Cooperation is a very common, yet not fully-understood phenomenon in natural and human
systems. The introduction of a network structure within the population is known to affect the
outcome of cooperative dynamics, as described by the Game Theory paradigm, allowing for the
survival of cooperation in adverse scenarios. Recently, the introduction of multilayered networks
has yet again modified the expectations for the outcome of the Prisoner’s Dilemma game, compared
to the monoplex case. However, much remains to be studied regarding other games in the plane
of social dilemmas on multiplex, as well as the unexplored microscopic underpinnings of it. In
this paper, we systematically and carefully study the evolution and outcome of all four games
in the S − T plane (Prisoner’s Dilemma, Stag-Hung, Snow Drift and Harmony) on multiplex, as
a function of the number of layers. More importantly, we find some remarkable and previously
unknown features in the microscopic organization of the strategies, that are at the root of the
important differences between cooperative dynamics in monoplex and multiplex. Specifically, we
find that in the stationary state, there are individuals that play the same strategy in all layers
(coherent), and others that don’t (incoherent). This second group of players is responsible for the
surprising fact of a non full-cooperation in the Harmony Game on multiplex, never observed before,
as well as a higher-than-expected survival of cooperation in some regions of the other three social
dilemmas.

Human mobility in a city represents a fascinating complex system that combines social interactions, daily constraints and random explorations. New collections of data that capture human mobility not only help us to understand their underlying patterns but also to design intelligent systems. Bringing us the opportunity to reduce traffic and to develop other applications that make cities more adaptable to human needs. In this paper, we propose an adaptive routing strategy which accounts for individual constraints to recommend personalized routes and, at the same time, for constraints imposed by the collectivity as a whole. Using big data sets recently released during the Telecom Italia Big Data Challenge, we show that our algorithm allows us to reduce the overall traffic in a smart city thanks to synergetic effects, with the participation of individuals in the system, playing a crucial role.

Human mobility and social structure are at the basis of disease spreading. Disease containment strategies are usually devised from coarse-grained assumptions about human mobility. Cellular networks data, however, provides finer-grained information, not only about how people move, but also about how they communicate.

In this paper, using cellular network data, we analyze the behavior of a large number of individuals in Ivory Coast. We model mobility and communication between individual by means of an interconnected multiplex structure where each node represents the population in a geographic area (i.e. a \textit{sous-pr\'efecture}, a third-level administrative region). We present a model that describes how diseases circulate around the country as people move between regions. We extend the model with a concurrent process of relevant information spreading. This process corresponds to people disseminating disease prevention information, e.g. hygiene practises, vaccination campaign notices and other, within their social network. Thus, this process interferes with the epidemic. We then evaluate how restricting the mobility or using an adverse information spreading process affects the epidemic. We find that restricting mobility does not delay the occurrence of an endemic state and that an information campaign might be an effective countermeasure.

Assessing the navigability of interconnected networks (transporting information, people, or goods) under eventual random failures is of utmost importance to design and protect critical infrastructures. Random walks are a good proxy to determine this navigability, specifically the coverage time of random walks, which is a measure of the dynamical functionality of the network. Here, we introduce the theoretical tools required to describe random walks in interconnected networks accounting for structure and dynamics inherent to real systems. We develop an analytical approach for the covering time of random walks in interconnected networks and compare it with extensive Monte Carlo simulations. Generally speaking, interconnected networks are more resilient to random failures than their individual layers per se, and we are able to quantify this effect. As an application, which we illustrate by considering the public transport of London, we show how the efficiency in exploring the multiplex critically depends on layers’ topology, interconnection strengths, and walk strategy. Our findings are corroborated by data-driven simulations, where the empirical distribution of check-ins and checks-out is considered and passengers travel along fastest paths in a network affected by real disruptions. These findings are fundamental for further development of searching and navigability strategies in real interconnected systems.

### Multilayer networks

**Journal of Complex Networks, Vol. 2, No. 3: 203-271 - DOI: 10.1093/comnet/cnu016 - 2014**

M. Kivela, A. Arenas, M Barthelemy, J.P. Gleeson, Y. Moreno and M. Porter

In most natural and engineered systems, a set of entities interact with each other in complicated patterns that can encompass multiple types of relationships, change in time and include other types of complications. Such systems include multiple subsystems and layers of connectivity, and it is important to take such ‘multilayer’ features into account to try to improve our understanding of complex systems. Consequently, it is necessary to generalize ‘traditional’ network theory by developing (and validating) a framework and associated tools to study multilayer systems in a comprehensive fashion. The origins of such efforts date back several decades and arose in multiple disciplines, and now the study of multilayer networks has become one of the most important directions in network science. In this paper, we discuss the history of multilayer networks (and related concepts) and review the exploding body of work on such networks. To unify the disparate terminology in the large body of recent work, we discuss a general framework for multilayer networks, construct a dictionary of terminology to relate the numerous existing concepts to each other and provide a thorough discussion that compares, contrasts and translates between related notions such as multilayer networks, multiplex networks, interdependent networks, networks of networks and many others. We also survey and discuss existing data sets that can be represented as multilayer networks. We review attempts to generalize single-layer-network diagnostics to multilayer networks. We also discuss the rapidly expanding research on multilayer-network models and notions like community structure, connected components, tensor decompositions and various types of dynamical processes on multilayer networks. We conclude with a summary and an outlook.

Epidemic-like spreading processes on top of multilayered interconnected complex networks reveal a rich
phase diagram of intertwined competition effects. A recent study by the authors [Granell et al. Phys. Rev.
Lett. 111, 128701 (2013)] presented the analysis of the interrelation between two processes accounting for
the spreading of an epidemics, and the spreading of information awareness to prevent its infection, on top of
multiplex networks. The results in the case in which awareness implies total immunization to the disease,
revealed the existence of a metacritical point at which the critical onset of the epidemics starts depending on
the reaching of the awareness process. Here we present a full analysis of these critical properties in the more
general scenario where the awareness spreading does not imply total immunization, and where infection does
not imply immediate awareness of it. We find the critical relation between both competing processes for a
wide spectrum of parameters representing the interaction between them. We also analyze the consequences of
a massive broadcast of awareness (mass media) on the final outcome of the epidemic incidence. Importantly
enough, the mass media makes the metacritical point to disappear. The results reveal that the main finding i.e.
existence of a metacritical point, is rooted on the competition principle and holds for a large set of scenarios.

We study explosive synchronization, a phenomenon characterized by first-order phase transitions between
incoherent and synchronized states in networks of coupled oscillators. While explosive synchronization has been
the subject of many recent studies, in each case strong conditions on either the heterogeneity of the network,
its link weights, or its initial construction are imposed to engineer a first-order phase transition. This raises
the question of how robust explosive synchronization is in view of more realistic structural and dynamical
properties. Here we show that explosive synchronization can be induced in mildly heterogeneous networks by
the addition of quenched disorder to the oscillators’ frequencies, demonstrating that it is not only robust to, but
moreover promoted by, this natural mechanism. We support these findings with numerical and analytical results,
presenting simulations of a real neural network as well as a self-consistency theory used to study synthetic
networks.

The ability to understand and eventually predict the emergence of information and
activation cascades in social networks is core to complex socio-technical systems research. However,
the complexity of social interactions makes this a challenging enterprise. Previous works on
cascade models assume that the emergence of this collective phenomenon is related to the activity
observed in the local neighborhood of individuals, but do not consider what determines the
willingness to spread information in a time-varying process. Here we present a mechanistic model
that accounts for the temporal evolution of the individual state in a simplified setup. We model
the activity of the individuals as a complex network of interacting integrate-and-fire oscillators.
The model reproduces the statistical characteristics of the cascades in real systems, and provides
a framework to study the time evolution of cascades in a state-dependent activity scenario.

We present the analysis of the interrelation between two processes accounting for the spreading of an
epidemic, and the information awareness to prevent its infection, on top of multiplex networks. This
scenario is representative of an epidemic process spreading on a network of persistent real contacts, and
a cyclic information awareness process diffusing in the network of virtual social contacts between the
same individuals. The topology corresponds to a multiplex network where two diffusive processes are
interacting affecting each other. The analysis using a microscopic Markov chain approach reveals the
phase diagram of the incidence of the epidemics and allows us to capture the evolution of the epidemic
threshold depending on the topological structure of the multiplex and the interrelation with the awareness
process. Interestingly, the critical point for the onset of the epidemics has a critical value (metacritical
point) defined by the awareness dynamics and the topology of the virtual network, from which the onset
increases and the epidemics incidence decreases

### Diffusion dynamics on multiplex networks

**Physical Review Letters, 110, 028701 - DOI: http://dx.doi.org/10.1103/PhysRevLett.110.028701 - 2013**

S. Gomez, A. Diaz-Guilera, J. Gomez-Gardenes, C.J. Perez-Vicente, Y. Moreno and A. Arenas

We study the time scales associated with diffusion processes that take place on multiplex networks, i.e., on a set of networks linked through interconnected layers. To this end, we propose the construction of a supra-Laplacian matrix, which consists of a dimensional lifting of the Laplacian matrix of each layer of the multiplex network. We use perturbative analysis to reveal analytically the structure of eigenvectors and eigenvalues of the complete network in terms of the spectral properties of the individual layers. The spectrum of the supra-Laplacian allows us to understand the physics of diffusionlike processes on top of multiplex networks.