# MuLaN: a MultiLayer Networks Alignment Algorithm

Marianna Milano  Pietro Cinaglia  Pietro Hiram Guzzi  Mario Cannataro
Department of Experimental and Clinical Medicine, Magna Graecia University of
Catanzaro, 88100 Catanzaro, Italy.  Department of Health Sciences, Magna
Graecia University of Catanzaro, 88100 Catanzaro, Italy.  Department of
Medical and Surgical Sciences, Magna Graecia University of Catanzaro, 88100
Catanzaro, Italy.  Data Analytics Research Center, Magna Graecia University of
Catanzaro, 88100 Catanzaro, Italy.

###### Abstract

A Multilayer Network (MN) is a system consisting of several topological levels
(i.e., layers) representing the interactions between the system’s objects and
the related interdependency. Therefore, it may be represented as a set of
layers that can be assimilated to a set of networks of its own objects, by
means inter-layer edges (or inter-edges) linking the nodes of different
layers; for instance, a biological MN may allow modeling of inter and intra
interactions among diseases, genes, and drugs, only using its own structure.
The analysis of MNs may reveal hidden knowledge, as demonstrated by several
algorithms for the analysis. Recently, there is a growing interest in
comparing two MNs by revealing local regions of similarity, as a counterpart
of Network Alignment algorithms (NA) for simple networks. However, classical
algorithms for NA such as Local NA (LNA) cannot be applied on multilayer
networks, since they are not able to deal with inter-layer edges. Therefore,
there is the need for the introduction of novel algorithms. In this paper, we
present MuLaN, an algorithm for the local alignment of multilayer networks. We
first show as proof of concept the performances of MuLaN on a set of synthetic
multilayer networks. Then, we used as a case study a real multilayer network
in the biomedical domain. Our results show that MuLaN is able to build high-
quality alignments and can extract knowledge about the aligned multilayer
networks.
MuLaN is available at https://github.com/pietrocinaglia/mulan.

###### keywords:

Multilayer Network , Network Alignment , Local Network Alignment

††journal: Computers in Biology and Medicine

##  1 Introduction

Networks are largely used to represent entities and their association in many
fields [1, 2, 3]. For instance, in computational biology and bioinformatics,
networks are used to model the set of association among genes, proteins and
other macromolecules [4, 5, 6]. More recently, it has been shown that a single
level of representation, i.e., considering only genes, or only proteins, may
result in a loss of information, so the need for introduction of other models
arise. Such models are based on a multilevel (or multilayered) view of a
system. As an example, in the biological scenario, each level may correspond
to a different view, e.g. genes, proteins, and diseases. In such a view, each
layer is composed by a set of homogeneous nodes linked by edges (i.e., intra-
layer edges). Associations between two different layers are modelled by cross
layer edges (i.e., inter-layer edges) [7, 8]. Figure 1 represents a non-
exhaustive example of a simple multilayer network having three layers. Each
layer is a different network. Figure 2 depicts an example of a biological
multilayer network representing the interplay between diseases and drugs.

Figure 1: A non-exhaustive example of multilayer network. The figure
represents a network in a multilayer prospective. The same entities are
modeled as nodes of a same layer, and their interactions consists of intra-
edges. Otherwise, the nodes of different layers are connected by inter-edges.
Figure 2: The figure shows a toy example of a biological multilayer network,
representing disease-drug associations. The nodes are the diseases and the
drugs, both discriminated by belonging to the respective layer. The intra-
edges represent the drug-drug and the disease-disease associations, while the
inter-edges the disease-drug associations. For instance, these may be analyzed
to investigate the mechanisms underlying the interaction between diseases and
drugs, or for computational drug repositoring.

Formally, a multilayer graph may be described as a tuple
$G_{M}=(V_{M},E_{M},V,L)$, where $V_{M}$ and $E_{M}$ are a set of nodes and
edges, respectively, and $L$ is a set of layers. Thus, given
$L=\\{L_{1},L_{2},...,L_{k}\\}$, with $k$ the number of layers in the
multilayer graph, the nodes as pairs $V_{M}\subseteq VxL_{1}x...L_{k}$, the
edges $E_{M}\subseteq V_{M}xV_{M}$ connect the pairs $(v,l)$,
$(v^{\prime},l^{\prime})$. In a multilayer graph, an edge is defined intra-
layer in case of $l$ is equal to $l^{\prime}$ or inter-layer, in case of $l$
and $l^{\prime}$ are different [9].

In general, the analysis of networks is rarely simple. Common challenges that
can be generalized for all types of networks concern i) the modelling of the
network, i.e., what it represents and ii) the analysis of the network, i.e.,
the metrics grabbing the biological phenomenon of interest. These aspects are
at least as important for multilayer networks, given their added complexity.
Furthermore, there are additional observations that are different to
multilayer networks with respect to classical networks. For instance, an
analysis should take into account the difference among layers. In particular,
since the strength of multilayer network analysis consists of the capability
to include information on different type of relationship, it would be
necessary to consider which layers should be included in the analysis, and the
interpretation of intra-layer and inter-layer edges values, since they
discriminate against the different relationships.

The analysis of networks enables to find interesting regions of similarity
between them [10, 11]. For simple networks, this task is accomplished by means
of Network Alignment (NA) algorithms. Unfortunately, as also evidenced in [12,
13, 14], existing alignment algorithms are unable to process multilayer
networks. Therefore, the need for defining a novel multilayer network
algorithm arises. We focused in particular on evidencing relatively small
regions of similarity among the multilayer networks, so we extended the Local
Network Algorithm approaches. In a previous work [14], we presented MultiLoAl
for the local alignment of multilayer networks. Despite the effectiveness of
the approach, it presented some limitations related to the scalability on
large datasets and on the topological structure of the region of the
similarity.

Therefore, we extended our methodology to admit further topological structures
of the found regions as well as better to obtain performances in terms of
quality and running time. Such a methodology has been implemented in a novel
tool, namely MuLaN (from MultiLAyer Network Alignment Algorithm). It consists
of a novel approach for the building of alignment graph 111the alignment graph
is the key data structure for local alignment algorithm who achieves better
performances, and it is integrated with a set of community discover algorithms
enabling to discover regions of similarity with different topological
structures.

Summarizing, MuLaN allows you to mainly benefit from the following advantages,
compared to MultiLoAl: (i) a reduction of running time for building the
alignment graph, and (ii) a deeper experimentation on a larger dataset.

MuLaN is based on the workflow depicted in Figure 3.

Figure 3: Workflow of the proposed algorithm.

It receives as input two multilayer networks ($M_{1}$ and $M_{2}$) having the
same number of layers $n$. Without loss of generality, we suppose the
existence of a bijective correspondence among layers of the networks, so the
layer $k$ in 1-$n$ of $M_{1}$ corresponds to the layer $k$ of $M_{2}$. MuLaN
also receives a multiset of seed nodes representing the similarity between the
node of the graph of the same layer $k$ of the two networks. Starting from
$M_{1}$ and $M_{2}$ and multiset of seed nodes, at first, MuLaN analyzes each
pair of $k$ layers in $M_{1}$ and $M_{2}$ independently. Then, it builds $n$
alignment graphs. After building each alignment graph for each $k$ layer
separately, MuLaN analyzes $M_{1}$ and $M_{2}$ to add inter-layer edges among
$n$ alignment graphs. Thus, the algorithms build $n-1$ alignment graphs, that
we call multilayer alignment graphs. Finally, MuLaN applies a community
detection algorithm suitable for multilayer networks on the final multilayer
alignment graph to detect communities representing local regions of
similarity, i.e. a single region of the local alignment. The current version
of MuLaN offer three different community discover algorithms: Louvain [15],
Infomap [16] and Greedy [17].

We implemented three versions of MuLaN by varying the community detection
method. The default version applies Louvain [15] to mine community from the
multilayer alignment graph, whereas, the other two versions use Infomap [16]
and Greedy [17] as community detection algorithms, respectively [18].

The rest of this paper is organized as follows. Section 2 discusses the
background on multilayer networks and multilayer community detection, Section
3 presents the MuLaN Algorithm, and Section 4 presents and discusses the
results. Finally, Section 5 concludes the paper.

##  2 Related Work

We report here some existing state-of-the-art approaches for the analysis of
MNs. We separate the algorithms that analyze a single network from those
comparing two networks [19].

###  2.1 Network Alignment Algorithms

The problem of graph alignment consists of finding a mapping between nodes of
two or more graphs to maximize an associated cost function. Formally, let
$G_{1}=\\{V_{1},E_{1}\\}$ and $G_{2}=\\{V_{2},E_{2}\\}$ two graphs, where
$V_{1,2}$ are sets of nodes and $E_{1,2}$ are sets of edges, the graph
alignment problem consists of finding a function (or a mapping)
$f:V_{1}\rightarrow V_{2}$ such that the similarity between mapped entities is
maximized. Among the others, algorithms may be categorised in global network
alignment (GNA) algorithms (i.e. algorithms that aim to find a single mapping
among all the nodes of the networks), or local network alignment (LNA)
algorithms (i.e. algorithms aiming at finding multiple small regions of
similarity among networks).

We here report some LNA algorithms, since MuLaN search for multiple regions of
similarity in multilayer networks.

An example of LNA algorithm is AlignNemo [20] that enables the discovery of
subnetworks of proteins related to biological function and topology. AlignMCL
[21] extends AlignNemo [20], by providing a formal definition of the alignment
graph, which is clustered by using the Markov cluster algorithm MCL [22].

Also, GLAlign (Global Local Aligner) [23] combines the topology information
from global alignment with biological information (e.g. homology
relationships) to build local network alignment. NetworkBLAST [24] aims to
find small dense regions in protein-protein interaction networks. Such
subgraphs represent protein complexes, i.e. groups of proteins performing a
similar function or involved in the same biological process. NetAligner [25]
presents a strategy to identify evolutionarily conserved interactions on the
basis of the consideration that interacting proteins evolve at rates
significantly closer than expected by chance. Furthermore, L-HetNetAligner
[12] extends the local alignment to the heterogeneous networks. Finally, some
recent papers focused on the alignment of dynamic or dual networks [26, 27,
5]. However, these network alignment algorithms do not perform very well for
multilayer networks [7, 28].

###  2.2 Community Detection in Multilayer Networks

Community detection algorithms aim to identify groups of nodes closely
connected with respect to the average of the network, starting from the
assumption that nodes in the same community have a similar role [29, 30, 31].
In detail, a community is defined as groups of nodes that are more densely
connected than the rest of the network; they represent significant
characteristics for understanding the functionalities and organizations of
complex systems modeled as a network.

In multilayer networks, the communities represent groups of well-connected
nodes considering whole layers, community detection methods should consider
the diversity between the layers[18].

To handle these issues, many community detection algorithms for multilayer
networks have been recently developed.

The Louvain algorithm, initially developed for simple networks, has been
extended to handle multilayer networks [15]. The algorithms are based on an ad
hoc defined notion of modularity for multilayer network, which is based on the
definition of community as a set of nodes and edges belonging to multiple
layers. Therefore, the algorithm applies an iterative greedy approach to build
communities.

Infomap [16, 32] discovers communities in multilayer networks by running
random walks, admitting also interlayer edges. Infomap can be used to find
both overlapping and non-overlapping communities.

Greedy [17] algorithm is a simple and fast community detection method that
works by greedily optimizing a quality function known as modularity. The
algorithm starts with each node in its own community and iteratively merges
communities that increase the modularity at the most.

ABACUS [33] algorithm enables to extract multidimensional communities by
mining frequent closed item sets on one-dimensional community memberships.
Initially, it extracts one-dimensional communities, looking at each dimension.
Then, ABACUS assigns a label to nodes that consists in a list of pair tags
(e.g., dimension and community). After that, ABACUS uses a frequent closed
item set mining algorithm by treating each pair of tags as an item. At the
end, the multidimensional communities described by the item sets consist of
frequent closed item sets.

Multi-Layer Clique Percolation [34] method applies the classical clique
percolation method on traditional networks. In the classical version, dense
regions are cliques, whereas the two cliques are adjacent if they have common
nodes. Multi-Layer Clique Percolation extends research of cliques by
considering multiple layers, and it redefines the adjacency metric by
considering both common nodes and common layers are expected. Thus, the
communities correspond to the combinations of adjacent cliques.

Multi-Dimensional Label Propagation [35] is an extension of the Label
Propagation algorithm, which is a semi-supervised learning algorithm used for
graph-based classification problems. The MDLP algorithm is designed to work
with multilayer networks.

In MDLP, each layer of the multilayer network is treated as a separate graph,
and the algorithm propagates labels across all layers simultaneously. It
allows incorporating information from multiple sources and to capture complex
relationships between nodes across layers. Furthermore, it can be
computationally expensive, especially for large multilayer networks, and may
require careful parameter tuning to achieve optimal results.

##  3 MuLaN Algorithm

The MuLaN algorithm builds the alignment through two steps:

  * 1.

Building of the Multilayer Alignment Graph: First, it integrates the input
networks into a multilayer alignment graph (MAG) using a set of initial pairs
of correspondences;

  * 2.

Analysis of the Multilayer Alignment Graph: Second, it reveals the regions of
similarity by analysing the topology of the multilayer alignment graph.

The core of the algorithm is the multilayer alignment graph, that is used to
integrate initial input networks and which is able to contains the regions of
similarity. The multilayer alignment graph is a multilayer network which has
the same layer of the input networks. Each layer ${K}$ of the MAG is an
alignment graph between the corresponding pairs of layers of the input
network. The edges between two layers of MAG are derived by analysing inter-
layer edges of the input networks, as shown in the following.

Figure 3 shows the workflow of the algorithm, while Algorithm 1 shows the
pseudocode of MuLaN.

Input: $G_{1}=(V1,E_{1},C_{1})$, and $G_{2}=(V2,E_{2},C2)$,$\Delta$, A set of
high-similar seed nodes

Result: A set of Aligned Regions

1 Initialization;

2 Step 1 – Building of the Multilayer Alignment Graph :
$G{al1}_{=}(V_{al},E_{al})$

3 $G{al2}_{=}(V_{al},E_{al})$

4 forall  _Pair of the input list of paired nodes_ do

5 add a node in Alignment Graph

6 end forall

7forall  _Nodes in $V_{al}$_ do

8 Add Edges Verifying Match-Mismatch- Conditions

9 end forall

10Building of the inter-layer Alignment Graph :
$G{inter_{l}}=(G{1_{a}l},G{2_{a}l},E_{inter_{l}})$

11 Step 2 – Community Detection Algorithm on:
$G{inter_{l}}=(G{1_{a}l},G{2_{a}l},E_{inter_{l}})$.

12 return _A set of subgraphs of
$G{inter_{l}}=(G{1_{a}l},G{2_{a}l},E_{inter_{l}})$ _

Algorithm 1 MuLaN Algorithm

We explain our contribution using a toy example of a multilayer network with
two layers without loss of generality; then we will discuss its formalization.
Let us consider two multilayer input networks $G_{1}$, and $G_{2}$ presenting
two layers: disease and drug, as reported in Figure 4. Node colors are used to
distinguish different types of nodes belonging to two different types of
layers. For simplicity, the two multilayer input networks have the same number
of nodes.

Figure 4: Example of two Multilayer networks. The nodes represent a set of
diseases and a set of drugs that belong to the Drug layer and the Disease
layer. Please note that each layer networks has different edges; moreover,
also, the inter-layer edges are different.

###  3.1  Step 1: Building of the multilayer alignment graph

###  3.2 Step (1.a). Building of intra-layer edges.

In the first step, MuLaN builds the MAG in two sub steps, initially it build
all the alignment graphs for each layer $k$ (step 1.a, see Appendix for
complete details), then it add all the inter-layer edges (Step 1.b).

###  3.3 Step (1.b). Building of inter-layer edges.

In the step 1.b, the algorithm adds the inter-layer edges. This step considers
all the pairs of layers in the MAG. For each pair of layers ${k,t}$, it
analyzes all the nodes. Each node of the graph into a single layer represents
a pair of nodes in the corresponding of nodes. For each pair of nodes of the
layers, MuLaN analyzes the corresponding layer of the input networks, and it
inserts and weights the edges considering two conditions: match, or mismatch.

Let us consider the nodes of the alignment graphs 1 and the alignment graphs
2; in particular, let us analyze the pair of nodes $(D1-Dr4)$ and $(D5-Dr2)$
in Figure 4. To determine the presence of an edge, we consider the edges
$(D1,Dr4)\in G_{1}$ network and $(D1,Dr4)\in G_{2}$ network. If $G_{1}$, and
$G_{2}$ contains these nodes, and the nodes are adjacent, there is a match,
that we call heterogeneous match because the node type are different, see
Figure 5 (a).

Otherwise, if $G_{1}$, and $G_{2}$ contain these nodes, and the nodes are
adjacent only in a single network, there is a mismatch which we call
heterogeneous mismatch Figure 5 (b).

Finally, the algorithm assigns the weight to each edge as follows:
heterogeneous match equal to 0.9, heterogeneous mismatch equal to 0.4. At the
end of this step, the multilayer alignment graph is built.

Figure 5: Example of a heterogeneous match, mismatch and building of inter-
layer Alignment Graph.

###  3.4 Step 2. Community Detection on the multilayer alignment graph

At this point, the algorithms analyzes the MAG to discover communities
employing one of the previously introduced community detection methods [36,
37, 27, 38]. Since our methodology presents a general design, it is possible
to mine the MAG by applying different community detection methods.

In the current version of MuLaN, we applied Louvain algorithm to mine the
communities on the alignment graph. However, the user can choose the community
detection algorithm by selecting among Louvain, Infomap and Greedy algorithms.

MuLaN generates three outputs: 1) a file containing the mined communities; 2)
a file of the multilayer alignment graph represented as edge list, where the
weight of the edge according to homogeneous/heterogeneous match,
homogeneous/mismatch, homogeneous gap cases and type of layer; 3) a file
containing the number of mined communities, the modularity value (where
modularity is the measure related to the network structure that detects the
density of connections within a module) and the run time (see an example of
output at https://github.com/pietrocinaglia/mulan).

Figure 6: Example of Community Detection Extraction on inter-layer Alignment
Graph.

###  3.5 Complexity of the algorithm

The asymptotic temporal complexity of the alignment process was estimated as
$NxM$, with $N$ the number of interactions defined into the source network and
$M$ that of the target one, calculated on the basis of the respective edge
list.

The latter was designed as a multi-line adjacency list with data values, in
order to store both node and edge attributes: node type, and the layer upon
which (or between) the interaction is modeled. This approach allows us to have
a flat view of the graph, which is therefore aligned using the same basic
principles of alignment between static networks; its complexity will be
proportionate to the resulting size.

It turns out to be quadratic, for multilayer networks consisting of edge lists
with a same size ($N=M$).

##  4 Results and Discussion

As a proof of principle, we experimented MuLaN on two datasets, consisting of
(i) ten synthetic multilayer networks, and (ii) one real multilayer network.

###  4.1 Dataset 1: Synthetic Networks

We build ten multilayer networks with two layers. Each layer is a Barabási-
Albert network [39] having $1000$ nodes ($n=1000$) and $1000$ edges ($m=1$,
with $m$ the number of edges to attach from a new node to existing ones).

The number of inter-edges was defined as $30\%$ (i.e., $300$) of the intra-
edges, and these were randomly generated. The resulting edge list consisted of
$2300$ interactions among $2000$ nodes, for the whole network.

According to the described approach, we modeled a set of $10$ initial multi-
layer networks with $2$ layers, and five noisy versions for each one. The
noisy version was built by removing $5\%$, $10\%$, $15\%$, $20\%$, and $25\%$
of interactions, randomly selected from the set of inter\- and intra\- edges.
We have generated the pairs to be aligned by using the initial networks and
the respective noisy versions.

###  4.2 Dataset 2: Real Network

We considered the following datasets of the Stanford Biomedical Network
Dataset Collection (BioSNAP) [40]:

  * 1.

Drug-Drug Interaction (DDI) network of interactions between drugs, approved by
the U.S. Food and Drug Administration (FDA): $1514$ nodes and $48514$ edges.

  * 2.

Disease-Disease (DD) network of interaction between inherited diseases: $6878$
nodes and $6877$ edges.

  * 3.

Disease-Drug Association (DDA) network, a set of curated relationships between
diseases and drugs: $5535$ disease nodes, $1662$ chemical nodes, and $466656$
edges.

We build a multilayer network with two layers obtained from the DDI and DD
database. Then, we added inter-layer edges by considering the DDA database.
Subsequently, a preprocessing was also performed to remove (i) zero degree
nodes, (ii) duplicate edges, and (iii) objects outside the intersection set
from the DDI, DD, and DDA networks. Finally, the resulting multilayer network
consisted of $83,92$ nodes and $128,200$ interaction, of which $72,809$ inter-
edges.

Similarly to our own synthetic networks, we generated five noised versions of
the real network by removing $5\%$, $10\%$, $15\%$, $20\%$, and $25\%$ of
interactions, randomly selected from the set of inter\- and intra\- edges.

###  4.3 Alignment Parameters

We set $\Delta$distance equal to 2, and following weights:

  * 1.

Homogeneous Match : 1

  * 2.

Homogeneous Mismatch: 0.5

  * 3.

Homogeneous Gap: 0.2

  * 4.

Heterogeneous Match: 0.9

  * 5.

Heterogeneous Mismatch: 0.4.

###  4.4 Performance Evaluation

We evaluated MuLaN alignments by considering:

  * 1.

the alignment of a network with respect to itself to show the ability to find
known regions of similarity;

  * 2.

the alignment of the network with respect to an altered version of the network
obtained by adding different levels of noise ($5\%$, $10\%$, $15\%$, $20\%$,
and $25\%$) by randomly removing edges from the network.

We aimed to demonstrate the ability of our algorithm to build high-quality
alignments with edge conservation of about $90\%$.

The proposed solution aligned high-confidence synthetic networks to themselves
and to their noisy counterparts. Overall, we computed a set of $60$ local
alignments.

In addition to the experimentation on synthetic data, we aligned the high-
confidence real network with itself and its noisy counterparts, by producing a
total of $6$ local alignments.

All experiments have been conducted on a 64-bit workstation with the following
specifications: AMD Ryzen 3 (2.6 GHz Dual-Core) and 8 GB of RAM.

MuLaN aligned two multilayer networks consisting of $2000$ nodes and $2298$
edges, each one, in $\sim 1.25$ seconds. To the latter, it was then necessary
to include the time required by the individual methods for community
detection; in details, Louvain, Greedy, and Infomap take $\sim 0.17$, $\sim
1.89$, and $\sim 14.09$ seconds, respectively.

According to our results, the most performing method in terms of modularity is
Louvain (see Table 1 and Table 2); therefore, we defined the latter by
default. However, Greedy and Infomap can be chosen as other options.

In order to evaluate the effectiveness of MuLaN, we generated the same local
alignment built with Louvain by using Infomap and Greedy, and we analyzed the
results. We selected those algorithms for the best and fast performance to
extract community in multilayer network according to literature [18].

Since, to the best of our knowledge, MultiLoAl and its extension MuLaN are the
only local alignment algorithms of multilayer networks in the literature, we
evaluated the effectiveness of MuLaN, by comparing the results obtained with
default MuLaN version, in which Louvain is used as community detection
algorithm and other MuLaN versions in which are applied Infomap and Greedy
algorithms.

Then, we measured the performance of the alignments built with different
versions of MuLaN by evaluating the quality of the results. In particular, we
measure the topological quality of alignments and the quality of communities
found.

####  4.4.1 Topological quality

At first, the results are evaluated by analyzing the topological quality. We
recall that intuitively an alignment is of high topological quality if it
reconstructs the underlying true node mapping well (when such mapping is
known) and if it conserves many edges. For example, for simple networks the
${F-NC}$ (F-score node correctness) is applied to measure the node
correctness, and it is defined as $\frac{M\cap N}{N}$ where $M$ is the set of
node pairs that are mapped under the true node mapping and $N$ the set of node
pairs that are aligned under an alignment $f$.

Otherwise, NCV-G$S^{3}$ (high node coverage (NCV) and Generalized $S^{3}$
(G$S^{3}$)) is applied to measure the edge correctness, and it is defined as
the geometric mean of high node coverage (NCV) and generalized $S^{3}$
(G$S^{3}$) measures. NCV is the percentage of nodes from $G_{1}$ and $G_{2}$
that are also in $G^{\prime}_{1}$ and $G^{\prime}_{2}$ and G$S^{3}$measures
how well edges are conserved between $G^{\prime}_{1}$ and $G^{\prime}_{2}$
where $G_{1}$ and $G_{2}$ are two graphs and $G^{\prime}_{1}$ and
$G^{\prime}_{2}$ are subgraphs of $G_{1}$ and $G_{2}$ that are induced by the
mapping.

In a previous work ([14]), we extended such measures in the multilayer case,
due to, to the best of our knowledge, there are not any other available
measures.

In particular, we compute the multilayer ${F-NC_{m}}$ as the average of
${F-NC_{i}}$ estimated for each layer. Likewise, we compute the multilayer
${NCV-GS^{3}_{m}}$ as the average of ${NCV-GS^{3}_{m}}$ estimated for each
layer.

Finally, we consider the edge correctness for the inter-layer edges. Without
loss of information, we consider all the inter-layer edges as a whole, and we
calculate the correctness of all the inter-layer edges as ${NCV-
GS^{3}_{i}nter}$.

We computed multilayer ${NCV-GS^{3}_{m}}$ and multilayer ${F-NC_{m}}$ measures
for all alignments built for each synthetic network and for real network by
considering the intra-layer and inter-layer. Figure 7, Figure 8, Figure 9,
Figure 10 report the results for synthetic multilayer networks, whereas Figure
11, Figure 12, Figure 13, Figure 14 report the results for real multilayer
network. The Tables related to ${NCV-GS^{3}_{m}}$ and multilayer ${F-NC_{m}}$
measures for all alignments built for each synthetic network and for real
network by considering the intra-layer and inter-layer are reported in
appendix.

Figure 7: NCV-G$S^{3}$ values computed on intra-layer edges for all the local
alignments built on synthetic networks by applying Louvain, Infomap, Greedy
community detection algorithms. Figure 8: NCV-G$S^{3}$ values computed on
inter-edges for all the local alignments built on synthetic networks by
applying Louvain, Infomap, Greedy community detection algorithms. Figure 9:
F-NC values computed on intra-layer edges for all the local alignments built
on synthetic networks by applying Louvain, Infomap, Greedy community detection
algorithms. Figure 10: F-NC values computed on inter-layer edges for all the
local alignments built on synthetic networks by applying Louvain, Infomap,
Greedy community detection algorithms. Figure 11: NCV-G$S^{3}$ values computed
on intra-edges for all the local alignments built on real network by applying
Louvain, Infomap, Greedy community detection algorithms. Figure 12:
NCV-G$S^{3}$ values computed on inter-edges for all the local alignments built
on real network by applying Louvain, Infomap, Greedy community detection
algorithms. Figure 13: F-NC values computed on intra-edges for all the local
alignments built on real network by applying Louvain, Infomap, Greedy
community detection algorithms. Figure 14: F-NC values computed on inter-edges
for all the local alignments built on real network by applying Louvain,
Infomap, Greedy community detection algorithms.

By analyzing the results, it is possible to notice that multilayer ${NCV-
GS^{3}_{m}}$ and multilayer ${F-NC_{m}}$ values decrease by increasing noise
level from 5 % to 25 %. The reduction of quality of the alignment is detected
for different MuLaN versions applied on both synthetic network and real
network. Otherwise, the results show that the quality of the alignment is
greater when Louvain is applied to extract communities.

####  4.4.2 Community Quality

We also evaluated the quality of mined communities by considering i) the
number of extracted communities and ii) the strength of the community
structure by using the modularity [41] as a community quality metric. In
detail, modularity is a measure related to the network structure that detects
the density of connections within a module. The modularity can be either
positive or negative, and positive values are indicative of the presence of
community structure. Thus, a network with a high modularity score will have
many connections within a community [41].

Thus, we first estimated the number of mined communities with Louvain, Infomap
and Greedy, for each local alignment built on synthetic and real networks,
then, we computed the modularity of the extracted communities.

Table 1 and Table 2 report the results obtained with different versions of the
MuLaN algorithm for synthetic and real networks. In particular, for each
network, the Tables report the number of the extracted communities and the
modularity value for each community. The result shows that, all the community
detection algorithms are able to mine more communities by aligning the
original network with itself. Otherwise, the number of communities decreases
on the alignment built with the synthetic counterparts with 5 %, 10%, 15%, 20%
and 25% of added noise. Furthermore, it is possible to notice that Louvain is
able to identify a higher number of communities in the multilayer network with
respect to other community detection algorithms.

By analyzing the results, it is possible to see that the higher modularity
values are obtained by applying Louvain as a community extraction algorithm.
This means that Louvain is able to extract communities that reveal the
relationships among nodes in various layers, with respect to other community
detection algorithms.

Table 1: The Table reports for each synthetic network: the level of noise,
number of Communities extracted and Modularity values by applying Louvain ,
Infomap , Greedy algorithms respectively.

Network |  Noise |  $\\#$ Communities with Louvain |  Modularity with Louvain |  $\\#$ Communities with Infomap |  Modularity with Infomap |  $\\#$ Communities with Greedy |  Modularity with Greedy
---|---|---|---|---|---|---|---
N1 |  0 |  37 |  0,844 |  28 |  0,839 |  38 |  0,844
5 |  40 |  0,850 |  26 |  0,845 |  42 |  0,851
10 |  40 |  0,857 |  29 |  0,855 |  44 |  0,857
15 |  40 |  0,860 |  30 |  0,857 |  45 |  0,861
20 |  42 |  0,872 |  28 |  0,867 |  46 |  0,873
25 |  45 |  0,872 |  29 |  0,869 |  46 |  0,873
N2 |  0 |  39 |  0,840 |  27 |  0,836 |  40 |  0,841
5 |  40 |  0,848 |  28 |  0,844 |  42 |  0,849
10 |  40 |  0,859 |  32 |  0,857 |  42 |  0,859
15 |  41 |  0,861 |  30 |  0,858 |  43 |  0,859
20 |  41 |  0,870 |  27 |  0,856 |  46 |  0,861
25 |  42 |  0,859 |  30 |  0,866 |  46 |  0,871
N3 |  0 |  38 |  0,843 |  28 |  0,840 |  37 |  0,843
5 |  39 |  0,846 |  26 |  0,844 |  40 |  0,848
10 |  37 |  0,868 |  31 |  0,854 |  42 |  0,856
15 |  41 |  0,857 |  27 |  0,856 |  47 |  0,859
20 |  43 |  0,859 |  28 |  0,865 |  44 |  0,867
25 |  46 |  0,877 |  26 |  0,870 |  48 |  0,877
N4 |  0 |  41 |  0,843 |  29 |  0,839 |  41 |  0,843
5 |  39 |  0,851 |  31 |  0,850 |  43 |  0,852
10 |  42 |  0,859 |  29 |  0,856 |  40 |  0,859
15 |  42 |  0,870 |  30 |  0,854 |  44 |  0,859
20 |  43 |  0,867 |  32 |  0,862 |  50 |  0,867
25 |  44 |  0,858 |  29 |  0,866 |  44 |  0,870
N5 |  0 |  38 |  0,842 |  29 |  0,839 |  41 |  0,841
5 |  41 |  0,867 |  29 |  0,844 |  42 |  0,849
10 |  42 |  0,849 |  30 |  0,852 |  43 |  0,854
15 |  42 |  0,863 |  31 |  0,856 |  45 |  0,863
20 |  42 |  0,874 |  34 |  0,863 |  47 |  0,867
25 |  45 |  0,854 |  30 |  0,871 |  48 |  0,874
N6 |  0 |  40 |  0,841 |  27 |  27,000 |  39 |  0,841
5 |  40 |  0,845 |  29 |  0,842 |  40 |  0,845
10 |  42 |  0,856 |  27 |  0,842 |  46 |  0,856
15 |  45 |  0,889 |  28 |  0,867 |  46 |  0,872
20 |  46 |  0,872 |  35 |  0,887 |  50 |  0,888
25 |  60 |  0,900 |  38 |  0,894 |  62 |  0,900
N7 |  0 |  37 |  0,845 |  29 |  0,841 |  39 |  0,844
5 |  38 |  0,854 |  28 |  0,850 |  39 |  0,854
10 |  43 |  0,867 |  30 |  0,859 |  43 |  0,867
15 |  49 |  0,880 |  31 |  0,874 |  50 |  0,881
20 |  52 |  0,904 |  35 |  0,889 |  55 |  0,893
25 |  53 |  0,892 |  37 |  0,900 |  59 |  0,904
N8 |  0 |  40 |  0,843 |  27 |  0,840 |  41 |  0,844
5 |  40 |  0,849 |  27 |  0,848 |  40 |  0,849
10 |  39 |  0,861 |  28 |  0,856 |  43 |  0,861
15 |  42 |  0,878 |  28 |  0,874 |  50 |  0,878
20 |  46 |  0,887 |  29 |  0,879 |  58 |  0,888
25 |  49 |  0,902 |  37 |  0,897 |  58 |  0,903
N9 |  0 |  36 |  0,852 |  24 |  0,839 |  40 |  0,844
5 |  40 |  0,852 |  28 |  0,850 |  40 |  0,851
10 |  39 |  0,861 |  30 |  0,858 |  44 |  0,861
15 |  42 |  0,874 |  33 |  0,872 |  47 |  0,874
20 |  44 |  0,887 |  32 |  0,883 |  47 |  0,886
25 |  52 |  0,900 |  35 |  0,895 |  52 |  0,901
N10 |  0 |  38 |  0,840 |  29 |  0,838 |  41 |  0,841
5 |  37 |  0,849 |  26 |  0,847 |  41 |  0,849
10 |  39 |  0,861 |  28 |  0,858 |  42 |  0,862
15 |  44 |  0,877 |  32 |  0,874 |  48 |  0,876
20 |  52 |  0,890 |  33 |  0,884 |  56 |  0,890
25 |  53 |  0,897 |  36 |  0,892 |  55 |  0,898

Table 2: The Table reports for the real network: the level of noise, number of
Communities extracted and Modularity values by applying Louvain algorithm,
Infomap algorithm, Greedy algorithm.

Network |  Noise |  $\\#$ Communities with Louvain |  Modularity with Generalized Louvain |  $\\#$ Communities with Infomap |  Modularity with Infomap |  $\\#$ Communities with Greedy |  Modularity with Greedy
---|---|---|---|---|---|---|---
N1 |  0 |  267 |  0.335 |  50 |  0.036 |  302 |  0.270
5 |  268 |  0.331 |  49 |  0.034 |  301 |  0.268
10 |  266 |  0.331 |  60 |  0.036 |  297 |  0.269
15 |  260 |  0.331 |  46 |  0.035 |  340 |  0.250
20 |  260 |  0.329 |  52 |  0.034 |  312 |  0.265
|  25 |  263 |  0.327 |  57 |  0.031 |  307 |  0.260

####  4.4.3 Functional Quality Evaluation

Finally, we evaluate the functional quality of the results by considering the
biological relevance of the extracted communities. For classical networks, the
biological relevance evaluation consists in evaluating whether groups of
related entities have a similar biological role or share functions. Since,
MuLaN constructs the local alignment that includes associations between
different entities, i.e. disease-drug, we conducted the evaluation
differently. In particular, we evaluated the functional quality of local
alignment, considering whether our algorithm is able to extract drug-disease
associations still unobserved.

We recall that, drug-disease associations consist of the cases in which drugs
affect disease. Thus, in research field, drug-disease associations may bring
relevant information in the drug discovery, for example, the detection of
convenient relations between drugs and diseases. The literature contain a
large set of drug-disease associations already identified, however many
associations are unobserved or not detected. In the rest of the paper, we
refer to unobserved or not detected drug-disease associations as candidate
drug-disease associations.

Starting from this consideration, our goal is to demonstrate that MuLaN is
able to build a local alignment containing candidate drug-disease
associations, and it is able to extract knowledge about the aligned multilayer
networks.

For this aim, we consider the local alignment constructed by aligning the real
network with its noisy counterpart at 25% (because it is the most different
from the original network) and we analyze the drug-disease associations
forming the extracted communities.

To evaluate whether the associations contained in the communities are known in
the literature or still unobserved drug-disease associations, we used SCMFDD
[42], a similarity constrained matrix factorization method for the drug-
disease association prediction. SCMFDD reveals unobserved drug-disease
associations by using drug features, disease semantic information and known
associations incorporated into the matrix factorization frame (see [42] for
complete details).

SCMFDD computes: i) the drug-drug similarities by using Jaccard index [43]
taking into account diverse drug features (such as, targets, enzymes, pathways
and drug-drug interactions), ii) the disease-disease semantic similarity by
using MeSH information [44], iii) two low-rank feature matrices by applying a
factorization on observed drug-disease associations matrix. Then, it uses an
objective function based on Newton’s method [45] to compute a score
representing the probability that a drug and a disease have an association
(see [42] for complete details on SCMFDD). Thus, we applied SCMFDD to analyze
each couple of drug-disease associations forming the community extracted with
MuLaN. The results show that the extracted communities contain 46,498
candidate drug-disease associations. Among these, 711 drug-disease
associations present a score greater than $0.5$. We reported candidate drug-
disease associations with score equal to 1 in Table 3. The complete detected
associations are available at https://github.com/pietrocinaglia/mulan).

Furthermore, we manually evaluated the candidate drug-disease associations by
searching literature evidences. For example, by considering the drug-disease
association Fenofibrate-Cholestasis, the works in [46, 47] report the effect
of fenofibrate against cholestasis. In [47], Ghonem et al. discuss the
effectiveness and well tolerability of Orlistat treatment for hyperglycemia.
Also, [48] presents the successful use of rifampin in a patient with Stevens-
Johnson syndrome. We list the literature evidence of the top 10 detected
drugs-disease associations in Table 4.

In conclusion, the results demonstrate that our algorithm is capable of
discovering candidate drug-disease associations and, consequently, of
extracting new knowledge from multilayer networks.

Table 3: The Table reports an example of novel drug-disease associations
contained in the local alignment built with MuLaN.

Drug ID | Drug Name | Disease ID | Disease Name | Score
---|---|---|---|---
DB01039 | Fenofibrate | DOID:13580 | Cholestasis | 1
DB01083 | Orlistat | DOID:4195 | Hyperglycemia | 1
DB01220 | Rifaximin | DOID:0050426 | Stevens-Johnson syndrome | 1
DB00951 | Isoniazid | DOID:2355 | Anemia | 1
DB00570 | Vinblastine | DOID:2355 | Anemia | 1
DB01241 | Gemfibrozil | DOID:13580 | Cholestasis | 1
DB01602 | Bacampicillin | DOID:0050426 | Stevens-Johnson syndrome | 1
DB01042 | Melphalan | DOID:987 | Alopecia | 1
DB00535 | Cefdinir | DOID:13580 | Cholestasis | 1
DB01005 | Hydroxyurea | DOID:1227 | Neutropenia | 1
DB01593 | Zinc | DOID:2355 | Anemia | 1
DB00993 | Azathioprine | DOID:1227 | Neutropenia | 1
DB02659 | Cholic Acid | DOID:13580 | Cholestasis | 1
DB01103 | Quinacrine | DOID:13580 | Cholestasis | 1
DB00970 | Dactinomycin | DOID:1227 | Neutropenia | 1
DB00586 | Diclofenac | DOID:2355 | Anemia | 1
DB00290 | Bleomycin | DOID:2355 | Anemia | 1
DB00963 | Bromfenac | DOID:13580 | Cholestasis | 1
DB01137 | Levofloxacin | DOID:13580 | Cholestasis | 1
DB01262 | Decitabine | DOID:299 | Adenocarcinoma | 1
DB01008 | Busulfan | DOID:2355 | Anemia | 1
DB00549 | Zafirlukast | DOID:13580 | Cholestasis | 1
DB00763 | Methimazole | DOID:0050426 | Stevens-Johnson syndrome | 1
DB00591 | Fluocinolone Ace | DOID:1555 | Urticaria | 1
DB01022 | Phylloquinone | DOID:1555 | Urticaria | 1
DB00271 | Diatrizoate | DOID:10763 | Hypertension | 1
DB01143 | Amifostine | DOID:615 | Leukopenia | 1
DB01015 | Sulfamethoxazole | DOID:1555 | Urticaria | 1
DB00984 | Nandrolone phenp | DOID:10763 | Hypertension | 1
DB00296 | Ropivacaine | DOID:10763 | Hypertension | 1
DB00205 | Pyrimethamine | DOID:1588 | Thrombocytopenia | 1
DB01254 | Dasatinib | DOID:13250 | Diarrhea | 1
DB00291 | Chlorambucil | DOID:1588 | Thrombocytopenia | 1
DB00293 | Raltitrexed | DOID:615 | Leukopenia | 1
DB01406 | Danazol | DOID:2237 | Hepatitis | 1
DB00456 | Cefalotin | DOID:1555 | Urticaria | 1
DB01138 | Sulfinpyrazone | DOID:576 | Proteinuria | 1
DB00480 | Lenalidomide | DOID:615 | Leukopenia | 1
DB01161 | Chloroprocaine | DOID:10763 | Hypertension | 1
DB00287 | Travoprost | DOID:10763 | Hypertension | 1
DB00121 | Biotin | DOID:10763 | Hypertension | 1
DB00491 | Miglitol | DOID:10763 | Hypertension | 1
DB00459 | Acitretin | DOID:2237 | Hepatitis | 1
DB00905 | Bimatoprost | DOID:10763 | Hypertension | 1
DB01288 | Fenoterol | DOID:10763 | Hypertension | 1
DB04272 | Citric acid | DOID:10763 | Hypertension | 1
DB01086 | Benzocaine | DOID:1555 | Urticaria | 1

Table 4: Top 10 drug-disease associations contained in the local alignment
built with MuLaN.

Drug Name | Disease Name | Evidence
---|---|---
Fenofibrate | Cholestasis | [46, 47]
Orlistat | Hyperglycemia | [49]
Rifaximin | Stevens-Johnson syndrome | [48]
Isoniazid | Anemia | [50]
Vinblastine | Anemia | [51]
Gemfibrozil | Cholestasis | [52]
Bacampicillin | Stevens-Johnson syndrome | [50]
Melphalan | Alopecia | [53]
Cefdinir | Cholestasis | [54]
Hydroxyurea | Neutropenia | [55]

####  4.4.4 MuLaN vs MultiLoAl

To clarify the need for developing an extended version of MultiLoAl, in this
section, we present the comparison of MuLaN with MultiLoAl.The aim is to
demonstrate that MuLaN reports the best performance to analyze multilayer
networks with respect to previous version MultiLoAl. The dataset that we used
for the comparison consists of ten synthetic multilayer networks and one real
multilayer network used for MuLaN evaluation. Thus, we built the alignment of
a network with respect to itself, and considering the alignment of a network
with respect to an altered version by applying MultiLoAl. Our experimentation
shows that MultiLoAl builds the alignment between two synthetic networks,
having $2000$ nodes and $2298$ edges each one, in $\sim 120$ minutes, against
the $\sim 1.42$ seconds required by MuLaN for the same operation.

Results show clearly that MuLaN outperforms MultiLoAl, in terms of memory
efficiency and runtime.

##  5 Conclusions

Multilayer networks are a powerful tool for the modelling of complex data in
biology and medicine. In this work, we focused on the problem of analysis of
multilayer networks through the comparison of their internal structure, by
applying Network Alignment algorithms. Since classical network alignment
algorithms for simple networks do not perform well on multilayer networks, we
presented a deep extension of a previously developed network alignment
algorithm for multilayer network, named, MuLaN (Local Alignment Algorithm for
Multilayer Networks). We tested it on ten synthetic multilayer networks and on
a real multilayer network, and we evaluated the quality of results. Results
confirm that MuLaN is able to build alignment with high topological quality.
According to these analyses, we demonstrated the MuLaN methodology is suitable
for the comparison and hence, the analysis of multilayer networks.

##  6 Competing interests

No competing interest is declared.

##  7 Author contributions statement

Conceptualization, PHG. and MM.; methodology, PHG, PC, and MM; software, PC,
MM; data curation, MM and PC; writing–original draft preparation, MM, PC, PHG,
and MC.; writing–review and editing, MM, PC, PHG and MC.; funding acquisition,
MC. All authors have read and agreed to the published version of the
manuscript.

##  8 Funding

This work was funded by the Next Generation EU - Italian NRRP, Mission 4,
Component 2, Investment 1.5, call for the creation and strengthening of
’Innovation Ecosystems’, building ’Territorial R&D Leaders’ (Directorial
Decree n. 2021/3277) - project Tech4You - Technologies for climate change
adaptation and quality of life improvement, n. ECS0000009. This work reflects
only the authors’ views and opinions, neither the Ministry for University and
Research nor the European Commission can be considered responsible for them.

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##  Appendix A Sample Appendix Section

This appendix present the process in which MuLaN builds the MAG and results of
the computation of ${NCV-GS^{3}_{m}}$ and ${F-NC_{m}}$ measures for all
alignments built for each synthetic network and for real network by
considering the intra-layer and inter-layer.

###  A.1 Building of the Alignment Graph

It receives as input two multilayer networks ($M_{1}$ and $M_{2}$) having the
same number of layers $n$ and a multiset of seed nodes representing similarity
between the node of each graph of the same layer $k$ of the two networks. We
assume the existence of a bijective correlation among layers of the networks,
so the layer $k$ of $M_{1}$ correspond to the layer $k$ of $M_{2}$. Starting
from $M_{1}$ and $M_{2}$ and the multiset of seed nodes, in the step 1.a,
MuLaN analyzes each pair of $k$ layers in $M_{1}$ and $M_{2}$ independently,
see Figure 15. For each layer, MuLaN constructs an alignment graph on the
basis of the strategy suggested in L-HetNetAligner. [12]. At this stage, MuLaN
creates the nodes of the alignment graph. Each node $v_{al}\in V_{al}$
represents a match of nodes of the input graphs, so $V_{al}\subseteq
V_{1}\times V_{2}$. Thus, the choice of node pairs is conducted by the input
similarity relationships. Consequently, each node is matched with the most
similar node of the other network on the same layer; and each node of the
alignment graph represents a pair of similarity among nodes from the input
networks, see Figure 16.

Figure 15: MuLaN separates the input networks according to the layer type.
Figure 16: Building of alignment graph for the disease layer of the two input
multilayer networks: Node Definition. MuLaN takes the two networks in a layer
and a subset of node pairs matched according to a similarity function and
starts to build the alignment graph. In this step, the algorithm defines the
nodes of the alignment graph represented by the pair matched nodes. Figure 17:
Example of homogeneous match, homogeneous mismatch and homogeneous gap and
building of Alignment Graph.

When all nodes have been inserted into the alignment graph, MuLaN adds the
edges of the alignment graphs. For each pair of nodes, MuLaN analyzes the two
input graphs, and it inserts and weights the edges considering three
conditions: match, mismatch and gap. In the following, we present each
condition. So, let us consider the nodes of the alignment graphs; in
particular, let us analyze the pair of nodes $(D1-D1)$ and $(D2-D2)$ in Figure
4. To determine the presence of an edge, we we must consider the edges
$(D1,D2)\in G_{1}$ network and the edges $(D1,D2)\in G_{2}$ network. If
$G_{1}$, and $G_{2}$ contains these nodes, and the nodes are adjacent, we
identify this condition as match, that, for convenience, we call homogeneous
match, since the nodes of the two networks are of the same type i.e. disease,
(see Figure 17 (a)).

Let us consider $\Delta=2$ as node distance, i.e. the length of the shortest
connecting path threshold to discriminate between gaps and mismatches. Let us
consider the pair of nodes $(D1-D1)$ and $(D2-D2)$ (Figure 17 (b)). $G_{1}$
contains the edges $(D1,D2)$, whereas the nodes $(D1-D1)$ are disconnected in
$G_{2}$. We identify this condition as mismatch that we call homogeneous
mismatch. Let us consider the pair of nodes $(D1-D1)$ and $(D2-D2)$ (Figure 17
(c)). $G_{1}$ contains the edges $(D1,D2)$, whereas the nodes $(D1-D1)$ have a
distance equal to 2 in $G_{2}$. We identify this condition as gap that we call
homogeneous gap.

Once the edges of the alignment graph are added, a weight is assigned to each
edge by applying an ad-hoc scoring function $F$ and the gap threshold
$\Delta$. The function assigns a highest score to the matches, respect to
mismatch and gaps. The kind of scoring function has a large significance on
the resulting alignment graph and on the alignment itself. The algorithm
enables the user to choose several values of $\Delta$ to optimize the quality
of results. In this work we set the weight assigned to each edge as follows:
homogeneous match equal to $1$, homogeneous mismatch equal to $0.5$,
homogeneous gap equal to $0.2$.

###  A.2 Topological quality

Table 5, Table 6, Table 7, Table 8 report the results for synthetic multilayer
networks, whereas Table 9, Table 10, Table 11, Table 12 report the results for
real multilayer network.

Table 5: NCV-G$S^{3}$ values computed on intra-layer edges for all the local
alignments built on synthetic networks by applying Louvain, Infomap, Greedy
community detection algorithms.

Network | Noise | NCV-G$S^{3}$ with Louvain | NCV-G$S^{3}$ with Infomap | NCV-G$S^{3}$ with Greedy
---|---|---|---|---
N1 | 0 | 0.631 | 0.574 | 0.604
5 | 0.631 | 0.574 | 0.604
10 | 0.625 | 0.562 | 0.589
15 | 0.596 | 0.56 | 0.557
20 | 0.593 | 0.559 | 0.55
25 | 0.583 | 0.545 | 0.537
N2 | 0 | 0.635 | 0.599 | 0.6
5 | 0.635 | 0.599 | 0.6
10 | 0.628 | 0.581 | 0.583
15 | 0.592 | 0.54 | 0.58
20 | 0.573 | 0.527 | 0.576
25 | 0.559 | 0.527 | 0.533
N3 | 0 | 0.648 | 0.587 | 0.595
5 | 0.648 | 0.587 | 0.595
10 | 0.643 | 0.572 | 0.557
15 | 0.629 | 0.544 | 0.53
20 | 0.553 | 0.532 | 0.524
25 | 0.552 | 0.53 | 0.518
N4 | 0 | 0.641 | 0.598 | 0.594
5 | 0.641 | 0.598 | 0.594
10 | 0.621 | 0.592 | 0.583
15 | 0.61 | 0.554 | 0.566
20 | 0.609 | 0.549 | 0.545
25 | 0.57 | 0.502 | 0.52
N5 | 0 | 0.643 | 0.588 | 0.571
5 | 0.643 | 0.588 | 0.571
10 | 0.624 | 0.565 | 0.554
15 | 0.569 | 0.559 | 0.545
20 | 0.568 | 0.559 | 0.511
25 | 0.552 | 0.549 | 0.51
N6 | 0 | 0.632 | 0.559 | 0.589
5 | 0.632 | 0.559 | 0.589
10 | 0.613 | 0.54 | 0.577
15 | 0.587 | 0.539 | 0.565
20 | 0.581 | 0.499 | 0.564
25 | 0.57 | 0.495 | 0.539
N7 | 0 | 0.639 | 0.6 | 0.604
5 | 0.639 | 0.6 | 0.604
10 | 0.635 | 0.581 | 0.601
15 | 0.607 | 0.576 | 0.596
20 | 0.563 | 0.52 | 0.553
25 | 0.552 | 0.517 | 0.527
N8 | 0 | 0.625 | 0.578 | 0.587
5 | 0.625 | 0.578 | 0.587
10 | 0.616 | 0.569 | 0.545
15 | 0.594 | 0.556 | 0.538
20 | 0.559 | 0.548 | 0.524
25 | 0.558 | 0.491 | 0.513
N9 | 0 | 0.638 | 0.561 | 0.566
5 | 0.638 | 0.561 | 0.566
10 | 0.636 | 0.552 | 0.556
15 | 0.635 | 0.532 | 0.549
20 | 0.604 | 0.527 | 0.548
25 | 0.572 | 0.51 | 0.547
N10 | 0 | 0.65 | 0.592 | 0.6
5 | 0.65 | 0.592 | 0.6
10 | 0.649 | 0.56 | 0.56
15 | 0.637 | 0.56 | 0.559
20 | 0.604 | 0.558 | 0.543
25 | 0.566 | 0.515 | 0.53

Table 6: NCV-G$S^{3}$ values computed on inter-layer edges for all the local
alignments built on synthetic networks by applying Louvain, Infomap, Greedy
community detection algorithms.

Network |  Noise |  NCV-G$S^{3}$ with Louvain |  NCV-G$S^{3}$ with Infomap |  NCV-G$S^{3}$ with Greedy
---|---|---|---|---
N1 |  0 |  0.598 |  0.547 |  0.545
5 |  0.598 |  0.547 |  0.545
10 |  0.586 |  0.537 |  0.506
15 |  0.527 |  0.52 |  0.5
20 |  0.504 |  0.487 |  0.5
25 |  0.501 |  0.475 |  0.486
N2 |  0 |  0.583 |  0.524 |  0.546
5 |  0.583 |  0.524 |  0.546
10 |  0.563 |  0.504 |  0.523
15 |  0.555 |  0.502 |  0.516
20 |  0.542 |  0.484 |  0.508
25 |  0.536 |  0.472 |  0.492
N3 |  0 |  0.589 |  0.544 |  0.577
5 |  0.589 |  0.544 |  0.577
10 |  0.589 |  0.526 |  0.556
15 |  0.567 |  0.525 |  0.545
20 |  0.52 |  0.503 |  0.539
25 |  0.516 |  0.48 |  0.527
N4 |  0 |  0.593 |  0.55 |  0.541
5 |  0.593 |  0.55 |  0.541
10 |  0.592 |  0.539 |  0.495
15 |  0.581 |  0.532 |  0.494
20 |  0.507 |  0.515 |  0.482
25 |  0.504 |  0.477 |  0.482
N5 |  0 |  0.598 |  0.532 |  0.567
5 |  0.598 |  0.532 |  0.567
10 |  0.571 |  0.508 |  0.548
15 |  0.546 |  0.5 |  0.498
20 |  0.532 |  0.493 |  0.488
25 |  0.525 |  0.484 |  0.483
N6 |  0 |  0.583 |  0.539 |  0.574
5 |  0.583 |  0.539 |  0.574
10 |  0.564 |  0.509 |  0.549
15 |  0.552 |  0.475 |  0.516
20 |  0.544 |  0.475 |  0.516
25 |  0.525 |  0.464 |  0.489
N7 |  0 |  0.559 |  0.551 |  0.574
5 |  0.559 |  0.551 |  0.574
10 |  0.517 |  0.489 |  0.522
15 |  0.516 |  0.481 |  0.512
20 |  0.514 |  0.474 |  0.496
25 |  0.513 |  0.462 |  0.491
N8 |  0 |  0.586 |  0.557 |  0.577
5 |  0.586 |  0.557 |  0.577
10 |  0.571 |  0.549 |  0.576
15 |  0.537 |  0.528 |  0.556
20 |  0.533 |  0.489 |  0.531
25 |  0.519 |  0.468 |  0.512
N9 |  0 |  0.576 |  0.56 |  0.574
5 |  0.576 |  0.56 |  0.574
10 |  0.564 |  0.558 |  0.555
15 |  0.558 |  0.558 |  0.555
20 |  0.533 |  0.534 |  0.545
25 |  0.515 |  0.525 |  0.487
N10 |  0 |  0.576 |  0.533 |  0.532
5 |  0.576 |  0.533 |  0.532
10 |  0.524 |  0.53 |  0.519
15 |  0.521 |  0.5 |  0.519
20 |  0.515 |  0.499 |  0.493
25 |  0.504 |  0.468 |  0.49

Table 7: F-NC values computed on intra-layer edges for all the local
alignments built on synthetic networks by applying Louvain, Infomap, Greedy
community detection algorithms.

Network |  Noise |  F-NC with Louvain |  F-NC with Infomap |  F-NC with Greedy
---|---|---|---|---
N1 |  0 |  0.999 |  0.934 |  0.974
5 |  0.999 |  0.934 |  0.974
10 |  0.994 |  0.913 |  0.927
15 |  0.961 |  0.912 |  0.889
20 |  0.926 |  0.86 |  0.87
25 |  0.896 |  0.857 |  0.863
N2 |  0 |  0.986 |  0.863 |  0.971
5 |  0.986 |  0.863 |  0.971
10 |  0.957 |  0.768 |  0.907
15 |  0.951 |  0.729 |  0.857
20 |  0.927 |  0.63 |  0.76
25 |  0.896 |  0.618 |  0.71
N3 |  0 |  0.981 |  0.87 |  0.93
5 |  0.981 |  0.87 |  0.93
10 |  0.96 |  0.836 |  0.859
15 |  0.919 |  0.821 |  0.858
20 |  0.902 |  0.741 |  0.804
25 |  0.883 |  0.613 |  0.799
N4 |  0 |  0.997 |  0.891 |  0.921
5 |  0.997 |  0.891 |  0.921
10 |  0.992 |  0.889 |  0.879
15 |  0.945 |  0.835 |  0.825
20 |  0.927 |  0.791 |  0.785
25 |  0.893 |  0.683 |  0.747
N5 |  0 |  0.963 |  0.951 |  0.92
5 |  0.963 |  0.951 |  0.92
10 |  0.907 |  0.853 |  0.777
15 |  0.905 |  0.852 |  0.768
20 |  0.895 |  0.82 |  0.753
25 |  0.894 |  0.684 |  0.733
N6 |  0 |  0.938 |  0.917 |  0.877
5 |  0.938 |  0.917 |  0.877
10 |  0.928 |  0.905 |  0.868
15 |  0.914 |  0.816 |  0.86
20 |  0.891 |  0.699 |  0.854
25 |  0.881 |  0.619 |  0.724
N7 |  0 |  0.998 |  0.967 |  0.908
5 |  0.998 |  0.967 |  0.908
10 |  0.985 |  0.922 |  0.906
15 |  0.977 |  0.828 |  0.861
20 |  0.938 |  0.765 |  0.846
25 |  0.892 |  0.764 |  0.723
N8 |  0 |  0.96 |  0.97 |  0.932
5 |  0.96 |  0.97 |  0.932
10 |  0.945 |  0.932 |  0.881
15 |  0.928 |  0.797 |  0.864
20 |  0.893 |  0.755 |  0.755
25 |  0.884 |  0.709 |  0.721
N9 |  0 |  0.992 |  0.916 |  0.961
5 |  0.992 |  0.916 |  0.961
10 |  0.989 |  0.786 |  0.931
15 |  0.974 |  0.739 |  0.925
20 |  0.962 |  0.736 |  0.884
25 |  0.915 |  0.651 |  0.836
N10 |  0 |  0.986 |  0.967 |  0.894
5 |  0.986 |  0.967 |  0.894
10 |  0.971 |  0.832 |  0.834
15 |  0.963 |  0.798 |  0.806
20 |  0.94 |  0.792 |  0.766
25 |  0.914 |  0.704 |  0.736

Table 8: F-NC values computed on inter-layer edges for all the local
alignments built on synthetic networks by applying Louvain, Infomap, Greedy
community detection algorithms.

Network |  Noise |  F-NC with Louvain |  F-NC with Infomap |  F-NC with Greedy
---|---|---|---|---
N1 |  0 |  0.771 |  0.706 |  0.757
5 |  0.771 |  0.706 |  0.757
10 |  0.77 |  0.671 |  0.7
15 |  0.717 |  0.64 |  0.678
20 |  0.659 |  0.614 |  0.652
25 |  0.645 |  0.604 |  0.592
N2 |  0 |  0.796 |  0.698 |  0.743
5 |  0.796 |  0.698 |  0.743
10 |  0.723 |  0.695 |  0.7
15 |  0.69 |  0.68 |  0.691
20 |  0.674 |  0.605 |  0.681
25 |  0.607 |  0.569 |  0.623
N3 |  0 |  0.793 |  0.696 |  0.728
5 |  0.793 |  0.696 |  0.728
10 |  0.697 |  0.692 |  0.704
15 |  0.677 |  0.661 |  0.699
20 |  0.668 |  0.65 |  0.69
25 |  0.614 |  0.566 |  0.585
N4 |  0 |  0.763 |  0.666 |  0.775
5 |  0.763 |  0.666 |  0.775
10 |  0.724 |  0.621 |  0.74
15 |  0.679 |  0.621 |  0.698
20 |  0.639 |  0.588 |  0.653
25 |  0.6 |  0.586 |  0.601
N5 |  0 |  0.745 |  0.732 |  0.775
5 |  0.745 |  0.732 |  0.775
10 |  0.702 |  0.697 |  0.683
15 |  0.666 |  0.694 |  0.607
20 |  0.658 |  0.628 |  0.603
25 |  0.616 |  0.572 |  0.585
N6 |  0 |  0.748 |  0.678 |  0.745
5 |  0.748 |  0.678 |  0.745
10 |  0.738 |  0.678 |  0.727
15 |  0.697 |  0.642 |  0.625
20 |  0.62 |  0.627 |  0.603
25 |  0.612 |  0.591 |  0.593
N7 |  0 |  0.735 |  0.718 |  0.777
5 |  0.735 |  0.718 |  0.777
10 |  0.728 |  0.67 |  0.713
15 |  0.692 |  0.659 |  0.698
20 |  0.661 |  0.592 |  0.631
25 |  0.608 |  0.561 |  0.602
N8 |  0 |  0.758 |  0.719 |  0.753
5 |  0.758 |  0.719 |  0.753
10 |  0.739 |  0.7 |  0.743
15 |  0.718 |  0.695 |  0.735
20 |  0.707 |  0.669 |  0.699
25 |  0.627 |  0.597 |  0.683
N9 |  0 |  0.712 |  0.708 |  0.763
5 |  0.712 |  0.708 |  0.763
10 |  0.665 |  0.687 |  0.735
15 |  0.659 |  0.685 |  0.724
20 |  0.658 |  0.669 |  0.714
25 |  0.621 |  0.619 |  0.666
N10 |  0 |  0.765 |  0.724 |  0.764
5 |  0.765 |  0.724 |  0.764
10 |  0.754 |  0.628 |  0.761
15 |  0.692 |  0.612 |  0.725
20 |  0.681 |  0.571 |  0.712
25 |  0.618 |  0.542 |  0.691

Table 9: NCV-G$S^{3}$ values computed on intra-layer edges for all the local
alignments built on real network by applying Louvain, Infomap, Greedy
community detection algorithms.

Network |  Noise |  NCV-G$S^{3}$ with Louvain |  NCV-G$S^{3}$ with Infomap |  NCV-G$S^{3}$ with Greedy
---|---|---|---|---
N1 |  0 |  0.884 |  0.565 |  0.773
5 |  0.884 |  0.565 |  0.773
10 |  0.879 |  0.554 |  0.769
15 |  0.877 |  0.551 |  0.763
20 |  0.836 |  0.549 |  0.713
25 |  0.818 |  0.541 |  0.702

Table 10: NCV-G$S^{3}$ values computed on inter-layer edges for all the local
alignments built on real network by applying Louvain, Infomap, Greedy
community detection algorithms.

Network |  Noise |  NCV-G$S^{3}$ with Louvain |  NCV-G$S^{3}$ with Infomap |  NCV-G$S^{3}$ with Greedy
---|---|---|---|---
Real Network |  0 |  0.661 |  0.565 |  0.635
5 |  0.661 |  0.565 |  0.635
10 |  0.65 |  0.565 |  0.634
15 |  0.622 |  0.558 |  0.609
20 |  0.602 |  0.555 |  0.609
25 |  0.592 |  0.544 |  0.599

Table 11: F-NC values computed on intra-layer edges forfor all the local
alignments built on real network by applying Louvain, Infomap, Greedy
community detection algorithms.

Network |  Noise |  F-NC with Louvain |  F-NC with Infomap |  F-NC with Greedy
---|---|---|---|---
Real Network |  0 |  0.661 |  0.565 |  0.635
5 |  0.661 |  0.565 |  0.635
10 |  0.65 |  0.565 |  0.634
15 |  0.622 |  0.558 |  0.609
20 |  0.602 |  0.555 |  0.609
25 |  0.592 |  0.544 |  0.599

Table 12: F-NC values computed on inter-layer edges for all the local
alignments built on real network by applying Louvain, Infomap, Greedy
community detection algorithms.

Network |  Noise |  F-NC with Louvain |  F-NC with Infomap |  F-NC with Greedy
---|---|---|---|---
Real Network |  0 |  0.576 |  0.497 |  0.528
5 |  0.576 |  0.497 |  0.528
10 |  0.538 |  0.486 |  0.521
15 |  0.511 |  0.484 |  0.51
20 |  0.507 |  0.478 |  0.507
25 |  0.498 |  0.46 |  0.486