Statistical mechanics of complex networks

Network science is the field dedicated to the investigation and analysis of complex systems via their representations as networks. We normally model such networks as graphs: sets of nodes connected by sets of edges and a number of node and edge attributes. This deceptively simple object is the starting point of never-ending complexity, due to its ability to represent almost every facet of reality: chemical interactions, protein pathways inside cells, neural connections inside...

This paper is an extensive survey of literature on complex network communities and clustering. Complex networks describe a widespread variety of systems in nature and society especially systems composed by a large number of highly interconnected dynamical entities. Complex networks like real networks can also have community structure. There are several types of methods and algorithms for detection and identification of communities in complex networks. Several complex networks...

Physicists study a wide variety of phenomena creating new interdisciplinary research fields by applying theories and methods originally developed in physics in order to solve problems in economics, social science, biology, medicine, technology, etc. In their turn, these different branches of science inspire the invention of new concepts in physics. A basic tool of analysis, in such a context, is the mathematical theory of complexity concerned with the study of complex systems...

The study of complex networks has pursued an understanding of macroscopic behavior by focusing on power-laws in microscopic observables. Here, we uncover two universal fundamental physical principles that are at the basis of complex networks generation. These principles together predict the generic emergence of deviations from ideal power laws, which were previously discussed away by reference to the thermodynamic limit. Our approach proposes a paradigm shift in the physics o...

A general scheme for detecting and analyzing topological patterns in large complex networks is presented. In this scheme the network in question is compared with its properly randomized version that preserves some of its low-level topological properties. Statistically significant deviation of any measurable property of a network from this null model likely reflect its design principles and/or evolutionary history. We illustrate this basic scheme on the example of the correlat...

Recent developments in graph theoretic analysis of complex networks have led to deeper understanding of brain networks. Many complex networks show similar macroscopic behaviors despite differences in the microscopic details. Probably two most often observed characteristics of complex networks are scale-free and small-world properties. In this paper, we will explore whether brain networks follow scale-free and small-worldness among other graph theory properties.

Built upon the shoulders of graph theory, the field of complex networks has become a central tool for studying a wide variety of real systems across many fields of research. Represented as a graph, all those systems can be studied using the same analysis methods allowing for their comparison. In this perspective we challenge the extended idea of graph theory as being a data-driven analysis tool. Instead we show that classical graph metrics (e.g., degree, matching index, clust...

We present a novel way to characterize the structure of complex networks by studying the statistical properties of the trajectories of random walks over them. We consider time series corresponding to different properties of the nodes visited by the walkers. We show that the analysis of the fluctuations of these time series allows to define a set of characteristic exponents which capture the local and global organization of a network. This approach provides a way of solving tw...

Networks are ubiquitous in science and have become a focal point for discussion in everyday life. Formal statistical models for the analysis of network data have emerged as a major topic of interest in diverse areas of study, and most of these involve a form of graphical representation. Probability models on graphs date back to 1959. Along with empirical studies in social psychology and sociology from the 1960s, these early works generated an active network community and a su...

The recent discovery of universal principles underlying many complex networks occurring across a wide range of length scales in the biological world has spurred physicists in trying to understand such features using techniques from statistical physics and non-linear dynamics. In this paper, we look at a few examples of biological networks to see how similar questions can come up in very different contexts. We review some of our recent work that looks at how network structure ...