How random are complex networks (original) (raw)

Quantifying randomness in real networks

Nature communications, 2015

Represented as graphs, real networks are intricate combinations of order and disorder. Fixing some of the structural properties of network models to their values observed in real networks, many other properties appear as statistical consequences of these fixed observables, plus randomness in other respects. Here we employ the dk-series, a complete set of basic characteristics of the network structure, to study the statistical dependencies between different network properties. We consider six real networks-the Internet, US airport network, human protein interactions, technosocial web of trust, English word network, and an fMRI map of the human brain-and find that many important local and global structural properties of these networks are closely reproduced by dk-random graphs whose degree distributions, degree correlations and clustering are as in the corresponding real network. We discuss important conceptual, methodological, and practical implications of this evaluation of network ...

The Statistical Physics of Real-World Networks

Statistical physics is the natural framework to model complex networks. In the last twenty years, it has brought novel physical insights on a variety of emergent phenomena, such as self-organisation, scale invariance, mixed distributions and ensemble non-equivalence, which cannot be deduced from the behaviour of the individual constituents. At the same time, thanks to its deep connection with information theory, statistical physics and the principle of maximum entropy have led to the definition of null models reproducing some features of empirical networks, but otherwise as random as possible. We review here the statistical physics approach for complex networks and the null models for the various physical problems, focusing in particular on the analytic frameworks reproducing the local features of the network. We show how these models have been used to detect statistically significant and predictive structural patterns in real-world networks, as well as to reconstruct the network structure in case of incomplete information. We further survey the statistical physics frameworks that reproduce more complex, semi-local network features using Markov chain Monte Carlo sampling, and the models of generalised network structures such as multiplex networks, interacting networks and simplicial complexes. The science of networks has exploded in the Information Age thanks to the unprecedented production and storage of data on basically any human activity. Indeed, a network represents the simplest yet extremely effective way to model a large class of technological, social, economic and biological systems, as a set of entities (nodes) and of interactions (links) among them. These interactions do represent the fundamental degrees of freedom of the network, and can be of different types—undirected or directed, binary or valued (weighted)—depending on the nature of the system and the resolution used to describe it. Notably, most of the networks observed in the real world fall within the domain of complex systems, as they exhibit strong and complicated interaction patterns, and feature collective emergent phenomena that do not follow trivially from the behaviours of the individual entities [1]. For instance, many networks are scale-free [2–6], meaning that the number of links incident to a node (known as the node's degree) is fat-tailed distributed, sometimes following a power-law: most nodes have a few links, but a few nodes (the hubs) have many of them. The same happens for the distribution of the total weight of connections incident to a node (the node's strength) [7, 8]. Similarly, most real-world networks are organised into modules or feature a community structure [9, 10], and they possess high clustering—as nodes tend to create tightly linked groups, but are also small-world [11–13] as the distance (in terms of number of connections) amongst node pairs scales logarithmically with the system size. The observation of these universal features in complex networks has stimulated the development of a unifying mathematical language to model their structure and understand the dynamical processes taking place on them—such as the flow of traffic on the Internet or the spreading of either diseases or information in a population [14–16]. Two different approaches to network modelling can be pursued. The first one consists in identifying one or more microscopic mechanisms driving the formation of the network, and use them to define a dynamic model which can reproduce some of the emergent properties of real systems. The small-world model [11], the preferential attachment model [2], the fitness model [5], the relevance model [17] and many others follow this approach which is akin to kinetic theory. These models can handle only simple microscopic dynamics, and thus while providing good physical insights they need several refinements to give quantitatively accurate predictions. The other possible approach consists in identifying a set of characteristic static properties of real systems, and then building networks having the same properties but otherwise maximally random. This approach is thus akin to statistical mechanics and therefore is based on rigorous probabilistic arguments that can lead to accurate and reliable predictions. The mathematical framework is that of exponential random graphs (ERG), which has been first introduced in the social sciences and statistics [18–26] as a convenient formulation relying on numerical techniques such as Markov chain Monte Carlo algorithms. The interpretation of ERG in physical terms is due to Park and Newman [27], who showed how to derive them from the principle of maximum entropy and the statistical mechanics of Boltzmann and Gibbs. As formulated by Jaynes [28], the variational principle of maximum entropy states that the probability distribution best representing the current state of (knowledge on) a system is the one which maximises the Shannon entropy, subject in principle to any prior information on the system itself. This means making self-consistent inference assum

Statistical mechanics of complex networks

Computing Research Repository, 2001

Complex networks describe a wide range of systems in nature and society. Frequently cited examples include the cell, a network of chemicals linked by chemical reactions, and the Internet, a network of routers and computers connected by physical links. While traditionally these systems have been modeled as random graphs, it is increasingly recognized that the topology and evolution of real networks are governed by robust organizing principles. This article reviews the recent advances in the field of complex networks, focusing on the statistical mechanics of network topology and dynamics. After reviewing the empirical data that motivated the recent interest in networks, the authors discuss the main models and analytical tools, covering random graphs, small-world and scale-free networks, the emerging theory of evolving networks, and the interplay between topology and the network's robustness against failures and attacks.

A random interacting network model for complex networks

Scientific Reports, 2015

We propose a RAndom Interacting Network (RAIN) model to study the interactions between a pair of complex networks. The model involves two major steps: (i) the selection of a pair of nodes, one from each network, based on intra-network node-based characteristics, and (ii) the placement of a link between selected nodes based on the similarity of their relative importance in their respective networks. Node selection is based on a selection fitness function and node linkage is based on a linkage probability defined on the linkage scores of nodes. The model allows us to relate withinnetwork characteristics to between-network structure. We apply the model to the interaction between the USA and Schengen airline transportation networks (ATNs). Our results indicate that two mechanisms: degree-based preferential node selection and degree-assortative link placement are necessary to replicate the observed inter-network degree distributions as well as the observed internetwork assortativity. The RAIN model offers the possibility to test multiple hypotheses regarding the mechanisms underlying network interactions. It can also incorporate complex interaction topologies. Furthermore, the framework of the RAIN model is general and can be potentially adapted to various real-world complex systems. Complex networks provide a powerful approach for investigating real-world systems. It helps us to understand phenomena observed over a wide range of disciplines such as biology, climate, environment, social sciences, technology, and economics 1-4. Many real-world networks, however, are composed of subnetworks called communities which are more closely connected within each other than to the rest of the network 5,6. In other cases, networks of one kind (e.g., a power grid) interact with networks of another kind (e.g., communications systems), leading to a necessary extension of the complex network paradigm that incorporates different types of networks with different types of interactions between them. Initial studies dealing with interdependent networks considered the interrelations between the Internet communication network and the power grid network 7,8. There have been similar studies in other fields since then: such as that of a network of networks in climate 9 where individual isobaric layers of the atmosphere were represented as a complex network with the different isobar networks connected among themselves, epidemic spreading on interconnected networks 10,11 , and that of the European air transportation multiplex network 12 , where each airline comprised a layer with the set of all airports forming a common set of nodes to all layers. For the purposes of this study, all such scenarios involving two or more networks, are examples of interacting networks. We use here the prefix intra-to denote quantities defined within a chosen subnetwork and the prefix inter-to denote quantities defined between subnetworks. While modeling interacting networks most studies use a rather simple interaction structure between the subnetworks even though the topology of each subnetwork is quite complex. This is in contrast to several real-world systems where the interactions between communities are extremely relevant and complex, such as in the brain 13,14. Moreover, a few studies have also indicated the influence of intra-network topology on the inter-network behavior 15-17. Keeping this in mind, we put forward a much more general model of interacting networks, i.e., a RAndom Interacting Network (RAIN) model, which should offer the following features: (i) it should be able to consider any given form of interaction of each subnetwork as well as of the interactions between the subnetworks, irrespective of its complexity, (ii) it should be able to consider the dependence (if any) of inter-network interaction on the

The Structure and Function of Complex Networks

Siam Review, 2003

Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.

Quantitative modeling of degree-degree correlation in complex networks

Physical Review E, 2013

This paper presents an approach to the modeling of degree-degree correlation in complex networks. Thus, a simple function, (k', k), describing specific degree-todegree correlations is considered. The function is well suited to graphically depict assortative and disassortative variations within networks. To quantify degree correlation variations, the joint probability distribution between nodes with arbitrary degrees, P(k', k), is used. Introduction of the end-degree probability function as a basic variable allows using group theory to derive mathematical models for P(k', k). In this form, an expression, representing a family of seven models, is constructed with the needed normalization conditions. Applied to (k', k), this expression predicts a nonuniform distribution of degree correlation in networks, organized in two assortative and two disassortative zones. This structure is actually observed in a set of four modeled, technological, social, and biological networks. A regression study performed on 15 actual networks shows that the model describes quantitatively degree-degree correlation variations.

Statistical mechanics of networks

Physical review. E, Statistical, nonlinear, and soft matter physics, 2004

We study the family of network models derived by requiring the expected properties of a graph ensemble to match a given set of measurements of a real-world network, while maximizing the entropy of the ensemble. Models of this type play the same role in the study of networks as is played by the Boltzmann distribution in classical statistical mechanics; they offer the best prediction of network properties subject to the constraints imposed by a given set of observations. We give exact solutions of models within this class that incorporate arbitrary degree distributions and arbitrary but independent edge probabilities. We also discuss some more complex examples with correlated edges that can be solved approximately or exactly by adapting various familiar methods, including mean-field theory, perturbation theory, and saddle-point expansions.

Randomness and complexity in networks

2007

Abstract: I start by reviewing some basic properties of random graphs. I then consider the role of random walks in complex networks and show how they may be used to explain why so many long tailed distributions are found in real data sets. The key idea is that in many cases the process involves copying of properties of near neighbours in the network and this is a type of short random walk which in turn produce a natural preferential attachment mechanism.

Quantifying structure in networks

The European Physical Journal B, 2010

We investigate exponential families of random graph distributions as a framework for systematic quantification of structure in networks. In this paper we restrict ourselves to undirected unlabeled graphs. For these graphs, the counts of subgraphs with no more than k links are a sufficient statistics for the exponential families of graphs with interactions between at most k links. In this framework we investigate the dependencies between several observables commonly used to quantify structure in networks, such as the degree distribution, cluster and assortativity coefficients.