Interdisciplinary and physics challenges of network theory

Abstract

-Network theory has unveiled the underlying structure of complex systems such as the Internet or the biological networks in the cell. It has identified universal properties of complex networks, and the interplay between their structure and dynamics. After almost twenty years of the field, new challenges lie ahead. These challenges concern the multilayer structure of most of the networks, the formulation of a network geometry and topology, and the development of a quantum theory of networks. Making progress on these aspects of network theory can open new venues to address interdisciplinary and physics challenges including progress on brain dynamics, new insights into quantum technologies, and quantum gravity.Introduction. -Network theory has emerged almost twenty years ago, as a new field for characterizing interacting complex systems, such as the Internet, the biological networks of the cell, and social networks. It is now time to reflect on the maturity of the field, indicating the main results obtained so far and the big challenges that lie ahead.Initially, the physics perspective, in particular the statistical mechanics approach, has dominated the field of Network Theory [1][2][3][4][5][6][7]. This point of view has played a central role to characterize the universal properties of the structure of complex networks. It has been found that despite the diversity of complex networks, ranging from the Internet to the protein interaction networks in the cell, most networks obey universal properties: they are small-world [9], they are scale-free [8], and they have a non trivial community structure [7]. Moreover, over the years, special attention has been addressed to the interplay between network structure and dynamics [10,11]. In fact phase diagrams of critical phenomena and dynamical processes change drastically when the dynamics is defined on complex networks. Complex networks are responsible for significant changes in the critical behaviour of percolation, Ising model, random walks, epidemic spreading, synchronization, and controllability of networks [10][11][12][13].The need to characterize complex systems, to extract relevant information from them, and to understand how dynamical processes are affected by network structure, has never been more severe than in the XXI century when we are witnessing a Big Data explosion in social sciences, information and communication technologies and in biology.

The Kuramoto model in complex networks

“…In parallel with the advances in the study of the traditional Kuramoto model, over almost the last two decades one has witnessed the rapid development of the new field of network science [5], which not only brought new insights into the characterization of real networks, but also introduced a new dimension in the study of dynamical systems [6,7]. Researchers were puzzled by the question of how the connectivity pattern between elements in a network can influence the performance of dynamical processes, such as epidemic spreading, percolation, diffusion, opinion formation and synchronization.…”

Extracting information from multiplex networks

“…The vast majority of complex interacting systems, from social networks to the brain and the biological networks of the cell, are multilayer networks [1][2][3] . Multilayer networks are formed by several networks (layers) describing interactions of different nature and connotation.…”

A dynamical approach to identify vertices' centrality in complex networks

“…Consider a network of N vertices joined in pairs by E links, which represent the interaction between vertices, for example, the acquaintance or collaborations between individual in social networks. In graph description, the network can (1) where, i is the subgraph of vertex i's nearest neighbors, k i is the connectivity of vertex i and λ is the positive coupling strength between vertices. Note that vertices are represented by their corresponding oscillators in the following part of our paper.…”

“…Recent decades have witnessed a vigorous development of the network science, which is an interdisciplinary academic field to understand the behavior of natural, social and technical systems under the fundamental framework of complex networks [1]. Empirical analysis shows that many real complex networks, where vertices represent the elementary units of a given system and links describe the interactions between units [2,3], exhibit some nontrivial properties, such as the heterogeneous nature of vertices (e.g., the power-law distribution of vertex's connectivity) and the community structure (also called clustering or module).…”