This document discusses the formal representation and analysis of multiplex networks. It begins by introducing complex networks science and the concept of abstracting real-world systems as graphs to study structure and interactions. It then defines multiplex networks as networks with multiple types of interactions or relations between nodes that can be represented as multiple layer-graphs. The document provides formal definitions and representations of multiplex networks using concepts like participation graphs, layer-graphs, coupling graphs, and supra-adjacency matrices. It also discusses analyzing and coarse-graining multiplex networks through measures like structural metrics, walks, and quotient graphs.

1. Multiplex Networks: Structure and Dynamics
Emanuele Cozzo
Tesis doctoral
Director: Yamir Moreno
Universidad de Zaragoza
February 2, 2016

2. The complex networks view of complex systems
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3. In the beginning were networks, and networks were
everywhere
Structural approach
shift
Metaphor =⇒ substantial notion
⇓
Contemporary Complex Networks Science
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4. In the beginning were networks, and networks were
everywhere
Structural approach
shift
Metaphor =⇒ substantial notion
⇓
Contemporary Complex Networks Science
Science of Complex Networks
Interdisciplinary point of view on complex systems → unifying language
Abstraction from the details of a system
Focus on the structure of interactions.
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5. Hypothesis
Structure and Function are intimately related
Abastraction =⇒ Graph Model of the System
Paraphrasing Wellman: It is a comprehensive paradigmatic way of taking structure seriously by studying directly how
patterns of ties determine the functioning of a system.
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6. A physicist point of view
Complex networks are systems that display a strong disorder with
large fluctuations of the structural characteristics
Four steps:
Step 1: formal representation
Step 2: topological characterization
Step 3: statistical characterization
Step 4: functional characterization.
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7. From simple networks to multiplex networks
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8. The concept of multiplex network has been around for many decades:
1962 Max Gluckman (antropology) - 1969 Kapferer (sociology of
work)
Concept of multiplex networks
• communication media
• multiplicity of roles and milieux
communication media constituents continuously switch among
a variety of media
roles interactions are always context dependent
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9. Contemporary debate
Internet and mobile communications ↔ social and technological
revolution
⇓
new steam for the formal and quantitative study on multiplex
networks
Botler and Gusin media
Rainie and Wellman roles
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10. Not only social...
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11. Not only social...
Biology integration of multiple set of omic data
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12. Not only social...
Biology integration of multiple set of omic data
Transportation different modes
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13. Not only social...
Biology integration of multiple set of omic data
Transportation different modes
Engineering interdependence of different lifelines
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14. Basic Definitions and Formalism
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15. 11 of 70

16. Multiplex networks as a primary object
• We propose a formal language intended to be general and
complete enough
A rigorous algebraic
formalism →
further more
complex reasonings
design data
structures and
algorithms
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17. Graph Model
Network
→
model
Graph: G(V , E)
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18. Graph Model
Network
→
model
Graph: G(V , E)
The notion of layer must be
introduced
Layer:
An index that represents a
particular type of interaction or
relation
L = {1, ..., m} index set
| L |= m the number of layers
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19. Nodes and node-layer pairs
Participation Graph:
• the set of nodes V ,
GP = (V , L, P): binary relation,
where P ⊆ V × L
Representative of node u in layer
α
(u, α) ∈ P, with u ∈ V , and
α ∈ L, is read node u participates
in layer α
define: node-layer pairs
• | P |= N number of node-layer pairs, | V |= n
number of nodes
(u,1)
(u,2)
(v,1)
(v,2)
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20. node-aligned multiplex networks
If each node u ∈ V has a representative in each layer we call the
multiplex a node-aligned multiplex and | P |= nm
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21. Layer-graphs
Each system of relations or interactions of different kind is naturally
represented by a graph
Gβ(Vβ, Eβ)
• Vβ ∈ P, Vβ = {(u, α) ∈ P | α = β} the set of all
the representatives of the node set in a particular layer
• | Vβ |= nβ the number of node-layer pairs in layer β
• Node-aligned multiplex networks: nα = n ∀α ∈ L.
• Eβ ⊆ Vβ × Vβ the set of edges. Interactions or
relations of a particular type
G1
G2
G3 G4
M = {Gα}α∈L, the set of all layer-graphs
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22. The Coupling Graph
GC (P, EC ) on P
EC = {((u, α), (v, β)) ⇐⇒ u =
v)}
Formed by n =| P | disconnected
components
(complete graphs or disconnected
nodes)
⇒ supra-nodes
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23. Multiplex Network Representation
A multiplex network is represented by :
M = (V , L, P, M):
• the node set V represents the components of the system
• the layer set L represents different types of relations or interactions
in the system
• the participation graph GP encodes the information about what
node takes part in a particular type of relation and defines the
representative of each component in each type of relation, i.e., the
node-layer pair
• the layer-graphs M represent the networks of interactions of a
particular type between the components, i.e., the networks of
representatives of the components of the system.
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24. 19 of 70

25. Synthetic Representation
The union of all the layer-graphs:
The intra-layer graph
Gl = α Gα
Define
The supra-graph
GM = Gl ∪ GC
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26. 21 of 70

27. Adjacencies Matrices
Adjacency matrix
G(V , E) → A, auv = 1u∼v
Layer adjacency matrix
Layer graph Gα → Aα, nα × nα symmetric matrix , with aα
ij = 1 iff
there is an edge between i and j in Gα
Coupling matrix
Coupling graph GC → C = {cij }, an N × N matrix , with cij = 1 iff
they are representatives of the same node in different layers
Standard labelling → C: block-matrix
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28. Supra-Adjacency Matrix
¯A =
α
Aα
+ C = A + C
By definition A is the adjacency matrix of Gl . ¯A the adjacency
matrix of GM
Node-aligned multiplex networks
¯A = A + Km ⊗ In
Identical layer-graphs
¯A = Im ⊗ A + Km ⊗ In,
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29. 0 1 0 1 0
1 0 1 0 1
0 1 0 0 0
1 0 0 0 1
0 1 0 1 0
1
2
3
4
5
1 2 3 4 5
A = =
A
1
A
2
C12
C21
0
0
C12
C21
=
A
1
A
2
0
0
A =
1 2
3
4
5
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30. Supra-Laplacian
¯L = ¯D − ¯A
By definition
¯L =
α
Lα
+ LC .
Node-aligned multiplex network
¯L =
α
(Lα
+(m−1)IN)−Km ⊗In
Identical layer-graphs
¯L = Im ⊗(L+(m −1)In)−Km ⊗In
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31. Multiplex Walk Matrices
A walk on a graph is a sequence of adjacent vertices. The length of a
walk is its number of edges.
Nij (k) = (Ak
)ij
Multiplex networks contain walks that can traverse different additional layers
Define
a supra-walk is a walk on a multiplex network in which, either before
or after each intra-layer step, a walk can either continue on the same
layer or change to an adjacent layer
C = αI + βC (1)
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32. • AC encodes the steps in which after each intra-layer step a walk
can change layer
• CA encodes the steps in which before each intra-layer step a walk
can change layer.
adjacency matrix of a directed (possible weighted) graph
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33. • AC encodes the steps in which after each intra-layer step a walk
can change layer
• CA encodes the steps in which before each intra-layer step a walk
can change layer.
adjacency matrix of a directed (possible weighted) graph
Define: Auxiliary supra-graph GM whose adjacency matrix is
M = M(A, C)
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34. Quotient graphs
It is natural to try to aggregate the interaction pattern of each layer
in a single network somehow
(a) (b)
(c)
The natural definition of an aggregate network is given by the notion
of quotient network
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35. Quotient graphs
Suppose that {V1, . . . , Vm} is a partition of the node set of a graph G with
adjacency matrix A(G)
ni =| Vi |
The quotient graph Q(G) is a coarsening of the network with respect to
that partition.
It has one node per cluster Vi , and an edge from Vi to Vj weighted by an
average connectivity from Vi to Vj
Exact results relate the adjacency and laplacian spectrum of the
quotient graph to the adjacency and laplacian spectrum of the parent
graph, respectively
The Laplacian of the quotient must be defined carefully
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36. Coarsening a Multiplex
Two natural partitions: supra-nodes and layers.
Define
• aggregate network: quotient
graph of the parent multiplex.
Partition according to
supra-nodes.
• network of layers: quotient
graph of the parent multiplex.
Partition according to layers.
(a) (b)
(c)
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37. Aggregate network
˜A = Λ−1
ST
n
¯ASn, (2)
• Sn = (siu) characteristic matrix
• Λ = diag{κ1, . . . , κn} the multiplexity degree matrix.
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38. Network of layers
The network of layers has adjacency matrix given by
˜Al = Λ−1
ST
l
¯ASl , (3)
• Sl = {siα} characteristic matrix
• Λ = diag{n1, . . . , nm} layer size matrix
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39. Supra-walk and Coarse-graining
we have a relation between the number of supra-walks in a multiplex
network and the weight of weighted walks in its aggregate network
when the multiplex is node-aligned and switching layer has no cost
ST
n (AC)l
Sn = ml+1 ˜Wl
= ml
Wl
. (4)
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40. Structural Metrics
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41. Structural Metric
Structural metric
Is a measure of some property directly dependent on the system of
relations between the components of the network: a measure of a
property that depends on the edge set
Graph ←→ Adjacency matrix
⇓
can be expressed as a function of the adjacency matrix
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42. How to, properly, generalize structural metrics to
multiplex networks?
We propose that a structural metric for multiplex networks should
• reduce to the ordinary single-layer metric (if defined) when layers
reduce to one
• be defined for node-layer pairs
• be defined for non-node-aligned multiplex networks
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43. How to, properly, generalize structural metrics to
multiplex networks?
We propose that a structural metric for multiplex networks should
• reduce to the ordinary single-layer metric (if defined) when layers
reduce to one
• be defined for node-layer pairs
• be defined for non-node-aligned multiplex networks
An additional requirement for intensive metrics:
• For a multiplex of identical layers when changing layer has no cost,
an intensive structural metric should take the same value when
measured on the multiplex network and on one layer taken as an
isolated network.
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44. How to, properly, generalize structural metrics to
multiplex networks?
We propose that a structural metric for multiplex networks should
• reduce to the ordinary single-layer metric (if defined) when layers
reduce to one
• be defined for node-layer pairs
• be defined for non-node-aligned multiplex networks
An additional requirement for intensive metrics:
• For a multiplex of identical layers when changing layer has no cost,
an intensive structural metric should take the same value when
measured on the multiplex network and on one layer taken as an
isolated network.
Start from first principles
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45. Structure of triadic relations in multiplex networks
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46. Walks as first principles
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47. In a monoplex network:
define
the local clustering coefficient Cu as the number of 3-cycles
(triangles) tu that start and end at the focal node u divided by the
number of 3-cycles du such that the second step of the cycle occurs
in a complete graph
tu = (A3
)uu, du = (AFA)uu (5)
local clustering coefficient
Cu =
tu
du
(6)
global clustering coefficient
C = u tu
u du
(7)
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48. Multiplex networks contain cycles that can traverse different additional
layers but still have 3 intra-layer steps.
A supra-step consists either of only a single intra-layer step or of a
step that includes both an intra-layer step changing from one layer to
another (either before or after having an intra-layer step)
tM,i = [(AC)3
+ (CA)3
]ii = 2[(AC)3
]ii (8)
dM,i = 2[ACFCAC]ii (9)
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49. Local and Global clustering coefficient for Multiplex
Networks
We can calculate a natural multiplex analog to the usual monoplex local
clustering coefficient for any node i of the supra-graph.
A node u allows an intermediate description for clustering between local
(node-layer pair) and the global (system level) clustering coefficients
c∗,i =
t∗,i
d∗,i
, (10)
C∗,u =
i∈l(u) t∗,i
i∈l(u) d∗,i
, (11)
C∗ = i t∗,i
i d∗,i
, (12)
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50. Layer-decomposed clustering coefficients
Our definition allows to decompose the previous expressions in terms
of the contributions from cycles that traverse exactly one, two, and
three layers (i.e., for m = 1, 2, 3) to give
t∗,ı = t∗,1,i α3
+ t∗,2,i αβ2
+ t∗,3,i β3
, (13)
d∗,i = d∗,1,i α3
+ d∗,2,i αβ2
+ d∗,3,i β3
, (14)
C
(m)
∗ = i t∗,m,i
i d∗,m,i
. (15)
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51. Clustering Coefficients in Erd˝os-R´enyi (ER) Multiplex
Networks
0.2
0.4
0.6
0.8
C∗
AC(1)
M
C(2)
M
C(3)
M
p
B C
0.2 0.4 0.6 0.8
x
0.2
0.4
0.6
0.8
c∗
DcAAA
cAACAC
cACAAC
cACACA
cACACAC
p
0.2 0.4 0.6 0.8
x
E
0.2 0.4 0.6 0.8
x
F
(A, B, C) Global and (D, E, F) local multiplex clustering coefficients in multiplex networks that consist of ER layers.
The markers give the results of simulations of 100-node ER node-aligned multiplex networks that we average over 10
realizations. The solid curves are theoretical approximations. Panels (A, C, D, F) show the results for three-layer
networks, and panels (B, E) show the results for six-layer networks. The ER edge probabilities of the layers are (A, D)
{0.1, 0.1, x}, (B, E) {0.1, 0.1, 0.1, 0.1, x, x}, and (C, F) {0.1, x, 1 − x}
Structure of triadic relations in multiplex networks EC, et al.- New Journal of Physics 2015
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52. Clustering Coefficient in Social Network is Context
Dependent
For each social network we analysed
CM < C
(1)
M and C
(1)
M > C
(2)
M > C
(3)
M
The primary contribution to the triadic structure in multiplex social
networks arises from 3-cycles that stay within a given layer.
Tailor Shop Management Families Bank Tube Airline
CM
orig. 0.319** 0.206** 0.223’ 0.293** 0.056 0.101**
ER 0.186 ± 0.003 0.124 ± 0.001 0.138 ± 0.035 0.195 ± 0.009 0.053 ± 0.011 0.038 ± 0.000
C
(1)
M
orig. 0.406** 0.436** 0.289’ 0.537** 0.013” 0.100**
ER 0.244 ± 0.010 0.196 ± 0.015 0.135 ± 0.066 0.227 ± 0.038 0.053 ± 0.013 0.064 ± 0.001
C
(2)
M
orig. 0.327** 0.273** 0.198 0.349** 0.043* 0.150**
ER 0.191 ± 0.004 0.147 ± 0.002 0.138 ± 0.040 0.203 ± 0.011 0.053 ± 0.020 0.041 ± 0.000
C
(3)
M
orig. 0.288** 0.192** - 0.227** 0.314** 0.086**
ER 0.165 ± 0.004 0.120 ± 0.001 - 0.186 ± 0.010 0.051 ± 0.043 0.037 ± 0.000
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53. Context Matter
Triadic-closure mechanisms in social networks cannot be considered
purely at the aggregated network level.
These mechanisms appear to be more effective inside of layers than
between layers.
0 0.4 0.8 1
cx
0.0
0.2
0.4
0.6
0.8
1.0
cy
c(1)
M,i / c(2)
M,i
c(2)
M,i / c(3)
M,i
c(1)
M,i / c(3)
M,i
0.5 0.0 0.5
cx − cx
0.5
0.0
0.5
cy−cy
A B
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54. • Existing definitions of multiplex clustering coefficients are mostly
ad hoc:difficult to interpret
• Starting from the basic concepts of walks and cycles →
transparent and general definition of transitivity.
• Clustering coefficients always properly normalized
• Reduces to a weighted clustering coefficient of an aggregated
network for particular values of the parameters
• Multiplex clustering coefficients decomposable by construction
• Do not require every node to be present in all layers
It is insufficient to generalize existing diagnostics in a na¨ıve manner.
One must instead construct their generalizations from first principles
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55. Spectra
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56. Important information on the topological properties can be extracted
from the eigenvalues of one of its associated matrix
like spectroscopy for condensed matter physics, graph spectra are
central in the study of the structural properties of a complex network
Eigendecomposition
A = XΛXT
Eigendecomposition
⇓
Topology ⇔ Dynamics (critical phenomena)
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57. The largest eigenvalue of the supra-adjacency
matrix
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58. Largest eigenvalue of the
adjacency matrix associated
to a network
⇒
• a variety of different
dynamical processes
• a variety of structural
properties (the entropy
density per step of the
ensemble of walks in a
network)
Perturbative approach
¯A as a perturbed version of A, C being the perturbation
|| C ||<|| A ||
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59. Dominant Layer
¯λ = λ + ∆λ
Call the layer δ for which λδ = λ the dominant layer
Approximation
∆λ ≈
φT Cφ
φT φ
+
1
λ
φT C2φ
φT φ
φT Cφ
φT φ
= 0
Effective multiplexity
z =
i
ci
(φ)2
i
φT φ
∆λ ≈
z
λ
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60. Structural and Dynamical consequences
The entropy production rate of the ensemble of paths {πij (l)} for
large length l depends only on the dominant layer and the effective
multiplexity
¯h = ln ¯λN ∼ ln(λ +
z
λ
)
Large walks on a multiplex are dominated by walks on the dominant
layer
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61. Structural and Dynamical consequences
0 0.2 0.4 0.6 0.8 1
β/µ
0
0.2
0.4
0.6
0.8
1
ρ
η = 0.25
η = 0.5
η = 1.0
η = 2.0
η = 3.0
0 0.1 0.2 0.3 0.4 0.5
β/µ
0
0.2
0.4
0.6ρ
η = 0.0
1/Λ1
1/Λ2
0 0.2 0.4 0.6 0.8 1
β/µ
0
0.2
0.4
0.6
0.8
1
ρ1
,ρ2
Layer 1
Layer 2
0 0.1 0.2
β/µ
0
0.2
0.4
ρ1
,ρ2
1/Λ2
1/Λ1
η =2.0
b)
a)
Contact-based social contagion in multiplex networks EC,
R.A. Banos, S. Meloni, Y. Moreno - Physical Review E,
2013
The dominant layer sets the
critical point for a contact-based
social contagion process on the
multiplex network
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62. Dimensionality reduction and spectral properties
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63. Interlacing results
Theorem The adjacency eigenvalues of a quotient network interlace the
adjacency eigenvalues of the parent network. The same result applies for
Laplacian eigenvalues.
¯µi ≤ ˜µi ≤ ¯µi+(N−n) (16)
¯µi ≤ ˜µ
(l)
i ≤ ¯µi+(N−m) (17)
An inclusion relation holds for equitable partition.
It holds for the network of layer in the case of node-aligned multiplex
network.
The spectrum of the network of layers IS INCLUDED in the spectrum
of the whole multiplex network
Dimensionality reduction and spectral properties of multilayer networks
R.J. S`anchez-Garc`ıa, E. Cozzo, Y. Moreno - Physical Review E, 2014
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64. The algebraic connectivity
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65. The algebraic connectivity
The algebraic connectivity of a graph G is the second-smallest
eigenvalue of the Laplacian matrix of G
Define the algebraic connectivity of a multiplex as the second-smallest
eigenvalue of its supra-Laplacian matrix
From the interlacing result we know that
¯µ2 ≤ ˜µ
(a)
2 (18)
¯µ2 ≤ m (19)
and
m is always an eigenvalue of the supra-Laplacian
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66. Look for the condition under which ¯µ2 = m holds
Conditions
if ˜µ
(a)
2 < m or µ2 > 1 then ¯µ2 = m,
This result points to a mechanism which can trigger a structural
transition of a multiplex network
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67. Structural organization and transitions
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68. theoretical question: Will critical phenomena behave differently on
multiplex networks with respect to traditional networks?
So far
Theoretical indication that such differences in the critical behaviours
indeed exists
Three different topological scales in a multiplex:
• the individual layers
• the network of layers
• the aggregate network
Quotient graphs give the connection in terms of spectral properties
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69. Eigengap
Gaps in the Laplacian spectrum are known to unveil a number of
structural and dynamical properties of the network related to the
presence of different topological scales in it
Introduce a weight parameter p for the coupling
→tune the relative strength of the coupling with respect to intra-layer
connectivity
¯L =
α
Lα
+ pLC
if node-aligned
¯L =
α
(L(α)
+ p(m − 1)In) − pKm ⊗ In
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70. gap
non-bounded
bounded
From the interlacing result we know:
• n bounded eigenvalues
• mp is always an eigenvalue of the system
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71. Scales separation
gk =
¯µk+1 − ¯µk
¯µk+1
(20)
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72. Aggregate Equivalent Multiplex
Define Aggregate Equivalent
Multiplex: A multiplex with the
same number of layers of the
original one with the aggregate
network in each layer.
{µAEM
k } = {˜µi + ˜µ
(l)
j } (21)
Very smooth transition
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73. • before p∗ structurally dominated by the network of layers
• after p structurally dominated by the aggregate network
• between those two points the system is in an effective multiplex
state
• VN-entropy shows a peak in the central region
• the relative entropy between the parent multiplex and its AEM
varies smoothly with p
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74. Conclusions
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75. • we have introduced the basic formalism to describe multiplex
networks in terms of graphs and associated matrices
• well defined structural metrics that unveils the functioning of the
system and its context dependent nature
• the effect of the coupling on the dynamical and topological
properties of the system
• we have introduced a coarse-grained representation of multiplex
networks in terms of quotient graphs
• exact results on the spectra unveil the interplay between different
topological scales in the system and associated structural
transitions
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76. Multiplex networks:
a challenge and an opportunity of innovation for the science of
complex networks
First challenge:
The need of a common formal language to represent them
An opportunity:
The necessity to reconsider the very foundations of the discipline
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77. A key step for the structure and function hypothesis
⇓
Natural evolution of complex networks science as a mature discipline
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78. The future
Different possibilities:
• the statistical characterization of the Laplacian and adjacency
spectra
• the generalization of more structural metrics in the common
framework settled up by the walk matrix representation
• a deeper understanding of structural transitions in multiplex
networks
especially with regard to the role played by symmetries and
correlations among and across layers
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79. Related Publications
• Stability of Boolean multilevel networks E. Cozzo, A. Arenas, Y. Moreno - Physical Review E, 2012
• Contact-based social contagion in multiplex networks E. Cozzo, R.A. Banos, S. Meloni, Y. Moreno - Physical
Review E, 2013
• Mathematical formulation of multilayer networks Manlio De Domenico, Albert Sol´e-Ribalta, Emanuele Cozzo,
Mikko Kivela, Yamir Moreno, Mason A Porter, Sergio G`o mez, Alex Arenas - Physical Review X, 2013
• Dimensionality reduction and spectral properties of multilayer networks R.J. S`anchez-Garc`ıa, E. Cozzo, Y. Moreno -
Physical Review E, 2014
• Multilayer networks: metrics and spectral properties E. Cozzo, G.F. de Arruda, F.A. Rodrigues, Y. Moreno - arXiv
preprint arXiv:1504.05567, 2015 (in press)
• Structure of triadic relations in multiplex networks E. Cozzo, M. Kivela, M. De Domenico, A. Sol`e-Ribalta, A.
Arenas, S. G`omez, M. A. Porter and Y. Moreno - New Journal of Physics 2015
• On degree-degree correlations in multilayer networks G.F. de Arruda, E. Cozzo, Y. Moreno, F.A. Rodrigues -
Physica D (in press)
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