}}(f(K),f(L))=d_{{\mathcal{C}}}(K,L)$ for all $K$,
$L\in{\mathcal{S}}$, or equivalently the cardinality of $K\triangle L$ equals
the cardinality of
$\Theta\big{(}{\mathcal{P}}(K)\big{)}\triangle\Theta\big{(}{\mathcal{P}}(L)\big{)}$
for all $K$, $L\in{\mathcal{S}}$. To show that $f$ is an isometry it therefore
suffices to establish a bijection $\phi_{(K,L)}$, for each $K$,
$L\in{\mathcal{S}}$, from $K\triangle L$ onto
$\Theta\big{(}{\mathcal{P}}(K)\big{)}\triangle\Theta\big{(}{\mathcal{P}}(L)\big{)}$.
Let $g\colon{\mathcal{R}}^{(m)}\to\big{[}|{\mathcal{R}}^{(m)}|\big{]}$ be the
bijective weight function for the flat filtration of the reference complex.
Define the function $\phi_{(K,L)}=g|_{K\triangle L}$ by restriction. Then
$\phi_{(K,L)}$ is a bijection onto its image, since simplices are added
individually in the flat filtration. It remains to show that the image of
$K\triangle L$ under $\phi_{(K,L)}$ is
$\Theta\big{(}{\mathcal{P}}(K)\big{)}\triangle\Theta\big{(}{\mathcal{P}}(L)\big{)}$.
Indeed, $\sigma\in K\triangle L$ if and only if
$a\in\Theta\big{(}{\mathcal{P}}(K)\big{)}\triangle\Theta\big{(}{\mathcal{P}}(L)\big{)}$
where $g|_{K\triangle L}(\sigma)=a$. ∎

If we know the relationships between two pairs of simplicial representations
with a complex in common, say between $J$ and $K$ and between $K$ and $L$,
then we can infer the relationship between $J$ and $L$. Specifically, the
distance between $J$ and $L$ can be determined either from $J\triangle K$ and
$K\triangle L$ or, if the complexes are all subcomplexes of
${\mathcal{R}}^{(m)}$ with a flat filtration, from the multisets of
persistence intervals. Formally,

######  Proposition 2.19.

Let ${\mathcal{C}}$ be the set of all components of models under
consideration, and let ${\mathcal{R}}^{(m)}$ be the associated reference
complex with a flat filtration. Let ${\mathcal{S}}$ be a collection of
labelled simplicial complexes corresponding to models generated by subsets
from ${\mathcal{C}}$. For $J$, $K$, $L\in{\mathcal{S}}$ we have

$d_{{\mathcal{C}}}(J,L)=|(J\triangle K)\triangle(K\triangle L)|,$

and

$\hat{d}_{{\mathcal{C}}}\big{(}{\mathcal{P}}(J),{\mathcal{P}}(L)\big{)}=\Big{|}\Big{(}\Theta\big{(}{\mathcal{P}}(J)\big{)}\triangle\Theta\big{(}{\mathcal{P}}(K)\big{)}\Big{)}\triangle\Big{(}\Theta\big{(}{\mathcal{P}}(K)\big{)}\triangle\Theta\big{(}{\mathcal{P}}(L)\big{)}\Big{)}\Big{|}.$

###### Proof.

Both equations follow immediately from the observation that the symmetric
difference satisfies $X\triangle Z=(X\triangle Y)\triangle(Y\triangle Z)$ for
sets $X$, $Y$, and $Z$. ∎

###  2.5 Model comparison by equivalence

We now consider model comparison by equivalence, which accounts for particular
conceptual similarities between models that we may regard as essentially
identical. So rather than comparing the simplicial complexes associated to
models based on the presence or absence of a particular labelled simplex, as
in model comparison by distance, here we consider the equivalence of models in
terms of an equivalence of the corresponding labelled simplicial complexes. As
we show below, ‘equivalence’ as used here is restricted to five narrowly-
defined operations. Moreover, equivalence between two models is only possible
(but not guaranteed) for models that have high similarity in terms of the
associated simplicial complexes.

For this we need to specify a _predicate_ , $\operatorname{Pr}$, which is a
statement that contains a finite number of variables with the specified domain
$\operatorname{Dom}$, such that $\operatorname{Pr}$ becomes a (Boolean)
proposition when instantiated. We can consider a predicate to be a Boolean-
valued function $\operatorname{Pr}\colon\operatorname{Dom}\to\\{\text{true,
false}\\}$. Predicates are often described in terms of the number of their
variables, so that for an integer $n\geq 1$ an $n$-place predicate
$\operatorname{Pr}(x_{1},\ldots,x_{n})$ has $n$ variables $x_{1},\ldots,x_{n}$
with domain $\operatorname{Dom}\subseteq D_{1}\times\cdots\times D_{n}$, where
$x_{i}\in D_{i}$ for $i=1,\ldots,n$.

######  Definition 2.20 (Equivalent simplicial complexes).

Let ${\mathcal{C}}$ be the set of all components of models under
consideration, let ${\mathcal{S}}$ be a collection of labelled simplicial
complexes with labels from ${\mathcal{C}}$, and let $\operatorname{Pr}$ be a
two-place predicate on ${\mathcal{S}}\times{\mathcal{S}}$ such that

$R_{\text{eq}}=\\{\,(K,L)\in{\mathcal{S}}\times{\mathcal{S}}\mid\operatorname{Pr}(K,L)\,\\}$
(5)

is an equivalence relation on ${\mathcal{S}}$. Then the simplicial complexes
$K$, $L\in{\mathcal{S}}$ are _equivalent_ if and only if $(K,L)\in
R_{\text{eq}}$.

To determine the similarity of two models, represented as simplicial
complexes, we first need to formulate a predicate on which equivalence is
based. There is not a unique predicate for this purpose of comparison, and the
established equivalence of models must always be considered with regard to the
particular choice of predicate. Note further that equivalence is not a
quantitative comparison, but rather a qualitative observation that models have
similar features with respect to the specified predicate. Informally, we want
to say that $K$, $L\in{\mathcal{S}}$ are equivalent when their labelled
simplices are conceptually the same, so in particular we need to identify the
labelled subcomplexes in $K\setminus L$ with those in $L\setminus K$. For this
we employ five operations on a labelled simplicial complex $K$. Since the
1-skeleton of a labelled simplicial complex is a labelled undirected graph, we
refer to the 0-simplices and 1-simplices as vertices and edges, respectively.

We first recall the definition of a simplicial map [33, 34]:

######  Definition 2.21 (Vertex map, simplicial map).

Let $K$ and $L$ be two simplicial complexes. A _vertex map_ is a function
$\psi\colon K^{(0)}\to L^{(0)}$, and a _simplicial map_ $\psi\colon K\to L$ is
a function such that $\psi|_{K^{(0)}}$ is a vertex map and whenever
$W=\\{w_{i}\\}_{i=0}^{n}$ spans a simplex in $K$ then
$\psi(W)=\\{\psi(w_{i})\\}_{i=0}^{n}$ spans a simplex in $L$.

We also require the notion of a simplicial multivalued map, which we define as
follows:

######  Definition 2.22 (Simplicial multivalued map).

Let $K$ and $L$ be two simplicial complexes, and let $F\colon
K^{(0)}\rightrightarrows L^{(0)}$ be a left-total binary relation such that
$F(w)$ is a nonempty subset of $L^{(0)}$ for each $w\in K^{(0)}$. Then $F$ is
a _simplicial multivalued map_ , denoted $F\colon K\rightrightarrows L$, if
whenever a set of vertices $W=\\{w_{i}\\}_{i=0}^{n}$ spans a simplex in $K$
then the set of vertices $F(W)=\\{F(w_{i})\\}_{i=0}^{n}$ spans a simplex in
$L$, and for each $i$ we possibly have $\lvert F(w_{i})\rvert>1$. We say that
$F$ is _absolutely injective_ when $F(v)\cap F(w)\neq\emptyset$ implies $v=w$,
for any $v$, $w\in K^{(0)}$, and $F$ is _surjective_ when
$F(K^{(0)})=L^{(0)}$.

We now define the five operations on simplicial complexes. Recall that two
distinct vertices in a simplicial complex are _adjacent_ if they belong to the
same simplex, and for a vertex $u$ in a simplicial complex $K$ we denote the
set of all vertices adjacent to $u$ as $V_{K}(u)$, noting that $u\notin
V_{K}(u)$.

######  Definition 2.23 (Operation 1: Adjacent-vertex identification).

Let ${\mathcal{C}}$ be the set of all components of models under
consideration, and let $K$ and $L$ be two labelled simplicial complexes with
labels from ${\mathcal{C}}$. Let $\\{u,v\\}$ be a pair of adjacent vertices in
$K$ such that the following all hold:

  * •

$V_{K}(u)\setminus\\{v\\}=V_{K}(v)\setminus\\{u\\}$.

  * •

For any nonempty subset $W\subseteq V_{K}(u)\setminus\\{v\\}$, the vertices
$W\cup\\{u\\}$ span a simplex in $K$ if and only if the vertices
$W\cup\\{v\\}$ span a simplex in $K$.

A simplicial map $\pi_{1}\colon K\to L$ is an _adjacent-vertex identification_
if $\pi_{1}$ is surjective, and is injective and label preserving on every
vertex except at the pair of vertices $\\{u,v\\}$ that are mapped to a single
vertex $c\in L^{(0)}$. That is, for $z\in K^{(0)}$,

$\pi_{1}(z)=\begin{cases}z\quad&\text{if $z\notin\\{u,v\\}$},\\\
c\quad&\text{if $z\in\\{u,v\\}$}.\\\ \end{cases}$ (6)

######  Definition 2.24 (Operation 2: Nonadjacent-vertex identification).

Let ${\mathcal{C}}$ be the set of all components of models under
consideration, and let $K$ and $L$ be two labelled simplicial complexes with
labels from ${\mathcal{C}}$. Let $\\{u,v\\}$ be a pair of nonadjacent vertices
in $K$ such that the following all hold:

  * •

$V_{K}(u)=V_{K}(v)$.

  * •

For any nonempty subset $W\subseteq V_{K}(u)$, the vertices $W\cup\\{u\\}$
span a simplex in $K$ if and only if the vertices $W\cup\\{v\\}$ span a
simplex in $K$.

A simplicial map $\pi_{2}\colon K\to L$ is a _nonadjacent-vertex
identification_ if $\pi_{2}$ is surjective, and is injective and label
preserving on every vertex except at the pair of vertices $\\{u,v\\}$ that are
mapped to a single vertex $c\in L^{(0)}$. That is, for $z\in K^{(0)}$,

$\pi_{2}(z)=\begin{cases}z\quad&\text{if $z\notin\\{u,v\\}$},\\\
c\quad&\text{if $z\in\\{u,v\\}$}.\\\ \end{cases}$ (7)

######  Definition 2.25 (Operation 3: Vertex split).

Let ${\mathcal{C}}$ be the set of all components of models under
consideration, and let $K$ and $L$ be two labelled simplicial complexes with
labels from ${\mathcal{C}}$. Further, let $u\in K^{(0)}$ and $c$, $d\in
L^{(0)}\setminus K^{(0)}$ be three vertices with mutually distinct labels such
that the following all hold:

  * •

The vertices $c$, $d\in L$ are adjacent.

  * •

$V_{L}(c)\setminus\\{d\\}=V_{L}(d)\setminus\\{c\\}$.

  * •

If $W\subseteq V_{K}(u)$ is a nonempty subset such that the vertices
$W\cup\\{u\\}$ span a simplex in $K$ then there exists a subset $X\subseteq
V_{L}(c)\setminus\\{d\\}$ such that $|W|=|X|$, the vertices in $W$ and $X$
have the same labels, the vertices $X\cup\\{c\\}$ span a simplex in $L$, and
the vertices $X\cup\\{d\\}$ span a simplex in $L$.

  * •

If $X\subseteq V_{L}(c)\setminus\\{d\\}$ is a nonempty subset such that the
vertices $X\cup\\{c\\}$ span a simplex in $L$ and the vertices $X\cup\\{d\\}$
span a simplex in $L$ then there exists a subset $W\subseteq V_{K}(u)$ such
that $|W|=|X|$, the vertices in $W$ and $X$ have the same labels, and the
vertices $W\cup\\{u\\}$ span a simplex in $K$.

A simplicial multivalued map $\pi_{3}\colon K\rightrightarrows L$ is a _vertex
split_ if $\pi_{3}$ is surjective, absolutely injective, and single valued and
label preserving on every vertex except at the vertex $u$ which is mapped to
the pair of vertices $\\{c,d\\}$. That is, for $z\in K^{(0)}$,

$\pi_{3}(z)=\begin{cases}z\quad&\text{if $z\neq u$},\\\
\\{c,d\\}\quad&\text{if $z=u$}.\\\ \end{cases}$ (8)

######  Definition 2.26 (Operation 4: Inclusion).

Let ${\mathcal{C}}$ be the set of all components of models under
consideration, and let $K$ and $L$ be two labelled simplicial complexes with
labels from ${\mathcal{C}}$. A simplicial map $\pi_{4}\colon K\to L$ is an
_inclusion_ if $\pi_{4}$ is injective and preserves all labels. That is, for
$z\in K^{(0)}$, $\pi_{4}(z)=z$.

######  Definition 2.27 (Operation 5: Vertex substitution).

Let ${\mathcal{C}}$ be the set of all components of models under
consideration, and let $K$ and $L$ be two labelled simplicial complexes with
labels from ${\mathcal{C}}$. A simplicial map $\pi_{5}\colon K\to L$ is a
_vertex substitution_ if $\pi_{5}$ is bijective and preserves all labels
except for one whereby the labelled vertex $u\in K^{(0)}$ is mapped to the
labelled vertex $c\in L^{(0)}$. That is, for $z\in K^{(0)}$,

$\pi_{5}(z)=\begin{cases}z\quad&\text{if $z\neq u$},\\\ c\quad&\text{if
$z=u$}.\end{cases}$ (9)

Note that, for appropriate simplicial complexes, an adjacent-vertex
identification is mutually inverse with a corresponding vertex split, a
nonadjacent-vertex identification is mutually inverse with an inclusion, and a
vertex substitution is mutually inverse with another vertex substitution.

To rigorously establish an equivalence between models through application of
Operations 1–5, we must ensure that each operation preserves the
representation of the general physical system. It is important to note that an
equivalence between models is based on the components of the models that we
allow to be identified as equivalent, and therefore the corresponding
operations on the simplicial complexes that we regard as admissible. Thus, in
addition to the formal requirements (Operations 1–5) the notion of equivalence
also incorporates tight domain-specific constraints. The existence of an
equivalence between models is therefore dependent on the level of model detail
that we include in the simplicial representations, and the model components
that we allow to be equivalent at this level of model detail. This approach
provides great flexibility to compare models at various levels of conceptual
detail, and whether or not models are equivalent or inequivalent will depend
on the perspective with which we want to view the models. Model equivalence is
not an absolute determination, but rather a perspective that is relative to
the operations regarded as admissible.

The application of both Operations 1 and 3 is intrinsically restrictive, since
they require similar interconnections between the vertices. Here we explicitly
specify the form in which the five operations are admissible.

######  Definition 2.28 (Admissible operations on labelled simplicial
complexes).

An _admissible operation_ on a labelled simplicial complex associated with a
model is one of the following:

  1. 1.

Operation 1 (Adjacent-vertex identification): the two identified vertices and
the new vertex must have labels that represent conceptually-equivalent
components of the model, which should intrinsically hold for the two
identified vertices since they are adjacent so have a direct interconnection.

  2. 2.

Operation 2 (Nonadjacent-vertex identification): the two identified vertices
and the new vertex must have labels that represent conceptually-equivalent
components of the model, which may not hold for the two identified vertices
since they are not adjacent and so lack a direct interconnection.

  3. 3.

Operation 3 (Vertex split): the original vertex and the two new vertices must
have labels that represent conceptually-equivalent components of the model,
which should intrinsically hold for the two new vertices since they are
adjacent.

  4. 4.

Operation 4 (Inclusion): the inclusion already implies that the original
simplicial complex is conceptually related to the supercomplex.

  5. 5.

Operation 5 (Vertex substitution): the new and old vertices must have labels
that represent conceptually-equivalent components of the model.

We are now in a position to define the required predicate.

######  Definition 2.29 (Predicate for the equivalence of models).

The two-place predicate $\operatorname{Pr}$ on
${\mathcal{S}}\times{\mathcal{S}}$ is as follows: for
$(K,L)\in{\mathcal{S}}\times{\mathcal{S}}$ there exists a sequence, which may
be empty, of the five admissible operations on simplicial complexes, say
$(f_{i})_{i=0}^{n}$, such that each $f_{i}$ is invertible and
$f_{n}\circ\cdots\circ f_{1}\circ f_{0}(K)=L$.

So for $(K,L)\in{\mathcal{S}}\times{\mathcal{S}}$, the proposition
$\operatorname{Pr}(K,L)$ is true when a sequence of the five admissible
operations can transform $K$ to $L$ and the corresponding sequence of inverse
operations can transform $L$ to $K$, otherwise if no such sequence exists then
$\operatorname{Pr}(K,L)$ is false. The following theorem establishes that the
specified predicate gives the necessary equivalence relation.

######  Theorem 2.30.

If $\operatorname{Pr}$ is the two-place predicate on
${\mathcal{S}}\times{\mathcal{S}}$ given in Definition 2.29 then it follows
that the relation
$R_{\text{eq}}=\\{\,(x,y)\in{\mathcal{S}}\times{\mathcal{S}}\mid\operatorname{Pr}(x,y)\,\\}$
is an equivalence relation on ${\mathcal{S}}$.

###### Proof.

$R_{\text{eq}}$ is reflexive, since we can apply the empty sequence of
operations to any simplicial complex. Symmetry of $R_{\text{eq}}$ follows from
the required invertibility of each operation, and transitivity follows from
the associativity of composition of the operations. Therefore, $R_{\text{eq}}$
is an equivalence relation on ${\mathcal{S}}$. ∎

We reiterate that the model equivalence that we consider here is quite
restrictive, so only models that have a high level of similarity would be
identified as equivalent. There are many other possible definitions of model
equivalence that may be more appropriate when comparing different collections
of models. It is important to note that our notion of equivalence of models
occurs infrequently, and the demonstration of an equivalence of two models
reveals fundamental and nontrivial similarities between the models.

##  3 Results and discussion

We now apply our methodology for model comparison to two different collections
of models: first, we examine the model equivalence of three enzyme-catalysed
reaction mechanisms involving two substrates; and second, we employ our
methodology for model comparison to the two main categories of models for
developmental pattern formation.

###  3.1 Comparison of bisubstrate reactions

Here we consider the equivalence of three models for enzyme-catalysed reaction
mechanisms involving two substrates: the ordered sequential bisubstrate
reaction and the random sequential bisubstrate reaction, which are both
ternary-complex mechanisms whereby both substrates bind to the enzyme
simultaneously; and, the ping-pong bisubstrate reaction, which involves a
chemically-modified intermediate form of the enzyme [35].

####  3.1.1 Ordered sequential bisubstrate reaction

In an ordered sequential bisubstrate reaction, two substrates A and B first
combine with the enzyme E to form a ternary complex EAB, followed by the
reaction, and then the release of the products P and Q. This reaction is shown
schematically in Figure 2(a), along with the corresponding simplicial
representation for our chosen level of conceptual detail.

Figure 2: Bisubstrate reactions represented schematically and as associated
labelled simplicial complexes. (a) Ordered sequential bisubstrate reaction.
(b) Random sequential bisubstrate reaction. (c) Ping-pong bisubstrate
reaction.

Note that the substrates combine with the enzyme in a particular order, here A
followed by B, and the products are released in a particular order, here P
then Q, resulting in a single reaction path.

####  3.1.2 Random sequential bisubstrate reaction

In a random sequential bisubstrate reaction, two substrates A and B first
combine with the enzyme E to form a ternary complex EAB, followed by the
reaction, and then the release of the products P and Q. This reaction is shown
schematically in Figure 2(b), along with the corresponding simplicial
representation for our chosen level of conceptual detail. Note that, in this
case, there is no required order for substrate combination with the enzyme and
for product release, resulting in four possible reaction paths. This
simplicial representation consists of four components, from a homological
perspective, since it represents four different potential reaction paths.

####  3.1.3 Ping-pong bisubstrate reaction

In a ping-pong bisubstrate reaction, also called a double-displacement
reaction, the substrate A combines with the enzyme resulting in the release of
product P and the formation of the intermediate E*, and then substrate B
combines with E* resulting in the release of product Q and regeneration of the
enzyme $E$. This reaction is shown schematically in Figure 2(c), along with
the corresponding simplicial representation for our chosen level of conceptual
detail. Note that the ping-pong mechanism requires the release of the first
product before the second substrate can react.

####  3.1.4 Equivalence of bisubstrate reactions

We first demonstrate an equivalence between the ordered sequential bisubstrate
reaction and the random sequential bisubstrate reaction. To transform the
complex for the ordered reaction to the complex for the random reaction, apply
vertex substitutions so that E, A, EA, B, EAB, EPQ, P, EQ, and Q become the
corresponding E1, A1, E1A1, B1, E1A1B1, E1P1Q1, P1, E1Q1, and Q1,
respectively, and then apply an inclusion operation. Conversely, to transform
the complex for the random reaction to the complex for the ordered reaction,
apply nonadjacent-vertex identifications so that
$\\{\textrm{E}_{i}\\}_{i=1}^{4}$, $\\{\textrm{A}_{i}\\}_{i=1}^{4}$,
$\\{\textrm{E}_{i}\textrm{A}_{i}\\}_{i=1}^{4}$,
$\\{\textrm{B}_{i}\\}_{i=1}^{4}$,
$\\{\textrm{E}_{i}\textrm{A}_{i}\textrm{B}_{i}\\}_{i=1}^{4}$,
$\\{\textrm{E}_{i}\textrm{P}_{i}\textrm{Q}_{i}\\}_{i=1}^{4}$,
$\\{\textrm{P}_{i}\\}_{i=1}^{4}$,
$\\{\textrm{E}_{i}\textrm{Q}_{i}\\}_{i=1}^{4}$, and
$\\{\textrm{Q}_{i}\\}_{i=1}^{4}$ are identified with E, A, EA, B, EAB, EPQ, P,
EQ, and Q, respectively.

Now we demonstrate that the ordered sequential bisubstrate reaction is not
equivalent to the ping-pong bisubstrate reaction. We cannot change the
2-simplices with vertex sets $\\{\textrm{E},\textrm{A},\textrm{EA}\\}$ and
$\\{\textrm{E},\textrm{Q},\textrm{EQ}\\}$ in each bisubstrate reaction, since
they correspond to identical components and interactions. The ping-pong
bisubstrate reaction contains a 1-simplex with vertex set
$\\{\textrm{EA},\textrm{E*P}\\}$, which is not in the ordered sequential
bisubstrate reaction and would require a vertex split operation applied to EA
in the ordered reaction. The new vertex from the vertex split operation,
labelled E*P, would however be a vertex in several new simplices, in
particular the simplex with vertex set
$\\{\textrm{E},\textrm{EA},\textrm{E*P}\\}$, which is not in the ping-pong
reaction. Therefore, we cannot transform the ordered sequential bisubstrate
reaction into the ping-pong bisubstrate reaction, so they are not equivalent.

###  3.2 Comparison of Turing-pattern and positional-information models

Here we apply our methods for model comparison to the two main categories of
models for developmental pattern formation, namely Turing-pattern models and
positional-information models. We consider the patterning to occur throughout
a two-dimensional rectangular domain, where the boundary conditions are always
zero-flux on the two opposite sides parallel to the morphogen concentration
gradient. We assume that the velocities of the cytoplasm and the growing
tissue are negligible, and we therefore assume no advection.

To represent the models as simplicial complexes we first determine the set of
components ${\mathcal{C}}$ on which the models under consideration are based.
We consider five positional-information models and four Turing-pattern models.
In this case the general components are: agents, namely morphogens,
modulators, and substrates; reactions involving the agents, such as self-
activation, activation, inhibition, and annihilation; agent degradation;
influx and outflux boundary conditions; agent diffusion; profile of the
morphogen gradient; and scale invariance of the morphogen gradient. The
ordered set of components is shown in Figure 3. Note that the components can
be given in any order since our definition of distance is invariant with
respect to changes in the flat filtration (see Lemma 2.17). The distance
between models is dependent on the level of conceptual detail used to
represent the models, so the distances obtained must be considered with
respect to the representative simplicial complexes. The most straightforward
way to obtain meaningful comparisons is to use a consistent level of
conceptual detail for all models. Here, we take the reference complex as the
simplex spanned by the complete set of components for all nine models.

Figure 3: Ordered set of components for the four Turing-pattern and five
positional-information models.

To construct the simplicial representation of each Turing-pattern and
positional-information model we first specify the 0-simplices, which represent
the model components, and the 1-simplices, which represent direct
interconnections between the components. While the 0- and 1-simplices are
specified by the model, to give a combinatorial graph, the higher-dimensional
simplices are obtained by forming cliques [36], where possible, incrementally
in dimensions 2 and higher. These higher-dimensional simplices indicate
higher-dimensional interactions between the corresponding model components.

To illustrate the barcode representation, we use a particular flat filtration
called the shortlex filtration, which we first describe. Note that while all
flat filtrations can be extended (see Proposition 2.13), an extension of the
shortlex filtration is not necessarily another shortlex filtration. We begin
by applying the shortlex order to the simplices of the simplicial complex with
respect to the assigned vertex order, where the shortlex order is defined as
follows [37, Chapter 0, page 14]:

######  Definition 3.1 (Shortlex order).

Let ${\mathcal{A}}$ be a totally-ordered finite set, called the _alphabet_ ,
and let ${\mathcal{W}}$ be the set of all _words_ that are finite sequences of
symbols from ${\mathcal{A}}$. The _shortlex order_ on ${\mathcal{W}}$ orders
the words as follows:

  1. 1.

Two different words in ${\mathcal{W}}$ with equal length are ordered according
to the alphabetic order of ${\mathcal{A}}$, therefore lexicographically.

  2. 2.

For two words with unequal lengths, the shorter word precedes the longer word.

Note that, since ${\mathcal{A}}$ is totally-ordered, the shortlex order is
also a total order. Applying the shortlex order to a collection of simplicies,
we first order the simplices by increasing dimension, and then the simplices
of the same dimension are ordered lexicographically. Since the set of vertices
is totally ordered as specified by $\operatorname{Ord}$, so is the shortlex
order for the simplicies.

We now define the _shortlex filtration_ :

######  Definition 3.2 (Shortlex filtration).

Let ${\mathcal{C}}$ be the set of all components that appear in the collection
of models under consideration for comparison, and let the reference complex
${\mathcal{R}}^{(m)}$ have the shortlex ordering. Define the bijective weight
function $w\colon{\mathcal{R}}^{(m)}\to\big{[}|{\mathcal{R}}^{(m)}|\big{]}$ by
$w(\sigma)=j$, where $j$ is the index of $\sigma$ in the shortlex ordering for
${\mathcal{R}}^{(m)}$. The _shortlex filtration_ of ${\mathcal{R}}^{(m)}$ is
the nested sequence of subcomplexes as defined in Definition 2.5.

With this filtration it is possible to define the distances of the models that
we have described under the umbrella of the reference complex
${\mathcal{R}}^{(m)}$. Note that, since our definition of distance is
invariant with respect to changes in the flat filtration (see Lemma 2.17), the
components indicated in Figure 3 can be given an arbitrary ordering, resulting
in an alternative shortlex filtration.

Here we describe one positional-information model and one Turing-pattern
model, and provide the descriptions of the four additional positional-
information models and three additional Turing-pattern models in the
Electronic Supplementary Material document. We begin by describing the
positional-information annihilation model.

####  3.2.1 Positional-information annihilation model

The annihilation model, and also the opposing gradients model (Electronic
Supplementary Material), are mechanisms whereby two opposing morphogen
gradients provide size information for developmental patterning. In the
annihilation model, the target gene responds to the concentration of Morphogen
1, to which Morphogen 2 irreversibly binds and thereby inhibits the action of
Morphogen 1 on activity of transcription, so that the gradient of Morphogen 2
provides size information to the concentration field of Morphogen 1 [38].

The sources of each morphogen are at opposite ends of the domain, and the
morphogens interact by an annihilation reaction with rate $k$ that results in
global scale-invariant patterning [39, 38]. Mathematically, the two morphogen
gradients with concentrations $m(\bf{x},t)$ and $c(\bf{x},t)$ can be modelled
as

$\displaystyle\frac{\partial m}{\partial t}$
$\displaystyle=D_{m}\nabla^{2}m-k_{m}m-kmc,$ (10) $\displaystyle\frac{\partial
c}{\partial t}$ $\displaystyle=D_{c}\nabla^{2}c-k_{c}c-kmc,$ (11)

where $D_{m}$ and $D_{c}$ are diffusivities, and $k_{m}$ and $k_{c}$ are
degradation rates. The boundary conditions for each morphogen are Neumann at
both boundaries, with influx at the source and zero flux at the opposite
boundary.

For the simplicial representation of the annihilation model, the vertices and
corresponding model components are:
• $v_{1}\longleftrightarrow\text{Morphogen 1}$ •
$v_{2}\longleftrightarrow\text{Diffusion 1}$ •
$v_{3}\longleftrightarrow\text{Degradation 1}$ •
$v_{6}\longleftrightarrow\text{Influx 1}$ •
$v_{9}\longleftrightarrow\text{Morphogen 2}$ •
$v_{10}\longleftrightarrow\text{Diffusion 2}$ •
$v_{11}\longleftrightarrow\text{Degradation 2}$ •
$v_{13}\longleftrightarrow\text{Influx 2}$ •
$v_{26}\longleftrightarrow\text{Annihilation between Morphogens 1 and 2}$ •
$v_{40}\longleftrightarrow\text{Monotonic gradient}$ •
$v_{43}\longleftrightarrow\text{Global scale-invariance}$

For the annihilation model, the simplicial complex and corresponding
persistence barcode is shown in Figure 4(a)–(b). Note that the simplicial
complex is 4-dimensional, however we only show the 1-skeleton of the
simplicial complex for simplicity.

Figure 4: Simplicial-complex representations and corresponding persistence
barcodes. Positional-information annihilation model: (a) 1-skeleton of the
4-dimensional simplicial representation; (b) Persistence barcode. Turing-
pattern activator-inhibitor model: (c) 1-skeleton of the 5-dimensional
simplicial representation; (d) Persistence barcode. Note that the $H_{n}$
labels on the persistence barcodes refer to the $n$-dimensional homology
classes, and the arrows at the right endpoints of certain persistence
intervals indicate that the right endpoints are $+\infty$, corresponding to
homology classes that are not annihilated in the filtration.

We now describe the Turing-pattern activator-inhibitor model.

####  3.2.2 Turing-pattern activator-inhibitor model

The activator-inhibitor system [8, 40, 41] consists of two diffusible
morphogens, an autocatalytic activator with concentration $m(\bf{x},t)$ and a
rapidly diffusing inhibitor with concentration $c(\bf{x},t)$. This model can
be represented mathematically as

$\displaystyle\frac{\partial m}{\partial t}$
$\displaystyle=D_{m}\nabla^{2}m+\frac{\rho
m^{2}}{c(1+\mu_{m}m^{2})}-k_{m}m+\rho_{m},$ (12) $\displaystyle\frac{\partial
c}{\partial t}$ $\displaystyle=D_{c}\nabla^{2}c+\rho m^{2}-k_{c}c+\rho_{c},$
(13)

where $D_{m}$ and $D_{c}$ are diffusivities, $\rho_{m}$ and $\rho_{c}$ are
basal production rates, $k_{m}$ and $k_{c}$ are degradation rates, and
$\mu_{m}$ is a saturation constant. The parameter $\rho$ is the _source
density_ , which measures the general ability of the cells to perform the
autocatalytic reaction. The patterning arises through local self-enhancement
of the activator, activation of the inhibitor, and long-range inhibition of
the activator. We assume that the boundary conditions are zero flux at both
boundaries.

For the simplicial representation of the activator-inhibitor model, the
vertices and corresponding model components are:
• $v_{1}\longleftrightarrow\text{Morphogen 1}$ •
$v_{2}\longleftrightarrow\text{Diffusion 1}$ •
$v_{3}\longleftrightarrow\text{Degradation 1}$ •
$v_{5}\longleftrightarrow\text{Basal production 1}$ •
$v_{9}\longleftrightarrow\text{Morphogen 2}$ •
$v_{10}\longleftrightarrow\text{Diffusion 2}$ •
$v_{11}\longleftrightarrow\text{Degradation 2}$ •
$v_{12}\longleftrightarrow\text{Basal production 2}$ •
$v_{27}\longleftrightarrow\text{Self-activation of Morphogen 1}$ •
$v_{28}\longleftrightarrow\text{Activation of Morphogen 2 by Morphogen 1}$ •
$v_{29}\longleftrightarrow\text{Inhibition of Morphogen 1 by Morphogen 2}$ •
$v_{41}\longleftrightarrow\text{Oscillatory gradient}$ •
$v_{43}\longleftrightarrow\text{Global scale-invariance}$

For the activator-inhibitor model, the simplicial complex and corresponding
persistence barcode is shown in Figure 4(c)–(d). Note that the simplicial
complex is 5-dimensional, however we only show the 1-skeleton of the
simplicial complex for simplicity.

####  3.2.3 Distances between models

We now apply Theorem 2.18 to calculate the distances between the simplicial
representations of the five positional-information and four Turing-pattern
models. These distances, shown in Table 2, indicate the total number of
labelled simplices that must be added and subtracted to transform one labelled
simplicial complex associated with a model to the labelled simplicial complex
of another model.

Table 2: Distances between the four Turing-pattern (TP) and five positional-information (PI) models. | PI 2 | PI 3 | PI 4 | PI 5 | TP 1 | TP 2 | TP 3 | TP 4
---|---|---|---|---|---|---|---|---
PI 1 (linear gradient) | 40 | 108 | 104 | 112 | 240 | 176 | 376 | 340
PI 2 (synthesis-diffusion-degradation) |  | 68 | 96 | 104 | 216 | 152 | 352 | 316
PI 3 (opposing gradients) |  |  | 120 | 172 | 270 | 220 | 406 | 370
PI 4 (annihilation) |  |  |  | 152 | 268 | 232 | 404 | 368
PI 5 (active modulation) |  |  |  |  | 304 | 240 | 440 | 404
TP 1 (activator-inhibitor) |  |  |  |  |  | 192 | 432 | 100
TP 2 (substrate depletion) |  |  |  |  |  |  | 424 | 292
TP 3 (inhibition of an inhibition) |  |  |  |  |  |  |  | 532
TP 4 (modulation) |  |  |  |  |  |  |  |

In Table 3 we also show the number of simplices in each simplicial
representation of the four Turing-pattern (TP) and five positional-information
(PI) models.

Table 3: Number of $n$-dimensional simplices in the labelled simplicial representations of the four Turing-pattern (TP) and five positional-information (PI) models. $n$ | 0 | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---|---
PI 1 (linear gradient) | 6 | 14 | 16 | 9 | 2 | 0
PI 2 (synthesis-diffusion-degradation) | 5 | 9 | 7 | 2 | 0 | 0
PI 3 (opposing gradients) | 10 | 27 | 32 | 18 | 4 | 0
PI 4 (annihilation) | 11 | 33 | 43 | 26 | 6 | 0
PI 5 (active modulation) | 13 | 39 | 47 | 24 | 4 | 0
TP 1 (activator-inhibitor) | 13 | 45 | 70 | 55 | 21 | 3
TP 2 (substrate depletion) | 12 | 37 | 50 | 33 | 10 | 1
TP 3 (inhibition of an inhibition) | 17 | 70 | 119 | 96 | 36 | 5
TP 4 (modulation) | 16 | 65 | 108 | 84 | 30 | 4

We observe that the distance between two models can be relatively large even
when the models have similar numbers of simplices of each dimension. For
example, the Turing-pattern inhibition-of-an-inhibition model (TP 3) and the
Turing-pattern modulation model (TP 4) have similar numbers of simplices in
each dimension, however they have the largest distance amongst the four
Turing-pattern models that we compare. This illustrates that the metric
accounts for not just the numbers of simplicies but also for the labelling of
the simplicies.

Further, our measure of distance between models is able to reveal when two
models are relatively similar, for example the Turing-pattern modulation model
(TP 4) is a modification of the Turing-pattern activator-inhibitor (TP 1)
model, and these two models have the smallest distance amongst the four
Turing-pattern models that we compare.

####  3.2.4 Equivalence of models

We now apply our notion of model equivalence to compare the positional-
information annihilation model with the Turing-pattern activator-inhibitor
model. The 1-skeletons of the simplicial representations of the annihilation
model and the activator-inhibitor model are shown in Figure 4(a) and (c),
respectively. We shall demonstrate the equivalence of the two models by
applying Operations 1, 3, and 5 to the Turing-pattern activator-inhibitor
model to obtain the positional-information annihilation model. Note that,
since the three operations are invertible, we could similarly apply operations
to the positional-information annihilation model to obtain the Turing-pattern
activator-inhibitor model. In accordance with the admissibility requirements
for the operations, we regard as equivalent the subsets of model components
indicated in Figure 5.

Figure 5: Model components considered as equivalent.

We consider ‘Influx 1’ to be equivalent to the combination of ‘Basal
production 1’ and ‘Self-activation of Morphogen 1’ since they are all a source
of ‘Morphogen 1’. Similarly, ‘Influx 2’ is equivalent to ‘Basal production 2’
since they are both a source of ‘Morphogen 2’. Since we are interested in the
existence of the pattern-forming gradient and not the particular profile of
the gradient, we consider ‘Monotonic gradient’ and ‘Oscillatory gradient’ as
equivalent. Finally, ‘Annihilation between Morphogens 1 and 2’, which reduces
the concentrations of both morphogens, is equivalent to the combination of
‘Activation of Morphogen 2 by Morphogen 1’ and ‘Inhibition of Morphogen 1 by
Morphogen 2’. Indeed, ‘Activation of Morphogen 2 by Morphogen 1’ and
‘Inhibition of Morphogen 1 by Morphogen 2’ form an inhibitory cycle whereby
‘Morphogen 2’ inhibits ‘Morphogen 1’, which in turn reduces the activation of
‘Morphogen 2’ by ‘Morphogen 1’. Note that these observed equivalences between
model components do not suggest that equivalent components are the same, but
rather that equivalent components have similar roles in the conceptual
frameworks of the corresponding models, which is demonstrated formally through
application of the admissible operations on simplicial complexes to ascertain
that the models are in the same equivalence class.

We now show that the positional-information annihilation model and the Turing-
pattern activator-inhibitor model are in the same equivalence class. While we
describe the application of the operations on vertices, there are also changes
to the higher-dimensional simplices in the simplicial complex. Note that it is
easy to automate the process on computer to transform one simplicial complex
to another by application of the operations on simplicial complexes. First
consider transforming the complex for the Turing-pattern activator-inhibitor
model to the complex for the positional-information annihilation model.
Perform an adjacent-vertex identification of the two vertices ‘Basal
production 1’ and ‘Self-activation of Morphogen 1’ to obtain the new vertex
‘Influx 1’. The vertex ‘Basal production 2’ is substituted with the vertex
‘Influx 2’. Similarly, we substitute the vertex ‘Monotonic gradient’ for the
vertex ‘Oscillatory gradient’. Finally, we perform an adjacent-vertex
identification of the two vertices ‘Activation of Morphogen 2 by Morphogen 1’
and ‘Inhibition of Morphogen 1 by Morphogen 2’ to obtain the vertex
‘Annihilation between Morphogens 1 and 2’. The resulting simplicial complex is
that which represents the positional-information annihilation model.

Conversely, to transform the complex for the positional-information
annihilation model to the complex for the Turing-pattern activator-inhibitor
model we apply the inverse operations. First perform a vertex split of the
vertex ‘Annihilation between Morphogens 1 and 2’ to obtain the two vertices
‘Activation of Morphogen 2 by Morphogen 1’ and ‘Inhibition of Morphogen 1 by
Morphogen 2’. Substitute the vertex ‘Oscillatory gradient’ for the vertex
‘Monotonic gradient’, and then substitute the vertex ‘Basal production 2’ for
the vertex ‘Influx 2’. Finally, apply a vertex split of the vertex ‘Influx 1’
to obtain the two vertices ‘Basal production 1’ and ‘Self-activation of
Morphogen 1’. The resulting simplicial complex is that which represents the
Turing-pattern activator-inhibitor model. Since we have only used admissible
and invertible operations to transform the simplicial representation of the
Turing-pattern model to the simplicial representation of the positional-
information model, the two models are in the same equivalence class, and
therefore are equivalent. We can also compare the persistence barcodes shown
in Figure 4(b) and (d), which have a high degree of similarity in accordance
with the established equivalence between the positional-information
annihilation model and the Turing-pattern activator-inhibitor model.

This analysis demonstrates that the Turing-pattern activator-inhibitor model
and the positional-information annihilation model, each one of the main models
for their patterning mechanism in developmental biology, are in fact closely
related in terms of structure and mechanism. This is not obvious given the
nature of the proposed models (e.g. [42] and [43]), has not previously been
reported, and is unexpected given the frequent historical acrimony between
proponents of the two mechanisms [30]. The main difference between the
positional-information model and the Turing-pattern model is the source of the
gradient-forming morphogen, whereby the positional-information mechanism takes
advantage of an influx of morphogen from outside the patterning domain,
whereas for the Turing-pattern mechanism the morphogen is produced uniformly
within the domain. The morphogen influx naturally produces a morphogen
gradient in the positional-information mechanism, so there is no requirement
for specialised dynamics. In contrast, the Turing-pattern mechanism requires a
Turing instability to generate an oscillating gradient from the spatially-
uniform production of the morphogen, requiring very particular conditions for
existence.

It is important to note here that very few models would be equivalent with
respect to our strict definition of model equivalence. Indeed, with the
various model components and their interconnections, along with higher-
dimensional interactions, it is unlikely that two simplicial representations
of two models could be transformed into each other by application of
admissible operations. Our established equivalence of the positional-
information annihilation model and the Turing-pattern activator-inhibitor
model, with only a few admissible operations required, therefore demonstrates
that the two models are conceptually very similar from our considered
perspective. This is further substantiated by the ability to employ adjacent-
vertex identification operations, or conversely vertex-split operations, which
require identical conceptual interconnections for the model concepts involved.

##  4 Conclusion

In this article we present a new methodology for comparing models based on the
relationships between various model aspects. Compared to some previous
attempts at comparing model structures [42] our approach is readily
automatable, and thus able to meet the demands of large-scale modelling
attempts and model-curation projects [44].

Representing models as labelled simplicial complexes allows us to determine
meaningful distances between models, and persistent homology provides an
alternative and often simplified representation of the models. In addition to
a measure of distance, we have also developed and applied the concept of
equivalence to compare model features. Model equivalence, as developed here,
gives more nuanced insights into the relationships between different models
and allows us to detect nontrivial similarities between mathematical models.
Our analysis of the Turing-pattern activator-inhibitor model and the
positional-information annihilation model shows that these two models are more
similar than had previously been suggested [30]. Given the conditions outlined
in the discussion they are in fact equivalent, something that we had not
expected and which to our knowledge has not been seen or considered before.
This is one example demonstrating the potential insights that can be gained
from our formalism for model comparison.

We foresee more need for this in the immediate future: modelling is increasing
in importance in the life and biomedical sciences, yet remains to be fallible
and often poorly grounded in reality [45, 46]. The ability to compare,
contrast, reconcile, and triage potentially large sets of models will aid in
making mathematical modelling more helpful in biology.

While we have applied our methodology to similar types of models, namely
systems of reaction-diffusion equations, another important aspect of our
formalism is that we can compare different models, irrespective of how
distinct they are in form and size: by identifying the model components and
their interconnections we can represent any model as a simplicial complex
which allows for direct comparison with any other simplicial representation of
a model. Identifying differences, but also highlighting similarities in model
structures, can aid our understanding of scientific problems. For example, in
the particular example of Turing-pattern versus positional-information
mechanisms this approach can really resolve long-standing problems and trigger
the search for a more synthetic approach [30, 42]. Because of its flexibility
and rigorous grounding our methodology is universally applicable to all
models.

## Code availability

The code (in both Julia and MATLAB languages) for our model-comparison
methodology which calculates the distances between both the simplicial
complexes and the persistence intervals is available on GitHub at:
https://github.com/DrSeanTVittadello/ModelComparison2021

## Author contributions

STV and MPHS conceived and planned this analysis; STV conducted the research,
performed the analysis, and drafted the manuscript; all authors reviewed,
edited, and approved the final version.

## Funding

The authors gratefully acknowledge funding through a “Life?” programme grant
from the Volkswagen Stiftung. MPHS is funded through the University of
Melbourne Driving Research Momentum program.

## Acknowledgments

We thank members of the Theoretical Systems Biology group at the University of
Melbourne and Imperial College London, Heike Siebert (FU Berlin), James
Briscoe (The Francis Crick Institute), and Mark Isalan (Imperial College
London) for helpful discussions on Turing patterns and model discrimination.

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