The physics of spreading processes in multilayer networks

The study of networks plays a crucial role in investigating the structure, dynamics, and function of a wide variety of complex systems in myriad disciplines. Despite the success of traditional network analysis, standard networks provide a limited representation of complex systems, which often include different types of relationships (i.e., "multiplexity") among their constituent components and/or multiple interacting subsystems. Such structural complexity has a significant effect on both dynamics and function. Throwing away or aggregating available structural information can generate misleading results and be a major obstacle towards attempts to understand complex systems. The recent "multilayer" approach for modeling networked systems explicitly allows the incorporation of multiplexity and other features of realistic systems. On one hand, it allows one to couple different structural relationships by encoding them in a convenient mathematical object. On the other hand, it also allows one to couple different dynamical processes on top of such interconnected structures. The resulting framework plays a crucial role in helping achieve a thorough, accurate understanding of complex systems. The study of multilayer networks has also revealed new physical phenomena that remain hidden when using ordinary graphs, the traditional network representation. Here we survey progress towards attaining a deeper understanding of spreading processes on multilayer networks, and we highlight some of the physical phenomena related to spreading processes that emerge from multilayer structure.Comment: 25 pages, 4 figure

Opinion-Based Centrality in Multiplex Networks: A Convex Optimization Approach

Most people simultaneously belong to several distinct social networks, in which their relations can be different. They have opinions about certain topics, which they share and spread on these networks, and are influenced by the opinions of other persons. In this paper, we build upon this observation to propose a new nodal centrality measure for multiplex networks. Our measure, called Opinion centrality, is based on a stochastic model representing opinion propagation dynamics in such a network. We formulate an optimization problem consisting in maximizing the opinion of the whole network when controlling an external influence able to affect each node individually. We find a mathematical closed form of this problem, and use its solution to derive our centrality measure. According to the opinion centrality, the more a node is worth investing external influence, and the more it is central. We perform an empirical study of the proposed centrality over a toy network, as well as a collection of real-world networks. Our measure is generally negatively correlated with existing multiplex centrality measures, and highlights different types of nodes, accordingly to its definition

Multilayer Networks in a Nutshell

Complex systems are characterized by many interacting units that give rise to emergent behavior. A particularly advantageous way to study these systems is through the analysis of the networks that encode the interactions among the system's constituents. During the last two decades, network science has provided many insights in natural, social, biological and technological systems. However, real systems are more often than not interconnected, with many interdependencies that are not properly captured by single layer networks. To account for this source of complexity, a more general framework, in which different networks evolve or interact with each other, is needed. These are known as multilayer networks. Here we provide an overview of the basic methodology used to describe multilayer systems as well as of some representative dynamical processes that take place on top of them. We round off the review with a summary of several applications in diverse fields of science.Comment: 16 pages and 3 figures. Submitted for publicatio

Interacting social processes on interconnected networks

We propose and study a model for the interplay between two different dynamical processes --one for opinion formation and the other for decision making-- on two interconnected networks $A$ and $B$. The opinion dynamics on network $A$ corresponds to that of the M-model, where the state of each agent can take one of four possible values ($S=-2,-1,1,2$), describing its level of agreement on a given issue. The likelihood to become an extremist ($S=\pm 2$) or a moderate ($S=\pm 1$) is controlled by a reinforcement parameter $r \ge 0$. The decision making dynamics on network $B$ is akin to that of the Abrams-Strogatz model, where agents can be either in favor ($S=+1$) or against ($S=-1$) the issue. The probability that an agent changes its state is proportional to the fraction of neighbors that hold the opposite state raised to a power $\beta$. Starting from a polarized case scenario in which all agents of network $A$ hold positive orientations while all agents of network $B$ have a negative orientation, we explore the conditions under which one of the dynamics prevails over the other, imposing its initial orientation. We find that, for a given value of $\beta$, the two-network system reaches a consensus in the positive state (initial state of network $A$) when the reinforcement overcomes a crossover value $r^*(\beta)$, while a negative consensus happens for $r<r^*(\beta)$. In the $r-\beta$ phase space, the system displays a transition at a critical threshold $\beta_c$, from a coexistence of both orientations for $\beta<\beta_c$ to a dominance of one orientation for $\beta>\beta_c$. We develop an analytical mean-field approach that gives an insight into these regimes and shows that both dynamics are equivalent along the crossover line $(r^*,\beta^*)$.Comment: 25 pages, 6 figure

Spreading processes in Multilayer Networks

Several systems can be modeled as sets of interconnected networks or networks with multiple types of connections, here generally called multilayer networks. Spreading processes such as information propagation among users of an online social networks, or the diffusion of pathogens among individuals through their contact network, are fundamental phenomena occurring in these networks. However, while information diffusion in single networks has received considerable attention from various disciplines for over a decade, spreading processes in multilayer networks is still a young research area presenting many challenging research issues. In this paper we review the main models, results and applications of multilayer spreading processes and discuss some promising research directions.Comment: 21 pages, 3 figures, 4 table