We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Social Network Analysis

The structure and dynamics of multilayer networks

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

Multilayer Networks

where A represents the magnetic vector potential, is an integral of the hydromagnetic equations. This -integral made it possible to formulate a variational principle for the force-free magnetic fields. The integral expresses the fact that motions cannot transform a given field in an entirely arbitrary different field, if the conductivity of the medium isconsidered infinite. In this paper we shall show that the full set of hydromagnetic equations admit five more integrals, besides the energy integral, if dissipative processes are absent. These integrals, as we shall presently verify, are I2 =fbHvdV, (2)

Proceedings of the National Academy of Sciences

European Commission. Community Research and Development Information Service (Project GROWTHCOM 611272)

Detecting early signs of the 2007–2008 crisis in the world trade

Disentangling spatial and non-spatial effects in real complex networks

This paper represents a contribution to the study of the brain functional connectivity from the perspective of complex networks theory. More specifically, we apply graph theoretical analyses to provide evidence of the modular structure of the mouse brain and to shed light on its hierarchical organization. We propose a novel percolation analysis and we apply our approach to the analysis of a resting-state functional MRI data set from 41 mice. This approach reveals a robust hierarchical structure of modules persistent across different subjects. Importantly, we test this approach against a statistical benchmark (or null model) which constrains only the distributions of empirical correlations. Our results unambiguously show that the hierarchical character of the mouse brain modular structure is not trivially encoded into this lower-order constraint. Finally, we investigate the modular structure of the mouse brain by computing the Minimal Spanning Forest, a technique that identifies subnetworks characterized by the strongest internal correlations. This approach represents a faster alternative to other community detection methods and provides a means to rank modules on the basis of the strength of their internal edges.

/pdf/hierarchical-organization-of-functional-connectivity-in-the-4wa7utpzh8.pdf

Hierarchical organization of functional connectivity in the mouse brain: a complex network approach

Exponential Random Graph Models (ERGMs) have gained increasing popularity over the years Rooted into statistical physics, the ERGMs framework has been successfully employed for reconstructing networks, detecting statistically significant patterns in graphs, counting networked configurations with given properties From a technical point of view, the ERGMs workflow is defined by two subsequent optimization steps: the first one concerns the maximization of Shannon entropy and leads to identify the functional form of the ensemble probability distribution that is maximally non-committal with respect to the missing information; the second one concerns the maximization of the likelihood function induced by this probability distribution and leads to its numerical determination This second step translates into the resolution of a system of $O(N)$ non-linear, coupled equations (with $N$ being the total number of nodes of the network under analysis), a problem that is affected by three main issues, ie accuracy, speed and scalability The present paper aims at addressing these problems by comparing the performance of three algorithms (ie Newton's method, a quasi-Newton method and a recently-proposed fixed-point recipe) in solving several ERGMs, defined by binary and weighted constraints in both a directed and an undirected fashion While Newton's method performs best for relatively little networks, the fixed-point recipe is to be preferred when large configurations are considered, as it ensures convergence to the solution within seconds for networks with hundreds of thousands of nodes (eg the Internet, Bitcoin) We attach to the paper a Python code implementing the three aforementioned algorithms on all the ERGMs considered in the present work

Fast and scalable likelihood maximization for Exponential Random Graph Models

The importance of identifying the presence of mesoscale structures in complex networks can be hardly overestimated. So far, much attention has been devoted to the detection of communities, bipartite and core-periphery structures on binary networks; such an effort has led to the definition of a unified framework based upon the score function called surprise, i.e. a p-value that can be assigned to any given partition of nodes, on both undirected and directed networks. Here, we aim at making a step further, by extending the entire framework to the weighted case: after reviewing the application of the surprise-based formalism to the detection of binary mesoscale structures, we present a suitable generalization of it for detecting weighted mesoscale structures, a topic that is still largely under-explored. To this aim, we analyze four variants of the surprise; from a technical point of view, this amounts at employing four variants of the hypergeometric distribution: the binomial one for the detection of binary communities, the multinomial one for the detection of binary `bimodular' structures and their negative counterparts for the detection of communities and `bimodular' structures on weighted networks. On top of that, we define two `enhanced' variants of surprise, able to encode both binary and weighted constraints and whose definition rests upon two suitable generalizations of the hypergeometric probability mass function. As a result, we present a general, statistically grounded approach to detect mesoscale structures on networks via a unified, suprise-based framework. To illustrate the performance of our methods, we report the results of their application to several real-world networks, including social, economic, financial and ecological ones. Moreover, we attach to the paper a Python code implementing all versions of the surprise considered in the present work.

Tiziano Squartini

Papers

Detecting early signs of the 2007–2008 crisis in the world trade

Disentangling spatial and non-spatial effects in real complex networks

Hierarchical organization of functional connectivity in the mouse brain: a complex network approach

Fast and scalable likelihood maximization for Exponential Random Graph Models

Detecting mesoscale structures by surprise