Approaches from statistical physics to model and study social networks (original) (raw)

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

New approaches to model and study social networks

New Journal of Physics, 2007

We describe and develop three recent novelties in network research which are particularly useful for studying social systems. The first one concerns the discovery of some basic dynamical laws that enable the emergence of the fundamental features observed in social networks, namely the nontrivial clustering properties, the existence of positive degree correlations and the subdivision into communities. To reproduce all these features we describe a simple model of mobile colliding agents, whose collisions define the connections between the agents which are the nodes in the underlying network, and develop some analytical considerations. The second point addresses the particular feature of clustering and its relationship with global network measures, namely with the distribution of the size of cycles in the network. Since in social bipartite networks it is not possible to measure the clustering from standard procedures, we propose an alternative clustering coefficient that can be used to extract an improved normalized cycle distribution in any network. Finally, the third point addresses dynamical processes occurring on networks, namely when studying the propagation of information in them. In particular, we focus on the particular features of gossip propagation which impose some restrictions in the propagation rules. To this end we introduce a quantity, the spread factor, which measures the average maximal fraction of nearest neighbors which get in contact with the gossip, and find the striking result that there is an optimal non-trivial number of friends for which the spread factor is minimized, decreasing the danger of being gossiped.

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.

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.

Clusterization, Frustration and Collectivity in Random Networks

International Journal of Modern Physics C, 2007

We consider the random Erd{\H o}s--R\'enyi network with enhanced clusterization and Ising spins s=pm1s=\pm 1s=pm1 at the network nodes. Mutually linked spins interact with energy JJJ. Magnetic properties of the system as dependent on the clustering coefficient CCC are investigated with the Monte Carlo heat bath algorithm. For J>0J>0J>0 the Curie temperature TcT_cTc increases from 3.9 to 5.5 when CCC increases from almost zero to 0.18. These results deviate only slightly from the mean field theory. For J<0J<0J<0 the spin-glass phase appears below TSGT_{SG}TSG; this temperature decreases with CCC, on the contrary to the mean field calculations. The results are interpreted in terms of social systems.

Assortative model for social networks

Physical Review E, 2004

In this paper we present a new version of a network growth model, generalized in order to describe the behavior of social networks. The case of study considered is the preprint archive at cul.arxiv.org. Each node corresponds to a scientist, and a link is present whenever two authors wrote a paper together. This graph is a nice example of degree-assortative network, that is to say a network where sites with similar degree are connected each other. The model presented is one of the few able to reproduce such behavior, giving some insight on the microscopic dynamics at the basis of the graph structure. PACS numbers: 05.40.-a, 64.60.-1, 87.10.+e Networks [1, 2] are present in different phenomena. The Internet [3, 4] is a graph composed by different computers, connected by cables; the WWW [5, 6] is a graph composed by HTML documents connected by hyperlinks, even social structures [7, 8] can be described as graphs.

Emergence of clustering, correlations, and communities in a social network model

arXiv preprint cond-mat/0309263, 2003

Abstract: We propose a simple model of social network formation that parameterizes the tendency to establish acquaintances by the relative distance in a representative social space. By means of analytical calculations and numerical simulations, we show that the model reproduces the main characteristics of real social networks: non-vanishing clustering coefficient, assortative degree correlations, and the emergence of a hierarchy of communities. Our results highlight the importance of communities in the understanding of ...

Statistical physics of social dynamics

Reviews of Modern Physics, 2009

Statistical physics has proven to be a very fruitful framework to describe phenomena outside the realm of traditional physics. The last years have witnessed the attempt by physicists to study collective phenomena emerging from the interactions of individuals as elementary units in social structures. Here we review the state of the art by focusing on a wide list of topics ranging from opinion, cultural and language dynamics to crowd behavior, hierarchy formation, human dynamics, social spreading. We highlight the connections between these problems and other, more traditional, topics of statistical physics. We also emphasize the comparison of model results with empirical data from social systems. * Electronic address: claudio.castellano@roma1.infn.it † Electronic address: fortunato@isi.it ‡ Electronic address: vittorio.loreto@roma1.infn.it

Correlation clustering on networks

Journal of Physics A: Mathematical and Theoretical, 2009

Random networks with co-existing positive and negative links are studied from the viewpoint of the NP hard correlation clustering problem. The task is to produce a clustering of the vertices which maximizes the number of positive edges within clusters and the number of negative edges between clusters. Simulated annealing, Monte Carlo renormalization and molecular dynamics optimization are used to find the optimal cluster structure. Recently, this problem was studied for globally coupled systems and an interesting phasetransition-like phenomenon was predicted: in the thermodynamic limit the relative size of the largest cluster, r, exhibits a step-like behavior as a function of the density of positive links q (r = 0 if q < 1/2 and r = 1 if q > 1/2). Here we prove that when considering random networks with a constant bond density, the same phase transition is expected. A totally different result emerges however, when networks with a fixed average number of connections per node are considered. In such cases a nontrivial spin-glass-type behavior is found, where the location of the critical point shifts toward q > 1/2 values. The results also suggest that instead of the simple step-like behavior, the r(q) curve has a more complex shape, which depends on the specific topology of the considered network.