Chapter 1 : A tipping point in the structural formation of interconnected networks
Alex Arenas and Filippo Radicchi
Abstract: The interaction substrate of many natural and synthetic systems
is well represented by a complex mesh of networks where information, people
and energy flows. These networks are interconnected with each other, and
present structural and dynamical features different from those observed in
isolated networks. While examples of such dissimilar properties are becom-
ing more abundant, for example diffusion, robustness and competition, it is
not yet clear where these differences are rooted. Here we show that the com-
position of independent networks into an interconnected network of networks
undergoes a structurally sharp transition, a tipping point, as the interconnec-
tions are formed. Depending on the relative importance of inter- and intra-
layer connections, we find that the entire interconnected system can be tuned
between two regimes: in one regime, the various layers are structurally de-
coupled and they act essentially as independent entities; in the other regime,
strong structural correlation arise, and network layers are indistinguishable
i.e. the whole system behaves as a single-level network. We analytically show
that the transition between the two regimes is discontinuous even for finite
size networks. Thus, any real-world interconnected system is potentially at
risk of abrupt changes in its structure, which may manifest new dynamical
properties.
Chapter 2: Multilayer networks: metrics and spectral properties
Emanuele Cozzo, Guilherme Ferraz de Arruda, Francisco A.Rodrigues and Yamir Moreno
Abstract: Multilayer networks represent systems in which there are several
topological levels each one representing one kind of interaction or interde-
pendency between the systems’ elements. These networks have attracted a
lot of attention recently because their study allows considering different dy-
namical modes concurrently. Here, we revise the main concepts and tools
developed up to date. Specifically, we focus on several metrics for multilayer
network characterization as well as on the spectral properties of the system,
which ultimately enable for the dynamical characterization of several criti-
cal phenomena. The theoretical framework is also applied for description of
real-world multilayer systems.
Chapter 3: An ensemble perspective on multi-layer networks
Nicolas Wider, Antonios Garas, Ingo Scholtes, Frank Schweitzer
Abstract: We study properties of multi-layered, interconnected networks
from an ensemble perspective, i.e. we analyze ensembles of multi-layer net-
works that share similar aggregate characteristics. The ensemble perspective
allows us to cope with uncertainties in the specific realization of the multi-
layer network. We focus on a diffusion process that evolves on a multi-layer
network, and we analyze how the speed of diffusion depends on the aggre-
gate characteristics of both intra- and inter-layer connectivity. We distinguish
cases where either the network topologies of single layers or the coupling be-
tween layers is given. To better understand their impact on the diffusion
process we vary the link density within or across layers. We show how one
can estimate the expected properties of multi-layer networks using a mean-
field approach that can be defined based on partial information available on
real-world multi-layer complex systems. We use a block-matrix model that
represents the layers to construct transition matrices of random walkers on
multi-layer networks. By analyzing statistical ensembles of these matrices we
show that a precise decomposition of the spectrum of the transition matrix
according to the topology of links inside and across layers is possible. Fur-
ther, we quantify and explore conditions on the link topology that allow to
Chapter 4: Interconnecting networks: the role of connector links
J. M. Buldu, R. Sevilla-Escoboza, J. Aguirre, D. Papo and R. Gutierrez
Abstract: Recently, some studies have started showing how global struc-
tural properties or dynamical processes such as synchronization, robustness,
cooperation, transport or epidemic spreading change dramatically when con-
sidering a network of networks, as opposed to networks in isolation. In this
Chapter we examine the effects that the particular way in which networks get
connected exerts on each of the individual networks. We describe how choos-
ing the adequate connector links between networks may promote or hinder
different structural and dynamical properties of a particular network. We
show how different connecting strategies have consequences on the distribu-
tion of network centrality, population dynamics or spreading processes. The
importance of designing adequate connection strategies is illustrated with
examples of social and biological systems. Finally, we discuss how this new
approach can be translated to other dynamical processes, such as synchro-
nization in an ensemble of networks.
Chapter 5: Vulnerability of interdependent networks and networks of networks
Michael M. Danziger, Louis M. Shekhtman, Amir Bashan, Yehiel Berezin and Shlomo Havlin
Abstract: Networks interact with one another in a variety of ways. Even
though increased connectivity between networks would tend to make the sys-
tem more robust, if dependencies exist between networks, these systems are
highly vulnerable to random failure or attack. Damage in one network causes
damage in another. This leads to cascading failures which amplify the origi-
nal damage and can rapidly lead to complete system collapse.
Understanding the system characteristics that lead to cascading failures
and support their continued propagation is an important step in develop-
ing more robust systems and mitigation strategies. Recently, a number of
important results have been obtained regarding the robustness of systems
composed of random, clustered and spatially embedded networks.
Here we review the recent advances on the role that connectivity and
dependency links play in the robustness of networks of networks. We fur-
ther discuss the dynamics of cascading failures on interdependent networks,
including cascade lifetime predictions and explanations of the topological
properties which drive the cascade.
Chapter 6: A unified approach to percolation processes on multiplex networks
G. J. Baxter, D. Cellai, S. N. Dorogovtsev, A. V. Goltsev and J.F. F. Mendes
Abstract: Many real complex systems cannot be represented by a single
network, but due to multiple sub-systems and types of interactions, must
be represented as a multiplex network. This is a set of nodes which exist
in several layers, with each layer having its own kind of edges, represented
by different colors. An important fundamental structural feature of networks
is their resilience to damage, the percolation transition. Generalization of
these concepts to multiplex networks requires careful definition of what we
mean by connected clusters. We consider two different definitions. One, a
rigorous generalization of the single-layer definition leads to a strong non-
local rule, and results in a dramatic change in the response of the system to
damage. The giant component collapses discontinuously in a hybrid transition
characterized by avalanches of diverging mean size. We also consider another
definition, which imposes weaker conditions on percolation and allows local
calculation, and also leads to different sized giant components depending on
whether we consider an activation or pruning process. This ’weak’ process
exhibits both continuous and discontinuous transitions.
Chapter 7: How much interconnected should networks be for cooperation to thrive?
Zhen Wang, Attila Szolnoki, Matjaz Perc
Abstract: While the consensus is that interconnectivity between networks
does promote cooperation by means of organizational complexity and en-
hanced reciprocity that is out of reach on isolated networks, we here address
the question just how much interconnectivity there should be. The more the
better according to naive intuition, yet we show that in fact only an interme-
diate density of sufficiently strong interactions between networks is optimal
for the evolution of cooperation. This is due to an intricate interplay between
the heterogeneity that causes an asymmetric strategy flow because of the
additional links between the networks, and the independent formation of co-
operative patterns on each individual network. Presented results are robust
to variations of the strategy updating rule, the topology of interconnected
networks, and the governing social dilemma, and thus indicate a high de-
gree of universality. We also outline future directions for research based on
coevolutionary games and survey existing work.
Chapter 8: The Cacophony of Interconnected Networks
V. H. P. Louzada, N. A. M. Araujo, J. S. Andrade Jr, and H. J. Herrmann
Abstract: The harmony of an orchestra emerges from the individual effort of
musicians towards mutual synchronization of their tempi. When the orchestra
is split between two concert halls communicating via Internet, a time delay is
imposed which might hinder synchronization. We present this type of system
as two interconnected networks of oscillators with a time delay and analyze its
dynamics as a function of the couplings and communication lag. We describe
a breathing synchronization regime, namely, for a wide range of parameters,
two groups emerge in the orchestra within the same concert hall playing at
different tempi. Each group has a mirror in the other hall, one group is in
phase and the other in anti-phase with their mirrors. For strong couplings,
a phase shift between halls might occur. The implications of our findings on
other interconnected systems are also discussed.
Chapter 9: Several multiplexes in the same city: The role of socioeconomic differences in urban mobility
Laura Lotero, Alessio Cadillo, Rafael Hurtado and Jesus Gomez-Gardenes
Abstract: In this work we analyze the architecture of real urban mobility net-
works from the multiplex perspective. In particular, based on empirical data
about the mobility patterns in the cities of Bogot ́a and Medell ́ın, each city
is represented by six multiplex networks, each one representing the origin-
destination trips performed by a subset of the population corresponding to a
particular socioeconomic status. The nodes of each multiplex are the differ-
ent urban locations whereas links represent the existence of a trip from one
node (origin) to another (destination). On the other hand, the different layers
of each multiplex correspond to the different existing transportation modes.
By exploiting the characterization of multiplex transportation networks com-
bining different transportation modes, we aim at characterizing the mobility
patterns of each subset of the population. Our results show that the socioe-
conomic characteristics of the population have an extraordinary impact in
the layer organization of these multiplex systems.
Chapter 10: The weak core and the structure of elites in social multiplex networks
Bernat Corominas-Murtra and Stefan Thurner
Abstract: Recent approaches on elite identification highlighted the impor-
tant role of intermediaries, by means of a new definition of the core of a mul-
tiplex network, the generalised K-core. This newly introduced core subgraph
crucially incorporates those individuals who, in spite of not being very con-
nected, maintain the cohesiveness and plasticity of the core. Interestingly,
has been shown that the performance on elite identification of the generalised
K-core is sensibly better that the standard K-core. Here we go further: Over
a multiplex social system, we isolate the community structure of the gener-
alised K-core and we identify the weakly connected regions acting as bridges
between core communities, ensuring the cohesiveness and connectivity ofcore region.
This gluing region is the Weak core of the multiplex system.test the suitability
of our method on data from the society of 420.000 players of the Massive Multiplayer
Online Game Pardus. Results show that the generalised K-core displays a clearly
identifiable community structure and that the weak core gluing the core communities
shows very low connectivity and clustering. Nonetheless, despite its low connectivity,
the weak core formsunique, cohesive structure. In addition, we find that members
populating the weak core have the best scores on social performance, when compared
to other elements of the generalised K-core. The weak core provides a new angle on
understanding the social structure of elites, highlighting those subgroups
of individuals whose role is to glue different communities in the core.
Chapter 11: Interbank markets and multiplex networks: centrality measures and statistical null models
Leonardo Bargigli, Giovanni di Iasio, Luigi Infante, Fabrizio Lillo and Federico Pierobon
Abstract: The interbank market is considered one of the most impor-
tant channels of contagion. Its network representation, where banks and
claims/obligations are represented by nodes and links (respectively), has re-
ceived a lot of attention in the recent theoretical and empirical literature,
for assessing systemic risk and identifying systematically important financial
institutions. Different types of links, for example in terms of maturity and
collateralization of the claim/obligation, can be established between finan-
cial institutions. Therefore a natural representation of the interbank struc-
ture which takes into account more features of the market, is a multiplex,
where each layer is associated with a type of link. In this paper we review
the empirical structure of the multiplex and the theoretical consequences of
this representation. We also investigate the betweenness and eigenvector cen-
trality of a bank in the network, comparing its centrality properties across
different layers and with Maximum Entropy null models.
Chapter 12: The financial system as a nexus of interconnected networks
Stefano Battiston, Guido Caldarelli, Marco D’Errico
Abstract: In this Chapter, we describe the phenomenology of multilevel fi-
nancial networks. Network analysis represents a useful tool for the analysis
of financial systems, allowing, in particular, for a better understanding of the
mechanics of systemic distress. However, the level of complexity reached by
the financial system, coupled with the linkages arising to and from other eco-
nomic sectors, calls for a more integrated approach that takes into account a
whole series of networks. In this Chapter, we therefore describe the financial
systems as a nexus of interconnected networks. By reviewing selected theo-
retical and empirical works and describing two methodological extensions for
DebtRank, we show different arguments in favor of the adoption of a broader
view of the network approach to finance.
