Complex Networks from Classical to Quantum

Recent progress in applying complex network theory to problems in quantum information has resulted in a beneficial crossover. Complex network methods have successfully been applied to transport and entanglement models while information physics is setting the stage for a theory of complex systems with quantum information-inspired methods. Novel quantum induced effects have been predicted in random graphs---where edges represent entangled links---and quantum computer algorithms have been proposed to offer enhancement for several network problems. Here we review the results at the cutting edge, pinpointing the similarities and the differences found at the intersection of these two fields.Comment: 12 pages, 4 figures, REVTeX 4-1, accepted versio

Mathematical Formulation of Multi-Layer Networks

A network representation is useful for describing the structure of a large variety of complex systems. However, most real and engineered systems have multiple subsystems and layers of connectivity, and the data produced by such systems is very rich. Achieving a deep understanding of such systems necessitates generalizing "traditional" network theory, and the newfound deluge of data now makes it possible to test increasingly general frameworks for the study of networks. In particular, although adjacency matrices are useful to describe traditional single-layer networks, such a representation is insufficient for the analysis and description of multiplex and time-dependent networks. One must therefore develop a more general mathematical framework to cope with the challenges posed by multi-layer complex systems. In this paper, we introduce a tensorial framework to study multi-layer networks, and we discuss the generalization of several important network descriptors and dynamical processes --including degree centrality, clustering coefficients, eigenvector centrality, modularity, Von Neumann entropy, and diffusion-- for this framework. We examine the impact of different choices in constructing these generalizations, and we illustrate how to obtain known results for the special cases of single-layer and multiplex networks. Our tensorial approach will be helpful for tackling pressing problems in multi-layer complex systems, such as inferring who is influencing whom (and by which media) in multichannel social networks and developing routing techniques for multimodal transportation systems.Comment: 15 pages, 5 figure

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

Walk entropies on graphs

Entropies based on walks on graphs and on their line-graphs are defined. They are based on the summation over diagonal and off-diagonal elements of the thermal Green’s function of a graph also known as the communicability. The walk entropies are strongly related to the walk regularity of graphs and line-graphs. They are not biased by the graph size and have significantly better correlation with the inverse participation ratio of the eigenmodes of the adjacency matrix than other graph entropies. The temperature dependence of the walk entropies is also discussed. In particular, the walk entropy of graphs is shown to be non-monotonic for regular but non-walk-regular graphs in contrast to non-regular graphs