# Global topological synchronization on simplicial and cell complexes

Timoteo Carletti  Department of Mathematics & naXys, Namur Institute for
Complex Systems, University of Namur, Rue Grafé 2, B5000 Namur, Belgium
Lorenzo Giambagli  Department of Mathematics & naXys, Namur Institute for
Complex Systems, University of Namur, Rue Grafé 2, B5000 Namur, Belgium
Department of Physics and Astronomy, University of Florence, INFN & CSDC,
Sesto Fiorentino, Italy  Ginestra Bianconi  School of Mathematical Sciences,
Queen Mary University of London, London, E1 4NS, United Kingdom  The Alan
Turing Institute, 96 Euston Road, London, NW1 2DB, United Kingdom

###### Abstract

Topological signals, i.e., dynamical variables defined on nodes, links,
triangles, etc. of higher-order networks, are attracting increasing attention.
However the investigation of their collective phenomena is only at its
infancy. Here we combine topology and nonlinear dynamics to determine the
conditions for global synchronization of topological signals defined on
simplicial or cell complexes. On simplicial complexes we show that topological
obstruction impedes odd dimensional signals to globally synchronize. On the
other hand, we show that cell complexes can overcome topological obstruction
and in some structures, signals of any dimension can achieve global
synchronization.

Synchronization is a widespread phenomenon at the root of several biological
rhythms or human made technological systems [1, 2]. Synchronization refers to
the spontaneous ability of coupled oscillators to operate at unison and thus
exhibit a coherent collective behavior. Global synchronization is the
resulting phenomenon where all oscillators behave in the same way.

Traditionally synchronization has been studied when identical [3, 4] or non-
identical oscillators [5, 6] are defined on the nodes of a network and are
coupled by the network links. However, to capture the function of many complex
systems, e.g., brain networks [7, 8], social networks [9] and protein
interaction networks [10], it is important to go beyond pairwise interactions
and consider higher-order interactions [11] between two or more nodes instead.
For instance collaborations typically require the cooperation of a team of
more than two individuals, and to perform a function in the cell, proteins
form complexes formed by several different types of proteins. Many-body
interactions are captured by higher-order networks [12, 13] (such as
hypergraphs, simplicial and cell complexes) and are dramatically changing our
understanding of the interplay between structure and dynamics of complex
systems [13, 11, 14]. Note that higher-order networks by definition are
constructed by higher-order building blocks, but like networks they can have
different structure [13]. In particular a higher-order network can display a
very regular (lattice-like) structure, a fractal [15] or a very random
structure [16]. Moreover higher-order networks can be also built from network
motifs according to clique-complex construction, by identifying each clique
with a simplex, and its generalizations [13].

Lately, synchronization of identical and non-identical oscillators defined on
the nodes of higher-order networks has been a field of intense research
activity. Global synchronization of identical oscillators have been first
formulated for special topologies (a $p$-regular hypergraph) [17] and for a
peculiar Laplace operator obtained from the hyper-adjacency matrix [18] while
recently a general and comprehensive theoretical framework has been proposed
in [19] to study dynamical systems defined on hypergraph with heterogeneous
hyperedges size distribution, the latter influencing also the Laplace matrix.
Partial synchronization of non-identical nodes oscillators has been
investigated using a variation of the Kuramoto model leading to explosive
transitions [20].

Simplicial and cell complexes also sustain topological signals [13, 21, 22,
23], i.e., dynamical variables that can be defined not only on nodes, but also
on links, triangles and higher-dimensional simplices or cells. Examples of
real topological signals are edge signals such as biological transportation
fluxes or traffic signals [23], synaptic and brain edge signals [24], or
climate data such as currents in the ocean or speed of wind at different
locations [25]. In the framework of quantum systems, it has been shown that
synchronization and quantum entanglement are strongly intertwined [26, 27];
observation that can be relevant in quantum computations involving anyons [28]
or bosons [29] where signals defined on links of some array, should be
protected from noise. Finally, topological signals are attracting increasing
attention in signal processing and machine learning [23, 25, 30, 31, 23, 32].
However the study of their collective phenomena its only at its infancy [21,
22, 33, 34, 35, 36].

Recently the formulation of higher-order topological Kuramoto model [21, 22,
36] has demonstrated that topological signals of any dimension can synchronize
leading to either continuous or to explosive synchronization transitions.
These results concern partial synchronization while an important question is
whether global synchronization of topological signals can ever be achieved.

The aim of this work is to determine the topological and dynamical conditions
under which global topological synchronization of identical topological
oscillators can be observed. Relying on the use of higher-order Laplacian
matrices [37, 38, 39, 40] and the extension of the Master Stability Function
(MSF) to simplicial and cell complexes dynamics, we are able to tackle this
problem emphasizing the difference existing among the two frameworks.

Anticipating on our results we can state that on simplicial complex we observe
topological obstruction: given a simplicial complex of dimension $K$, if the
topological signal is defined on an odd-dimensional simplex of dimension $k<K$
then, global synchronization is not possible. On the other hand if the simplex
has an even dimension, then we can have global synchronization provided the
simplex is balanced (see hereafter) and the model parameters allow for it.
Interestingly we show that cell complexes can overcome topological obstruction
and some topologies can sustain global synchronization of signals of any
dimension.

Definition of simplicial and cell complexes.– Simplicial and cell complexes
are generalised network structures that beside nodes and links do also contain
triangles, polygons, tetrahedra, hypercubes, orthoplexes and so on. Given a
non-negative integer $k$, a $k$-simplex, $\sigma^{k}$, is a set of $k+1$
different nodes, $\sigma^{k}=[v_{0},v_{1},\ldots,v_{k}]$. A $(k-1)$-face,
$\sigma^{k-1}$, of a $k$-simplex is a $(k-1)$-simplex obtained by removing one
vertex from $\sigma^{k}$. Therefore a node is a $0$-simplex, a link is a
$1$-simplex, a $2$-simplex is a triangle and so on. A simplicial complex
$\mathcal{X}$ of dimension $K$ is a finite collection of simplices of
dimensions $k\leq K$, closed under the inclusion of faces, namely if
$\sigma\in\mathcal{X}$, then also all the faces of $\sigma$ should belong to
$\mathcal{X}$. Simplicial complexes are, thus a natural generalisation of
networks, which are recovered for $k=1$. In algebraic topology a simplex
$\sigma^{k}$ is assigned an orientation, typically induced by the node labels.
A coherent orientation of the face with the one of the simplex, will be
denoted by $\sigma^{k-1}\sim\sigma^{k}$, otherwise we will write
$\sigma^{k-1}\not\sim\sigma^{k}$.

A more general structure retaining the algebraic richness of simplicial
complexes, is given by the cell complexes [41, 42, 43, 44, 38]. The latter
differs from simplicial complexes because they are not just built by simplices
but instead they are obtained by gluing cells, (i.e., regular polytopes) along
their faces. In particular $0$-cells are nodes, $1$-cells links, while
$2$-cells are generic polygons,and $3$-cells are the Platonic polytopes.

In algebraic topology the boundary operator is defined on chains [13, 38],
linear combinations of oriented simplices of the simplicial complex. The $k$
boundary operator retains the information about the faces of a $k$-simplex
$\sigma^{k}$ and their relative orientation. The boundary operator is encoded
into the incidence matrices, $\mathbf{B}_{k}$ whose elements are given for all
$k=1,\dots,K$ by $B_{k}(i,j)=1$ if $\sigma_{i}^{k-1}\sim\sigma_{j}^{k}$, $-1$
if $\sigma_{i}^{k-1}\not\sim\sigma_{j}^{k}$ and $0$ otherwise 111Let us
observe that the $(i,j)$ entry of the matrix $\mathbf{B}_{k}$ has been denoted
by $B_{k}(i,j)$; a similar assumption has been used to denote the entries of
vectors., where $K$ is the dimension of the simplicial complex, namely the
size of the largest simplex. One key property of the boundary operator is that
the boundary of the boundary is null, hence
$\mathbf{B}_{k}\mathbf{B}_{k+1}=0$. To be coherent an oriented simplicial
complex needs to satisfy this condition for every $k$ with $0\leq k\leq K$.
Let us conclude this section by noticing that the boundary operators for cell
complexes have the same definition [23].

The higher-order Laplacians [13, 33, 38, 39, 40] allow to define diffusion
among $k$-simplices and capture the topology of the higher-order network; they
are defined in terms of the incidence matrix as:

$\mathbf{L}_{k}=\mathbf{B}_{k}^{\top}\mathbf{B}_{k}+\mathbf{B}_{k+1}\mathbf{B}_{k+1}^{\top}\,,\quad
k=1,\dots,K-1\,.$ (1)

For $k=0$ and $k=K$ we have instead
$\mathbf{L}_{0}=\mathbf{B}_{1}\mathbf{B}_{1}^{\top}$ and
$\mathbf{L}_{K}=\mathbf{B}_{K}^{\top}\mathbf{B}_{K}$. If we denote by $N_{k}$,
$k=0,\dots,K$, the number of $k$-simplices, then it follows that
$\mathbf{B}_{k}$ is a $N_{k-1}\times N_{k}$ matrix, while the size of
$\mathbf{L}_{k}$ is $N_{k}\times N_{k}$. The matrix $\mathbf{L}_{0}$ coincides
with the graph Laplacian. The higher-order Laplacian $\mathbf{L}_{k}$ is a
semidefinite operator and one of its most celebrated properties is that the
dimension of its kernel is given by the $k$-Betti number, i.e.,
$\mathrm{dim}(\ker\mathbf{L}_{k})=\beta_{k}$, where $\beta_{k}$ indicates the
number of $k$-dimensional cavity in the simplicial complex. Moreover there is
a basis of the kernel $\mathbf{L}_{k}$ which is formed by eigenvectors
localized on each of the $k$-dimensional cavities of the simplicial complex.

Simplicial and cell-complex dynamics.– Let us now consider a topological
$k$-dimensional signal encoded in a $k$ dimensional cochain
$\mathbf{x}:C_{k}\to\mathbb{R}^{d}$ which assigns to every chain $C_{k}$
(linear combination of $k$-simplices) values on $\mathbb{R}^{d}$. The
$k$-topological signal has elements
$\mathbf{x}_{i}=\mathbf{x}(\sigma_{i}^{k})=(x_{i}^{1},\dots,x_{i}^{d})$
defined on the $i$-th oriented $k$-simplex, $\sigma_{i}^{k}$ (see SM).
According to the properties of the $k$-cochains [38, 39] we have ${\bf
x}(-\sigma_{i}^{k})=-{\bf x}(\sigma_{i}^{k})$, being the discrete analogous of
differential forms on manifolds. For instance, for $k=1$ and $d=1$,
$x_{i}={x}(\sigma_{i}^{k})$ indicates a flux defined on the link $i$ that is
positive if going in the same direction of the positive orientation of the
link and negative otherwise, i.e., $x(-\sigma_{i}^{k})=-x_{i}$. Let us assume
the value of the topological signal on every simplex $i$ follows the same
dynamics and evolves according to
$\dot{\mathbf{x}}_{i}=\mathbf{f}(\mathbf{x}_{i})$, for some odd nonlinear
function $\mathbf{f}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$. Assume now the
$k$-simplex to belong to a $K$-simplicial complex, $K\geq k$, and assume the
existence of a diffusive–like nonlinear interaction among adjacent simplices
of the same dimension:

$\frac{d\mathbf{x}_{i}}{dt}=\mathbf{f}(\mathbf{x}_{i})-\sum_{j=1}^{N_{k}}L_{k}(i,j)\mathbf{h}(\mathbf{x}_{j})\quad\forall
i=1,\dots,N_{k}\,,$ (2)

where $\mathbf{h}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ is some odd
nonlinear coupling function. This equation generalizes the dynamics of
identical oscillators anchored to each node [3] to the scenario in which
identical oscillators are defined on higher-dimensional simplices or cells.
Please note that requiring odd functions $\mathbf{f}(\mathbf{x}_{i})$ and
$\mathbf{h}(\mathbf{x}_{i})$ is necessary for higher-order topological signals
with $k>0$ in order to ensure invariance under change of orientation of each
simplex $i$ (see SM). For node dynamics ($k=0$) the existence of a global
synchronized state is automatically determined by the properties of the graph
Laplacian whose kernel is spanned by $\mathbf{u}=(1,\ldots,1)^{\top}$, indeed
we have $\beta_{0}=1$, one connected component. Its stability is instead
determined by the celebrated MSF [3, 4].

Given the growing interest in topological signals, a key question is how these
classic results of nonlinear dynamics on networks extend to nonlinear dynamics
of topological signals on simplicial complexes. Anticipating our results, we
will show that topology and combinatorics of the higher-order Laplacian will
not always ensure existence of a globally synchronized state, and moreover
since the dimension of the kernel of $\mathbf{L}_{k}$ can be bigger than one,
also the MSF will differ from the network case.

Let us then fix a reference stable solution $\mathbf{s}(t)$ of the uncoupled
system, $\dot{\mathbf{x}}_{i}=\mathbf{f}(\mathbf{x}_{i})$. We are interested
in determining the conditions under which the state having each simplex $i$
either in the state $\mathbf{x}_{i}=\mathbf{s}(t)$ or in
$\mathbf{x}_{i}=-\mathbf{s}(t)$ is also a stable solution of the coupled
system (2). Namely the latter exhibits a global synchronous behavior in which
all simplices display the same dynamics of the isolated simplices when we
account for differences of sign, determined by their orientation (see SM).

Let us now introduce the vector
$\mathbf{v}=(v_{1},\dots,v_{N_{k}})^{\top}\in\\{-1,1\\}^{N_{k}}$, such that
the globally synchronized state is given by
$\mathbf{x}_{i}=v_{i}\mathbf{s}(t)$. A necessary condition to observe global
synchronization, is that $\sum_{j}L_{k}(i,j)\mathbf{v}_{j}=0$ (see SM). Let us
recall that
$\mathrm{ker}\mathbf{L}_{k}=\mathrm{ker}\mathbf{B}_{k}\cap\mathrm{ker}\mathbf{B}_{k+1}^{\top}$,
thus the latter condition ultimately returns to require
$\mathbf{B}_{k}\mathbf{v}=0$ (condition i) and
$\mathbf{v}^{\top}\mathbf{B}_{k+1}=0$ (condition ii) (see SM).

Figure 1: Schematic description of conditions i) (panels a,b,e,f) and ii)
(panels c and d) for topological signals defined on $1$-dimensional cells
(links, panels in top row) and $2$-dimensional cells (triangles or squares,
panels in bottom row) in which we assume that there is an orientation such
that $\mathbf{w}=\mathbf{u}=(1,1,\ldots)^{\top}$. In the case of simplicial
complexes (panels a, d) condition i) cannot be satisfied for signals defined
on $1$-dimensional simplices while in the case of cell complexes (panels e, f)
condition i) it can be satisfied. Condition ii) can be satisfied on simplicial
and cell complexes as long as the simplices are balanced (see panels c, d) for
the simplicial complex case).

The first condition has a striking consequence. If $k$ is an odd number,
because any $(k+1)$-simplex contains an odd number of $k$-faces, then
condition i) cannot be ever satisfied. On the contrary if $k$ is even, then
any $(k+1)$-simplex contains an even number of $k$-faces condition i) can be
realised (see Fig. 1a-b). On the other hand condition ii) can be satisfied by
imposing a suitable condition of the $(k-1)$-faces of the $k$-simplex, which
we call balanced condition. In particular if
$\mathbf{v}=\mathbf{u}=(1,\ldots,1)^{\top}$ this condition can be satisfied by
requiring every $(k-1)$-face to be adjacent to an even number of $k$-simplices
and to be coherently oriented with half of them (see Fig. 1c-d).

Therefore for even values of $k$, global synchronization can be achieved while
if $k$ is odd we observe, as long as $\mathbf{v}^{\top}{\bf B}_{k+1}\neq{\bf
0}$, a topological obstruction to the onset of global synchronization.
Interestingly for $K$-dimensional signals having ${\bf B}_{K+1}={\bf 0}$, only
the balanced condition remains (i.e., condition ii)) which is automatically
satisfied for the vector $\mathbf{v}=\mathbf{u}$ if the simplicial complex is
a closed manifold without boundary. Hence $K$-dimensional topological signals
defined on closed $K$ dimensional manifolds can always achieve global
synchronization for arbitrary value of $K$.

A similar derivation can be generalized and extended to topological signals
defined on the $k$-dimensional cells of cell complexes. In particular the
conditions to achieve global synchronization on a cell complex are unchanged
and given again by condition ${\bf i)}$ and ${\bf ii)}$. However the
combinatorics of cell complexes is different from the one of simplicial
complexes. Take for instance a cell complex whose network skeleton is formed
by a $d$-dimensional square lattice with periodic boundary conditions, i.e., a
regular tessellation of $d$-dimensional torus. Then every cell of dimension
$k+1>0$ has a even number of $k$-dimensional faces therefore condition ${\bf
i)}$ can be satisfied also if $k$ is odd (see Fig. 1e-f). This implies that on
cell complexes we can overcome topological obstruction. Until now we have
focused on the combinatorial implication of the conditions $\mathbf{i)}$ and
$\mathbf{ii)}$. However these conditions have also topological consequences.
In fact, since on manifolds the eigenvectors of the kernel of the Hodge
Laplacian $\mathbf{L}_{k}$ are localized on the $k$-dimensional holes,
manifolds that will display an eigenvector with the properties of the above
defined $\mathbf{v}$, are characterized by holes spanning the whole structure
as for instance $(k+1)$-dimensional hyperspheres or $d$-dimensional tori with
$d>k$.

Master Stability Equation for Topological Signals.– Let us now assume the
reference solution $\mathbf{s}(t)$ to be also a solution of the coupled system
(2), then by introducing the distance from the reference orbit,
$\delta\mathbf{x}_{i}=\mathbf{x}_{i}-\mathbf{s}(t)$, we can derive its time
evolution by linearizing Eq. (2):

$\frac{d\delta\mathbf{x}_{i}}{dt}=\mathbf{J}_{\mathbf{f}}(\mathbf{s})\delta\mathbf{x}_{i}-\sum_{j=1}^{N_{k}}L_{k}(i,j)\mathbf{J}_{\mathbf{h}}(\mathbf{s})\delta\mathbf{x}_{j}\quad\forall
i=1,\dots,N_{k}\,,$ (3)

being $\mathbf{J}_{\mathbf{f}}(\mathbf{s})$ (resp.
$\mathbf{J}_{\mathbf{h}}(\mathbf{s})$) the Jacobian of the function
$\mathbf{f}$ (resp. $\mathbf{h}$) evaluated on the reference solution.

The matrix $\mathbf{L}_{k}$ being symmetric, it admits an orthonormal basis,
$\pmb{\phi}_{k}^{(\alpha)}$, associated to eigenvalues
$\Lambda_{k}^{(\alpha)}$, $\alpha=1,\dots,N_{k}$, namely
$\mathbf{L}_{k}\pmb{\phi}_{k}^{(\alpha)}=\Lambda_{k}^{(\alpha)}\pmb{\phi}_{k}^{(\alpha)}$.
In particular, since we work under the assumption that the simplicial complex
is balanced,
$\pmb{\phi}_{k}^{(1)}\sim(1,\dots,1)^{\top}\in\mathbb{R}^{N_{k}}$,
$\Lambda_{k}^{(\alpha)}=0$ for $1\leq\alpha\leq\beta_{k}$ and
$\Lambda_{k}^{(\alpha)}>0$ for all $\alpha>\beta_{k}$.

Let us decompose the deviation vectors $\delta\mathbf{x}_{i}$ onto this
eigenbasis:
$\delta\mathbf{x}_{i}=\sum_{\alpha}\delta\mathbf{x}^{(\alpha)}\pmb{\phi}_{k}^{(\alpha)}(i)$.
Then linearizing the dynamical equation, we obtain

$\frac{d\delta\mathbf{x}^{(\alpha)}}{dt}=\left[\mathbf{J}_{\mathbf{f}}(\mathbf{s})-\Lambda_{k}^{(\alpha)}\mathbf{J}_{\mathbf{h}}(\mathbf{s})\right]\delta\mathbf{x}^{(\alpha)}\quad\forall\alpha=1,\dots,N_{k}\,.$
(4)

Perturbations aligned with the kernel do not change the stability of the
uncoupled system, therefore only the perturbations orthogonal to the kernel
can modify the stability of the reference solution. This is the MSF in the
framework of simplicial synchronization of topological signals. It is a
linear, in general non autonomous, ODE parametrized by the eigenvalues
$\Lambda_{k}^{(\alpha)}$, allowing to infer the stability character of the
reference solution by looking at its spectrum.

Figure 2: The Kuramoto order parameter $R$ is plotted versus time $t$ for the
Stuart-Landau model of topological oscillators of the balanced simplicial and
cell complexes represented in panels a) and d) respectively. Panels b) and c)
refer respectively to the order parameter of triangles and links of the
simplicial complex in panel a). Panels e) and f) refer respectively to the
order parameter of the squares and links of the cell complex in panel d). The
insets display the dynamical time series of the topological signals. It is
clear that while on the links of the simplicial complex, the oscillators do
not globally synchronize, the ones of the cell complex do support
synchronization. The model parameters are $\sigma=1.0+4.3i$, $\beta=1.0+1.1i$,
$\mu=1.0-0.5i$ and $m=3$ ensuring the negativity of the dispersion relation
(see SM).

Simplicial Stuart-Landau model.– As an application of the general theory above
introduced, let us consider the Stuart-Landau (SL) model [46, 47, 48] defined
for topological signals of dimension $k$ and $d=2$. For $k=0$ the model
describes a nonlinear oscillator anchored at each node, while for $k=1$ it can
describe an oscillating flux associated to an edge linking two nodes. More
precisely let us define $w_{j}=x_{j}^{1}+\mathrm{i}x_{j}^{2}$ and let us
consider the “local reaction” function $f({\bf x})=\tilde{f}(w)=\sigma w-\beta
w|w|^{2}$, where $\sigma=\sigma_{\Re}+i\sigma_{\Im}$ and
$\beta=\beta_{\Re}+i\beta_{\Im}$ are complex control parameters. We can prove
that the uncoupled dynamics $\dot{w}_{j}=f(w_{j})$ in each simplex $j$ admits
a limit cycle solution $\hat{z}(t)=\sqrt{\sigma_{\Re}/\beta_{\Re}}e^{i\omega
t}$, where $\omega=\sigma_{\Im}-\beta_{\Im}\sigma_{\Re}/\beta_{\Re}$, that is
stable provided $\sigma_{\Re}>0$ and $\beta_{\Re}>0$, conditions that we
hereby assume. We now consider the coupled dynamics Eq. (2) with nonlinear
coupling function $h({\bf x})=\tilde{h}(w)=\mu w|w|^{m-1}$, where $m$ is a
positive integer and $\mu=\mu_{\Re}+i\mu_{\Im}$ a complex parameter that sets
the coupling strength [49]. We study the stability of the reference limit
cycle solution $\hat{z}(t)$ (see SM) and we prove that the system can globally
synchronize, i.e., the dispersion relation is negative, only if the model
parameters do satisfy $\mu_{\Re}+\mu_{\Im}{\beta_{\Im}}/{\beta_{\Re}}>0\,,$
and the simplicial complex is such that $\mathbf{u}\in\ker\mathbf{L}_{k}$. To
measure global synchronization we compute the (generalised) Kuramoto order
parameter $R(t)=\Big{\lvert}\sum_{j}\rho_{j}(t)e^{i\theta_{j}(t)}\Big{\rvert}\
/N_{k}$, where $w_{j}(t)=\rho_{j}(t)e^{i\theta_{j}(t)}$ is the polar form of
the complex signal. Then $R(t)\rightarrow 1$ testifies the existence of phase
and amplitude synchronization. Results shown in Fig. 2 provide numerical
evidence of our theoretical predictions. In Fig. 2a-c we show the results
obtained by studying the SL model defined on top of a designed balanced
$3$-simplicial complex (see SM). The model parameters have been set to values
allowing for a negative dispersion relation (see SM) and indeed once the
complex amplitudes are defined on $2$-faces, i.e., triangles, the system
globally synchronizes (see Fig. 2b). On the other hand, once the SL
oscillators are defined on links the system cannot globally synchronize (see
Fig. 2c). In Fig. 2d-f we provide an example of a cell complex which overcomes
topological obstruction: a $3D$ square lattice with periodic boundary
conditions. Such cell complex is made of nodes, link, squares and cubes (see
SM). In this case case global synchronization can be achieved for signals of
every dimension (see Fig. 2e-f for global synchronization of links and
squares).

In conclusion in this work we have studied global synchronization of identical
topological oscillators on simplicial or cell complexes. We have found that
global synchronization of topological signals cannot be observed on arbitrary
simplicial or cell complexes but that only some special higher-order network
topologies can sustain such a dynamical state. This is in stark contrast with
the corresponding scenario in networks where global synchronization can be
observed in every network topology given proper dynamical conditions. By
combining topology, and in particular the spectral properties of higher-order
Laplacians, to nonlinear dynamics techniques such as the MSF, we have
identified the topological and dynamical conditions under which identical
topological oscillators can achieve global synchronization on simplicial or
cell complexes. We have proved that global synchronization of odd dimensional
topological signals is obstructed in simplicial complexes. This topological
obstruction implies that on a $K$-dimensional simplicial complex we can never
observe global synchronization of odd dimensional topological signals of
dimension $d<K$. However, such obstruction is not present in cell complexes.
In particular we show evidence that in specific topologies such as the
$d$-dimensional square lattice with periodic boundary conditions global
synchronization of topological signal of any dimension can be observed.

These results significantly enrich our understanding of the relation between
higher-order network topology and dynamics revealing collective phenomena of
topological signals. Our study is relevant, for its inherent simplicity, to a
wide spectrum of applications (neuroscience/biology/social sciences) where
many-body interactions involve higher-order interacting units. Therefore we
hope that this Letter will stimulate further research work in physics and
beyond.

G.B. acknowledges funding from the Alan Turing Institute and the Royal Society
(IEC\NSFC\191147).

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SUPPLEMENTAL MATERIAL

##  Appendix A Extension of the Master Stability Function Approach for
dynamics of topological signals

In this section we provide the mathematical background for the extension of
the Master Stability Function (MSF) defined on networks to the MSF for
topological signals of a simplicial or cell complex.

Let us start by defining a $k$-dimensional chain $c\in\mathcal{C}_{k}$ as a
linear combination of oriented simplices (or cells) $\sigma^{k}_{i}$, i.e.

$c=\sum_{i}c_{i}\sigma^{k}_{i}$ (S-1)

with coefficients $c_{i}\in\mathbb{R}$.

A $k$-dimensional topological signal $x$ is a $k$-dimensional cochain
$x:\mathcal{C}_{k}\to\mathbb{R}^{d}$, i.e., a linear function that associates
to every $k$-chain of the simplicial complex a value in $\mathbb{R}^{d}$.

Since the cochain indicates a linear function we have

$x(c)=\sum_{i}c_{i}x(\sigma^{k}_{i}).$ (S-2)

Note that thanks to the linearity of the cohain $x$ we always have
$x(\sigma^{k}_{i})=-x(-\sigma^{k}_{i})$. From Eq.(S-2) it follows that a
cochain $x$ is uniquely determined by the vector $\mathbf{x}$ of elements
$\mathbf{x}_{i}=x(\sigma^{k}_{i})$.

Let us consider a cochain $\mathbf{x}$ and a function $\mathbf{F}$ that
associated to $\mathbf{x}$ a new cochain denoted by $\mathbf{F}(\mathbf{x})$.
Based on the above, the latter is solely characterized by its values on
$\sigma_{i}^{k}$, namely its “components”,
$\mathbf{F}(\mathbf{x})=(\mathbf{f}_{1}(\mathbf{x}),\dots,\mathbf{f}_{N_{k}}(\mathbf{x}))$.
Let us finally assume each component to act componenent-wise, namely
$\mathbf{f}_{i}(\mathbf{x})=\mathbf{f}_{i}(\mathbf{x}_{i})$ for all $i$. Being
$\mathbf{x}$ and $\mathbf{F}(\mathbf{x})$ cochains they should be invariant
with respect to the change of orientation of $\sigma_{i}^{k}$, namely

${\mathbf{f}_{i}(\mathbf{x}_{i})=-\mathbf{f}_{i}(-\mathbf{x}_{i})\,.}$

We now want to study the evolution of the topological signals by considering
dynamical equations for the vector $\mathbf{x}$.

The uncoupled dynamics of topological signals is determined by

$\frac{d\mathbf{x}_{i}}{dt}=\mathbf{f}({\bf x}_{i}),$ (S-3)

where $\mathbf{f}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ is a nonlinear function
which should preserve the invariance of the dynamics under a change of the
orientations of the simplices. Requiring the invariance of this equation under
change of orientation of the simplex $i$, imposes that the function
$\mathbf{f}(\mathbf{x})$ must be odd. Indeed if the orientation of the simplex
$\sigma^{k}_{i}$ is reversed we have $\mathbf{x}_{i}\to-\mathbf{x}_{i}$ and
the dynamical Eq.(S-3) becomes

$-\frac{d\mathbf{x}_{i}}{dt}=\mathbf{f}(-{\bf x}_{i}).$ (S-4)

This is indeed the above presented case where $\mathbf{f}_{i}=\mathbf{f}$ for
all $i$. Therefore to ensure that the dynamical equation (S-3) obeys the
necessary invariance conditions for describing the dynamics of topological
signals of dimension $k>0$, we need to require $\mathbf{f}(-{\bf
x}_{i})=-\mathbf{f}({\bf x}_{i})$, hence that $\mathbf{f}(\mathbf{x}_{i})$ is
an odd function. Note that requiring odd nonlinear functions
$\mathbf{f}(\mathbf{x}_{i})$ is a necessary condition only for higher-order
topological signals of dimension $k>0$. Indeed since nodes do not have an
orientation this requirement is not necessary for treating topological signals
defined on nodes. Therefore this is an important difference with the dynamics
defined exclusively on nodes.

Let us now consider a stable solution $\mathbf{s}(t)$ of Eq.(S-3). Due to the
parity of $\mathbf{f}(\mathbf{x})$ we are automatically guaranteed that also
$-\mathbf{s}(t)$ is a stable solution of Eq.(S-3).

We define as global synchronization of topological signals a dynamical
configuration in which $\mathbf{x_{i}}=\mathbf{s}(t)$ or
$\mathbf{x_{i}}=-\mathbf{s}(t)$ for every $k$-dimensional simplex $i$ of the
simplicial or cell complex. This definition again differs from the one of
global synchronization defined for node signals and guarantees the invariance
under the orientation of the simplices. To give an intuition of this result,
take the case of link signals indicating fluxes: saying that a flux is $\phi$
on the link $[i,j]$ oriented from $i$ to $j$ is the same as saying that the
flux is $-\phi$ on the link $[i,j]$ oriented from $j$ to $i$. Therefore global
synchronization is observed where the dynamics of each simplex is the same,
when factoring out the sign due to the choice of orientation.

We now turn our attention to the case of interacting topological signals
coupled by the Hodge Laplacian operator whose dynamics is dictated by Eq.(3)
in the main text that we rewrite here for convenience

$\frac{d\mathbf{x}_{i}}{dt}=\mathbf{f}(\mathbf{x}_{i})-\sum_{j=1}^{N_{k}}L_{k}(i,j)\mathbf{h}(\mathbf{x}_{j})\quad\forall
i=1,\dots,N_{k}\,,$ (S-5)

where $\mathbf{h}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ is a nonlinear function
which should preserve the invariance under choice of the orientation of the
simplices (or cells) and where $\mathbf{L}_{k}$ is the $k$-dimensional Hodge
Laplacian. Taking into account that the latter has elements

$L_{k}(i,j)=\left\\{\begin{array}[]{ll}d(\sigma^{k}_{i})+(p+1),&i=j.\\\
1,&i\neq
j,\sigma^{k}_{i}\not\frown\sigma^{k}_{j},\sigma^{k}_{i}\smile\sigma^{k}_{j},\sigma^{k}_{i}\sim\sigma^{k}_{j}.\\\
-1,&i\neq
j,\sigma^{k}_{i}\not\frown\sigma^{k}_{j},\sigma^{k}_{i}\smile\sigma^{k}_{j},\sigma^{k}_{i}\nsim\sigma^{k}_{j}.\\\
0,&\text{otherwise}\end{array}\right.,$ (S-6)

where $\smile$ and $\frown$ indicate lower and upper incident neighbor
simplices respectively, it can be easily derived that in order to preserve the
mentioned invariance also $\mathbf{h}(\mathbf{x})$ must be an odd function.
Note that this is another example of a peculiar property of higher-order
topological signals with $k>0$ which is not present for the description of
dynamics of identical oscillators placed on the nodes (node signals: case
$k=0$).

We now address the question of where global synchronization of higher-order
topological signals can be observed. From Eq. (S-5) it is clear that a
necessary condition to observe global synchronization is that the kernel of
the Hodge Laplacian $\mathbf{L}_{k}$ admits an eigenvector $\mathbf{v}$ whose
elements are in $\\{-1,1\\}$, i.e., $|v_{i}|=1$, for all $i$. Note that since
the degeneracy of the zero eigenvalue of $\mathbf{L}_{k}$ is given by the
$k$-th Betti number, it follows that the Hodge Laplacian $\mathbf{L}_{k}$,
when $k>0$, can admit more than one eigenvectors of this form. However we are
not guaranteed that an arbitrary simplicial or cell complex will have at least
one such eigenvectors. In order to give a concrete example of such topologies
we mention that a $d$-dimensional regular cell complex which tessellates a
torus has $\text{Bin}(d,k)$ such eigenvectors while any simplicial or cell
complex without any $k$-dimensional hole will have $k$-th Betti number
$\beta_{k}=0$ and hence any such eigenvector. This is a property that is in
strike contrast with what happens for $k=0$ where any connected network has
one and only one constant eigenvector $(1,\ldots,1)^{\top}$ in its kernel.

Let us consider a simplicial or cell complex which has an eigenvector
$\mathbf{v}$ as above, therefore as stated in the main text, a necessary
condition to have global synchronization is that the kernel of the Hodge
Laplacian $\mathbf{L}_{k}$ admits the eigenvector $\mathbf{v}$. This necessary
condition is met under the combinatorial conditions i) and ii) discussed in
the main text and for more detail in the next section of the SM. Note that
these combinatorial constraints have also topological consequences as
discussed in the main text, indeed they imply that global synchronization is
allowed on topologies having holes that span the entire structure as
hypersphere and tori.

The Master Stability Equation (MSE) for topological signals is discussed in
the main text. We observe that in this case the major difference with the MSE
for node signals ($k=0$) is that the perturbations can also lie on the kernel
since the $k$-th Betti number $\beta_{k}$ of the simplicial complex can be
eventually also bigger than one. These perturbations however do not change the
stability of the dynamics, since the non-interacting solutions is assumed to
be stable.

##  Appendix B About the topological obstruction

Let us consider a $k$ dimensional topological signal whose time evolution is
given by Eq. (2) (main text), we have shown that a necessary condition for the
existence of the global synchronization manifold is that the vector
$\mathbf{v}$ of elements $|v_{i}|=1$ lies in $\ker\mathbf{L}_{k}$. Because
$\mathrm{ker}\mathbf{L}_{k}=\mathrm{ker}\mathbf{B}_{k}\cap\mathrm{ker}\mathbf{B}_{k+1}^{\top}$,
such condition is equivalent to require $\mathbf{v}^{\top}\mathbf{B}_{k+1}=0$
(condition i) and $\mathbf{B}_{k}\mathbf{v}=0$ (condition ii).

Condition i) imposes a constraint on the dimension of the simplex. It cannot
be verified on a $k$-simplex being $k$ an odd number smaller than the simplex
dimension. Here we consider a pedagogical example with $k=1$, i.e., the
topological signal is defined on a link. Because any $2$-simplex, i.e., a
triangle, contains three links, whichever the orientation they have and for
any choice of the vector $\mathbf{v}$, then $\mathbf{v}^{\top}\mathbf{B}_{2}$
cannot vanish, being a sum of an odd number of “$+1$” and “$-1$”, thus
condition i) cannot be satisfied see Fig. 1a). On the other hand if $k$ is
even we can build a suitable $k$-simplex verifying condition i). For
simplicity we propose an example with $k=2$, i.e., a triangle. Then any
tetrahedron must contain four triangles and we can always give them a suitable
orientation, two by two coherent each other, in such a way the sum of the
column of $\mathbf{B}_{3}$ is zero see Fig. 1b).

On the other hand condition ii) can be satisfied by imposing the balanced
condition on the $(k-1)$-faces of the $k$-simplex. Let us consider for
instance the case in which $\mathbf{v}=\mathbf{u}=(1,\ldots,1)^{\top}$. In
Fig. 1c we consider again, as an example, the case $k=1$, hence to satisfy
condition ii) is enough to impose that each $(k-1)$-simplex, i.e., a node in
this case, is incident with an even number of $1$-simplex (four in the
figure), half of them coherently oriented, i.e., entering in the node, and
half of them non coherently oriented, i.e., exiting from the node. The
validity of this condition does not depend on the dimension of the simplex,
indeed in panel d) we consider the case of $k=2$, i.e. a triangle, and it is
thus enough that each $(k-1)$-simplex, i.e., a link, is again incident with an
even number of $2$-simplex (two in the figure), having two by two opposite
orientation.

##  Appendix C A $3$-simplicial complex satisfying conditions i) and ii)

The aim of this section is to show how to build a $3$-simplicial complex,
$\mathcal{P}$, satisfying conditions i) and ii) for $k=2$ and $k=0$, and thus
admitting a synchronous manifold for topological signals defined on faces or
nodes. The nodes of this simplicial complex are placed on a $2$-dimensional
square grid with periodic boundary conditions. However each four nodes of any
(imaginary) square placquette of this grid are belonging to a distinct
tetrahedron. As it can be directly shown this structure admits an eigenvector
$\mathbf{v}$ with elements $|v_{i}|=1$ for any orientation induced by the node
labels, as long as $k\in\\{0,2\\}$.

This interesting structure can be obtained by looking at topologies that
satisfy conditions i) and ii) for an orientation such that
$\mathbf{v}=\mathbf{u}=(1,\dots,1)^{\top}$ will lie in $\ker\mathbf{L}_{2}$
(and of course in $\ker\mathbf{L}_{0}$). One can show that the resulting
oriented simplicial complex has the the same topological properties of the
simplicial complex oriented by using the node label order, the reason being
that their basis are related by an invertible linear transformation.

For sake of concreteness we will consider the case of topological signals
defined on $2$-simplices, belonging to a $3$-simplicial complex. The case
$k=0$ being associated to a standard network synchronization phenomenon fits
in the standard global synchronisation phenomenon on networks. Conditions i)
and ii) translate thus into $\mathbf{B}_{2}\mathbf{u}=0$ and
$\mathbf{B}_{3}^{\top}\mathbf{u}=0$. The sought simplicial complex will thus
be obtained by assembling together nodes, links, triangles and tetrahedra. Let
us notice that once such simplicial complex has been built we show that he can
support global synchronization also for topological signals defined on
$0$-simplex, i.e., on nodes, while this will not be the case for topological
signals defined on $1$-simplex, links, or $3$-simplex, tetrahedra.

Let us start by considering a tetrahedron (see Fig. S-1a) and let us fix an
orientation for its faces, $A$, $B$, $C$ and $D$. For sake of concreteness, we
define a face to be positively oriented if one circulates between its vertices
in a anticlockwise manner, and negatively oriented on the opposite (see Fig.
S-1b). To better appreciate the relative orientations of the faces we show the
same tetrahedron but “flattened”, in this way it is clear the presence of two
positively oriented ($A$ and $C$) and two negatively oriented ($B$ and $D$)
faces, which implies that condition i) is satisfied, i.e.,
$\mathbf{B}_{3}^{\top}\mathbf{u}=0$.

By labeling the vertices with $a$, $b$, $c$ and $d$, we can assign an
orientation to the links (Fig. S-1b); together with the faces orientation we
can compute the incidence matrix $\mathbf{B}_{2}$ (see Fig. S-1c). From the
latter we can appreciate the existence of four links (rows shaded in light
grey in the incidence matrix, also coloured in orange in Fig. S-1a) that are
coherently oriented with two faces, thus in the product
$\mathbf{B}_{2}\mathbf{u}$ they return a $+2$. There are also two links (white
rows in the incidence matrix) that are not coherently oriented with the two
faces to which they are incident, hence in the product
$\mathbf{B}_{2}\mathbf{u}$ they contribute with a $0$.

Figure S-1: An oriented tetrahedron. We consider an oriented tetrahedron,
i.e., $3$-simplex, both in a $3$D view (panel a)) and in a “flatten” $2$D one
(panel b)). Given the orientation of the faces and of the links proposed in
panel b), we can compute the incidence matrix $\mathbf{B}_{2}$ (panel c)). We
can observe the presence of four links whose contribution to
$\mathbf{B}_{2}\mathbf{u}$ is non zero, they correspond to the grey rows in
the matrix and to the orange links in panel a).

The idea to build the sought $3$-simplicial complex satisfying the constraints
$\mathbf{B}_{2}\mathbf{u}=0$ and $\mathbf{B}_{3}^{\top}\mathbf{u}=0$, is to
add another oriented tetrahedron with an opposite orientation for its faces
with respect to the previous one, but with the same orientation of the links
(see Fig. S-2a). This “opposite” tetrahedron will have two positively oriented
and two negatively oriented faces and thus it satisfies again
$\mathbf{B}_{3}^{\top}\mathbf{u}=0$, moreover its incidence matrix
$\mathbf{B}_{2}$ will be the opposite of the previous one (panel b) of Fig.
S-2). This means that we can select an edge in the first tetrahedron (say
$ad$) and a second edge in the opposite tetrahedron (say
$a^{\prime}d^{\prime}$) and “glue” them (see Fig. S-2c). In this way this edge
will be incident with four faces, and it will be coherently oriented with two
of them and non coherently oriented with the remaining two. Hence the row of
the incidence matrix of the “glued” tetrahedron,
$\mathbf{B}^{(\mathrm{glue})}_{2}$, will contain two “$+1$” and two “$-1$”,
the remaining entries being “$0$” (see Fig. S-2d). In conclusion this link
will contribute to $0$ in the product
$\mathbf{B}^{(\mathrm{glue})}_{2}\mathbf{u}$.

Figure S-2: An opposite oriented tetrahedron. We show a tetrahedron whose
faces have an opposite orientation with respect to the ones of Fig. S-1, while
the tetrahedron and its edges are oriented in the same way in both cases
(panels a) and b)). In panel c) we show the “gluing” process of these two
tetrahedra along two selected links, $ad$ and $a^{\prime}d^{\prime}$, in such
a way the shared link is now incident with four faces and its is coherently
oriented with two of them and non coherently with the two other ones (panels
c) and d)).

By construction the $3$-simplicial complex obtained by gluing the two opposite
tetrahedra contains a link, ($ad$) that is incident with four faces and it is
coherently oriented with two of them and incoherently with the other two (see
Fig. S-3a). Moreover such simplicial complex contains six links each one
belonging to two faces and coherently oriented with both of them, hence they
will contribute with a non zero entry in the product
$\mathbf{B}_{2}\mathbf{u}$. In panel a) of Fig. S-3 they are colored in blue
in the case of the first considered tetrahedron and in red for the second one;
let us for short say that such links have the “wrong property”. To tackle this
issue we repeat the construction show on Fig. S-2 by alternating “positively”
and “negatively” oriented tetrahedra along the two opposite sides of the
square obtained by “flattening” each tetrahedron and considering only the
sides with the “wrong property” (see Fig. S-3b). The resulting structure
resemble to a “waffle” (see Fig. S-3c) to which we imposed periodic boundary
conditions (see Fig. S-3c). Eventually taking $L$ squares along the
longitudinal direction and $M$ squares along the latitudinal direction, we
obtain the sought $3$-simplicial complex that by construction satisfies
$\mathbf{B}_{2}\mathbf{u}=0$ and $\mathbf{B}_{3}^{\top}\mathbf{u}=0$. It can
thus support a synchronous manifold for topological signals defined on
$2$-simplex but not on the $1$-simplex; indeed
$\mathbf{B}_{2}^{\top}\mathbf{u}\neq 0$, nor the $3$-simplex because
$\mathbf{B}_{3}\mathbf{u}\neq 0$. Let us observe that in this case
$\mathbf{B}_{4}^{\top}\mathbf{u}=0$, being the tetrahedron the simplex with
the largest dimension and thus $\mathbf{B}_{4}=0$, however we do not have
global synchronization of $3$-topological signals because the used simplex is
not a closed manifold without boundary.

Figure S-3: The $3$-simplicial complex $\mathcal{P}$. We schematically show
the construction of the $3$-simplicial complex obtained by gluing together
opposite tetrahedra and then by identifying the opposite sides of the square
composed by the tetrahedron links contributing with a non zero value to
$\mathbf{B}_{2}\mathbf{u}$ (the edges in blue and red in panels a-b) or in
yellow in panel c)).

##  Appendix D A $2$-simplicial complex satisfying conditions i) and ii) for
nodes and triangles but not for links

The aim of this section is to build a $2$-simplicial complex satisfying again
conditions i) and ii) for $k=2$ and $k=0$. In this case the simplex is formed
by triangles connected among them to pave a $2$-torus. We could have used the
orientation given by the node labels, but again we decided to introduce an
alternative orientation aimed at directly obtaining that
$\mathbf{u}=(1,\dots,1)^{\top}$ lies in $\ker\mathbf{L}_{2}$. Also in this
case this choice does not change the topological properties of the simplicial
complex with respect to those of the same simplicial complex oriented by using
the node labels orientation.

Consider the case of topological signals defined on $2$-simplices, i.e., the
faces of the $2$-simplicial complex. Conditions i) becomes thus
$\mathbf{B}_{2}\mathbf{u}=0$, while condition ii) is automatically satisfied
because $\mathbf{B}_{3}=0$, being the $3$-simplices not allowed. The basic
element is thus an oriented triangle, i.e., a $2$-simplex $A=[a\,c\,b]$ whose
links are oriented as $[a\,b]$, $[b\,c]$ and $[c\,a]$ (see Fig. S-4a).
Consider now a second triangle $B=[b\,c\,d]$ sharing the link $[b\,c]$ with
the previous one, and whose links are oriented as $[b\,c]$, $[c\,d]$ and
$[d\,b]$. Because of the chosen orientations the shared link determine a $+1$
and a $-1$ entries in the matrix $\mathbf{B}_{2}$, namely in the product
$\mathbf{B}_{2}\mathbf{u}$ they contribute with a $0$ (see Fig. S-4b). We can
then repeat such construction by alternating the two kinds of triangles above
defined and eventually identify among them the “vertical left” and “vertical
right” sides (red lines in Fig. S-4c) and also the “horizontal top” and
“horizontal bottom” sides (blue lines in Fig. S-4c). In this way we have
obtained a $2$-torus whose surface is paved with triangles and such that by
construction we have $\mathbf{B}_{2}\mathbf{u}=0$.

Figure S-4: A $2$-torus paved with triangles. We consider $2$-torus whose
surface is paved with oriented triangles (panel a)) organized in such a way
that once they share a common link then the latter is coherent with the
orientation of one triangle and incoherent with the second; they contribute
thus to the matrix $\mathbf{B}_{2}$ with entries $+1$ and $-1$ (panel b)).
Such construction is repeated in the longitudinal and vertical direction and
eventually including periodic boundary conditions (panel c)).

##  Appendix E A $3$-cell complex supporting global synchronization:
$3$D-torus

The aim of this appendix is to build a $3$-cell complex made of nodes, links,
squares (i.e., $2$-cells) and cubes (i.e., $3$-cells) capable to support
global synchronization for topological signals of any dimensions, $k=0,1,2,3$.

Let us consider a cube formed by six $2$-cells, the squares (see Fig. S-5a),
and we can orient the faces to have three positive and three negative
contributions to $\mathbf{B}_{3}^{\top}\mathbf{u}$ is in such a way the latter
sums to zero (see panels b)). It is important for the following construction
to incoherently orient opposite faces (see panels c)-d) and e)), this is
because once we will “glue” together several cubes, the shared faces should
have the right orientation to ensure $\mathbf{B}_{2}\mathbf{B}_{3}=0$ .

We then orient the links according to a left-right on the horizontal
direction, bottom-up in the vertical one and front-back in the transversal
direction (see again panels a) and b)). Once faces and links have been
oriented we can compute the matrices $\mathbf{B}_{k}$, $k=1,\dots,3$ and we
can observe that six links exist, three of which contributing with “$+2$”
(solid blue line) or “$-2$” (dashed red line) to $\mathbf{B}_{2}\mathbf{u}$.

Figure S-5: An oriented cube. We show an oriented cube, both in a $3$D view
(panel a)) then in a “flatten” $2$D one (panel b)). Given an orientation of
the faces and of the links (see panels a) and b)), we can compute the
incidence matrix $\mathbf{B}_{2}$ and identify the link whose contribution to
$\mathbf{B}_{2}\mathbf{u}$ is not zero. This allows to emphasise $6$ links
forming a closed loop (thicker dashed red and solid blue lines in panel a)) in
the cube. Let us observe that opposite faces of the cube must be incoherently
oriented (see panels c)-d) and e)).

Once we have obtained the oriented cube, we can “glue” together several copies
of the same cube along the three directions (see Fig. S-6). Because of the
orientations given to the faces (see Fig. S-5), the face shared by each couple
of cubes will have the right orientation. Moreover this gluing process ensures
also that $\mathbf{B}_{2}\mathbf{u}=0$; indeed as it can been observed from
Fig. S-6 the unique link incident with four cubes, hence eight faces, has a
zero contribution to $\mathbf{B}_{2}\mathbf{u}$. By considering for instance
the configuration shown in panel a), the vertical link common to the four
cubes A, B, C and D will contribute with $+2$ (it has been coloured in blue
according to the scheme presented in Fig. S-5) because of the two faces of the
B cube, with $-2$ (it has been coloured in red according to the scheme
presented in Fig. S-5) because of the two faces of the D cube and $0$ (it has
been coloured in black according to the scheme presented in Fig. S-5) to the
faces in the cubes A and C. The same analysis can be done for the
constructions in the vertical direction (panel b)) and in the transversal one
(panel c)).

Figure S-6: Gluing together four oriented cubes along the three dimensions. We
show how to glue together four cubes in the horizontal (panel a)), vertical
(panel b)) and vertical transversal (panel c)) plane. We can observe that in
all cases the link common to eight faces (and four cubes) returns a null
contribution to $\mathbf{B}_{1}\mathbf{u}$, indeed it contributes with $0$ to
two couples of faces (thin black line), with $+2$ to a couple of faces (blue
thick line) and with $-2$ to the remaining couple of faces (dashed red line).
The final construction is schematically depicted in panel d). The cube is
repeated along each one of the three spatial dimensions and periodic boundary
conditions are implemented connecting opposite faces of the cubic array
(represented with a lower opacity). The basic element of the construction is
shown in the centre. As in the other figures, the non zero contributing edges
are shown in orange and in dark grey all the others. The edges coloured with
the same color index as in panels a)-c) are also shown for orientation
reference.

Finally by imposing periodic boundary conditions in the three directions we
obtain a $3$D torus with the required properties. Let us observe that the
links orientation together with the periodic boundary conditions ensure
$\mathbf{B}_{1}\mathbf{u}=0$.

##  Appendix F Analysis of the Stuart-Landau model

The aim of this section is to provide the interested reader more details about
the global synchronization of identical Stuart-Landau systems defined on top
of simplicial and cell complexes introduced in the main text.

Let us thus assume to deal with a $k$ dimensional topological signal
associated to a nonlinear SL oscillator, whose time evolution is given by

$\frac{dw}{dt}=\sigma w-\beta w|w|^{2}\,,$

where $\sigma=\sigma_{\Re}+i\sigma_{\Im}$ and $\beta=\beta_{\Re}+i\beta_{\Im}$
are complex model parameters. One can easily prove that the former system
admits a limit cycle solution
$\hat{z}(t)=\sqrt{\sigma_{\Re}/\beta_{\Re}}e^{i\omega t}$, where
$\omega=\sigma_{\Im}-\beta_{\Im}\sigma_{\Re}/\beta_{\Re}$, is the signal
frequency. Such solution is stable provided $\sigma_{\Re}>0$ and
$\beta_{\Re}>0$, conditions that we hereby assume.

We now consider $N_{k}$ SL topological oscillators and we couple them using
the the $(k-1)$ and $(k+1)$-faces of a simplicial complex. Let also assume the
coupling to be given by the nonlinear function $h(w)=\mu w|w|^{m-1}$, where
$m$ is a positive integer and $\mu=\mu_{\Re}+i\mu_{\Im}$ is a complex
parameter defining the interaction strength. Observe that such coupling has
been recently introduced and studied in the framework on synchronization on
time varying networks in [49]. In conclusion we are interested in studying the
system

$\frac{dw_{i}}{dt}=\sigma w_{i}-\beta
w_{i}|w_{i}|^{2}+\mu\sum_{j=1}^{N_{k}}L_{k}(i,j)w_{j}|w_{j}|^{m-1}\quad\forall
i=1,\dots,N_{k}\,.$ (S-7)

We are now interested in studying the stability of the reference limit cycle
solution $\hat{z}(t)$. If this condition is realised, then the system shows
global synchronization. To achieve this goal we introduce real “small”
functions $\rho_{j}(t)$ and $\theta_{j}(t)$ and rewrite $w_{j}(t)$ as follows

$w_{j}(t)=\hat{z}(t)(1+\rho_{j}(t))e^{i\theta_{j}(t)}\,,$ (S-8)

where $\rho_{j}(t)$ and $\theta_{j}(t)$ are real valued functions. We now
insert the previous expression in the coupled Eq. (S-7). By using the
expression for $\hat{z}(t)$ and by expanding the resulting equation up to the
first order in $\rho_{j}$ and $\theta_{j}$, we eventually obtain

$\begin{dcases}\frac{d\rho_{j}}{dt}=-2\sigma_{\Re}\rho_{j}-\left(\frac{\sigma_{\Re}}{\beta_{\Re}}\right)^{\frac{m-1}{2}}\sum_{\ell=1}^{N_{k}}L_{k}(j,\ell)\left(m\mu_{\Re}\rho_{\ell}-\mu_{\Im}\theta_{\ell}\right)\\\
\frac{d\theta_{j}}{dt}=-2\beta_{\Im}\frac{\sigma_{\Re}}{\beta_{\Re}}\rho_{j}-\left(\frac{\sigma_{\Re}}{\beta_{\Re}}\right)^{\frac{m-1}{2}}\sum_{\ell=1}^{N_{k}}L_{k}(j,\ell)\left(m\mu_{\Im}\rho_{\ell}+\mu_{\Re}\theta_{\ell}\right)\,.\end{dcases}$
(S-9)

We can then decompose $\rho_{j}(t)$ and $\theta_{j}(t)$ on the orthonormal
eigenbasis $\pmb{\phi}_{k}^{(\alpha)}$, $\alpha=1,\dots,N_{k}$, of the Laplace
matrix $\mathbf{L}_{k}$:

$\rho_{j}=\sum_{\alpha}\hat{\rho}_{\alpha}\phi^{(\alpha)}_{k}(j)\text{ and
}\theta_{j}=\sum_{\alpha}\hat{\theta}_{\alpha}\phi^{(\alpha)}_{k}(j)\,,$
(S-10)

to eventually obtain

$\begin{dcases}\frac{d{\hat{\rho}}_{\alpha}}{dt}=-2\sigma_{\Re}\hat{\rho}_{\alpha}-\left(\frac{\sigma_{\Re}}{\beta_{\Re}}\right)^{\frac{m-1}{2}}\Lambda^{(\alpha)}_{k}\left(m\mu_{\Re}\hat{\rho}_{\alpha}-\mu_{\Im}\hat{\theta}_{\alpha}\right)\\\
\frac{d{\hat{\theta}}_{\alpha}}{dt}=-2\beta_{\Im}\frac{\sigma_{\Re}}{\beta_{\Re}}\hat{\rho}_{\alpha}-\left(\frac{\sigma_{\Re}}{\beta_{\Re}}\right)^{\frac{m-1}{2}}\Lambda^{(\alpha)}_{k}\left(m\mu_{\Im}\hat{\rho}_{\alpha}+\mu_{\Re}\hat{\theta}_{\alpha}\right)\,.\end{dcases}$
(S-11)

Let us observe that Eq. (S-11) is autonomous, hence one can compute its
eigenvalues and define the largest real part of the latter ones, say
$\lambda$, named in the literature dispersion relation. One can thus conclude
that if $\lambda<0$ the reference solution is stable and hence the system
globally synchronizes. The same reasoning can be done whenever the generic MSF
(4) is autonomous.

Let us also observe that a similar conclusion can be obtained if
$\mathbf{s}(t)$ is a periodic solution by resorting to Floquet analysis;
calling again $\lambda$ the largest real part of the Floquet eigenvalues we
can show that if $\lambda<0$ then the reference solution is stable and the
system globally synchronizes. In the general case, one has to (numerically)
compute the Lyapunov exponent of (4) and infer about the stability of the
reference solution using the Lyapunov theory. Let us observe that to stress
the dependence on the simplex eigenvalues we will also write
$\lambda_{\alpha}=\lambda(\Lambda_{k}^{(\alpha)})$.

Back to Eq. (S-11) one can infer the stability of the reference solution and
thus of the global simplicial synchronization by studying if the perturbations
$\rho_{j}$ and $\theta_{j}$ fade away, or equivalently if their projections
$\hat{\rho}_{\alpha}$ and $\hat{\theta}_{\alpha}$ vanish. Sufficient
conditions are obtained by assuming an exponential behavior, namely
$\hat{\rho}_{\alpha}\sim\tilde{\rho}_{\alpha}e^{\lambda_{\alpha}t}$ and
$\hat{\theta}_{\alpha}\sim\tilde{\theta}_{\alpha}e^{\lambda_{\alpha}t}$ with
time-independent $\tilde{\rho}_{\alpha}$ and $\tilde{\theta}_{\alpha}$.
Inserting this ansatz into (S-11) and imposing that
$(\tilde{\rho}_{\alpha},\tilde{\theta}_{\alpha})\neq(0,0)$, one gets the
following equation for $\lambda_{\alpha}$

$\lambda_{\alpha}^{2}+\lambda_{\alpha}\left(\left(\frac{\sigma_{\Re}}{\beta_{\Re}}\right)^{\frac{m-1}{2}}\mu_{\Re}\Lambda^{(\alpha)}_{k}(m+1)+2\sigma_{\Re}\right)+m\left(\frac{\sigma_{\Re}}{\beta_{\Re}}\right)^{{m-1}}\left(\Lambda^{(\alpha)}_{k}\right)^{2}(\mu_{\Re}^{2}+\mu_{\Im}^{2})+2\Lambda^{(\alpha)}_{k}\left(\frac{\sigma_{\Re}}{\beta_{\Re}}\right)^{\frac{m-1}{2}}\left(\mu_{\Re}\sigma_{\Re}+\mu_{\Im}\beta_{\Im}\frac{\sigma_{\Re}}{\beta_{\Re}}\right)=0\,.$
(S-12)

We eventually define the dispersion relation (or maximum Floquet exponent)
$\lambda=\max_{\alpha}\Re\lambda_{\alpha}$. Let us observe that
$\lambda_{1}=0$, that is $\lambda$ vanishes if evaluated on
$\Lambda^{(1)}_{k}=0$; this is because the reference solution is a limit
cycle. By considering then the behavior of $\lambda_{\alpha}$ for
$\Lambda^{(\alpha)}_{k}$ close to zero, one can develop the root
$\lambda_{\alpha}$ as follows

$\displaystyle\lambda_{\alpha}$ $\displaystyle\sim$
$\displaystyle\frac{1}{2}\left[-\left(\frac{\sigma_{\Re}}{\beta_{\Re}}\right)\mu_{\Re}\Lambda^{(\alpha)}_{k}(m+1)-2\sigma_{\Re}+2\sigma_{\Re}\left(1+\frac{1}{2}\left(\frac{\sigma_{\Re}}{\beta_{\Re}}\right)^{(m-1)/2}\Lambda^{(\alpha)}_{k}\left(\frac{\mu_{\Re}}{\sigma_{\Re}}(m-1)-2\frac{\beta_{\Im}\mu_{\Im}}{\beta_{\Re}\sigma_{\Re}}\right)\right)+\dots\right]$
$\displaystyle=$
$\displaystyle-\Lambda^{(\alpha)}_{k}\left(\frac{\sigma_{\Re}}{\beta_{\Re}}\right)^{(m-1)/2}\left(\mu_{\Re}+\frac{\beta_{\Im}\mu_{\Im}}{\beta_{\Re}}\right)+\dots\,.$

Hence there exists an interval of values for $\Lambda^{(\alpha)}_{k}$ such
that $\lambda_{\alpha}>0$, namely the global synchronization cannot be
achieved, if and only if

$\mu_{\Re}+\mu_{\Im}\frac{\beta_{\Im}}{\beta_{\Re}}<0\,.$

The above presented theory is applied to topological SL signals defined on top
of the $3$-simplicial complex and the $3$-cell complex previously introduced.
The model parameters have been set to some generic values allowing for a
negative dispersion relation (see Fig. S-7(b-c) for the simplicial complex and
Fig. S-7(e-f) for the cell complex). However once the complex amplitudes are
defined on $2$-faces, i.e., triangles or squares, the system globally
synchronizes as we can appreciate from the inset in panel b) and e), while SL
oscillators defined on links have a different behavior if we are dealing with
a simplicial complex where they cannot globally synchronize (see panel c)) or
a cell complex where global synchronization is achieved (see panel f)). Those
different behaviors result from the fact that
$\mathbf{u}\in\ker\mathbf{L}_{1}$ for the cell complex while
$\mathbf{u}\not\in\ker\mathbf{L}_{1}$ for the simplicial complex, in the
latter case synchronization cannot be achieved because of the presence of the
topological obstruction.

Figure S-7: Dispersion relation for topological Stuart-Landau model. The left
panels refer to the $3$-simplicial complex schematically represented in panel
a). In panel b) we report the dispersion relation $\lambda$ as a function of
the eigenvalues $\Lambda^{(\alpha)}_{2}$ in the case of topological complex
amplitudes defined on $2$-faces, i.e., triangles, while as a function of
$\Lambda^{(\alpha)}_{1}$ in panel c) dealing with signals defined on links.
Similar functions are reported in the right panels in the case of $3$-cell
complex schematically represented in panel d). The model parameters have been
fixed to some generic values, $\sigma=1.0+4.3i$, $\beta=1.0+1.1i$,
$\mu=1.0-0.5i$ and $m=3$, and are the same used to obtain the results reported
in Fig. 2.

Let us conclude this section by showing the dispersion relation for the
topological SL signals defined on the $2$-torus paved with triangles. By
assuming the same model parameters as in Fig. S-7 the numerical results
presented in Fig. S-8 confirm our theory, global synchronization can be
achieved only for topological signals defined on nodes, i.e., $k=0$ simplices
(see left panel), or triangles, i.e., $k=2$ simplices (see right panel).
Indeed we have $\mathbf{u}\in\ker\mathbf{L}_{0}$ and
$\mathbf{u}\in\ker\mathbf{L}_{2}$; on the other hand
$\mathbf{u}\not\in\ker\mathbf{L}_{1}$ and thus topological signals defined on
links cannot globally synchronize (see middle panel).

Figure S-8: Dispersion relation for topological Stuart-Landau model (II). We
consider the $2$-torus paved with triangles. The left panel refers to
topological signals defined on nodes, i.e., $0$-simplices, the middle panel to
the case of links, i.e., $1$-simplices and the right panel to the case of
faces, i.e., $2$-simplices panels. Top panels show the dispersion relation and
we can observe that in all cases the latter is negative except for the zero
value associated to $\Lambda^{(1)}_{k}=0$, $k=0,1,2$. Bottom panels present
the (generalized) order parameters while the inset report the real part of the
signals. The model parameters have been fixed to some generic values,
$\sigma=1.0+4.3i$, $\beta=1.0+1.1i$, $\mu=1.0-0.5i$ and $m=3$, and are the
same used to obtain the results reported in Fig. 2.