In the last 15 years, statistical physics has been a very successful framework to model complex networks. On the theoretical side, this approach has brought novel insights into a variety of physical phenomena, such as self-organisation, scale invariance, emergence of mixed distributions and ensemble non-equivalence, that display unconventional features on heterogeneous networks. At the same time, thanks to their deep connection with information theory, statistical physics and the principle of maximum entropy have led to the definition of null models for networks reproducing some features of real-world systems, but otherwise as random as possible. We review here the statistical physics approach and the various null models for complex networks, focusing in particular on the analytic frameworks reproducing the local network features. We then 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 models that reproduce more complex, semi-local network features using Markov chain Monte Carlo sampling, as well as the models of generalised network structures such as multiplex networks, interacting networks and simplicial complexes. arXiv:1810.05095v2 [physics.soc-ph]

We show that to explain the growth of the citation network by preferential attachment (PA), one has to accept that individual nodes exhibit heterogeneous fitness values that decay with time. While previous PAbased models assumed either heterogeneity or decay in isolation, we propose a simple analytically treatable model that combines these two factors. Depending on the input assumptions, the resulting degree distribution shows an exponential, log-normal or power-law decay, which makes the model an apt candidate for modeling a wide range of real systems.Over the years, models with preferential attachment (PA) were independently proposed to explain the distribution of the number of species in a genus [1], the power-law distribution of the number of citations received by scientific papers [2], and the number of links pointing to World Wide Web (WWW) pages [3]. A theoretical description of this class of processes and the observation that they generally lead to power-law distributions are due to Simon [4]. Notably, the application of PA to WWW data by Barabási and Albert helped to initiate the lively field of complex networks [5]. Their network model, which stands at the center of attention of this work, was much studied and generalized to include effects such as presence of purely random connections [6], nonlinear dependence on the degree [7], node fitness [8], and others ([9], Chap. 8).Despite its success in providing a common roof for many theoretical models and empirical data sets, preferential attachment is still little developed to take into account the temporal effects of network growth. For example, it predicts a strong relation between a node's age and its degree. While such first-mover advantage [10] plays a fundamental role for the emergence of scale-free topologies in the model, it is a rather unrealistic feature for several real systems (e.g., it is entirely absent in the WWW [11] and significant deviations are found in citation data [10,12]). This motivates us to study a model of a growing network where a broad degree distribution does not result from strong time bias in the system. To this end we assign fitness to each node and assume that this fitness decays with time-we refer it as relevance henceforth. Instead of simply classifying the vertices as active or inactive, as done in [13,14], we use real data to investigate the relevance distribution and decay therein and build a model where decaying and heterogeneous relevance are combined.Models with decaying fitness values (''aging'') were shown to produce narrow degree distributions (except for very slow decay) [15] and widely distributed fitness values were shown to produce extremely broad distributions or even a condensation phenomenon where a single node attracts a macroscopic fraction of all links [16]. We show that when these two effects act together, they produce various classes of behavior, many of which are compatible with structures observed in real data sets.Before specifying a model and attempting to solve it, we turn to data to provide sup...

We address a fundamental problem that is systematically encountered when modeling real-world complex systems of societal relevance: the limitedness of the information available. In the case of economic and financial networks, privacy issues severely limit the information that can be accessed and, as a consequence, the possibility of correctly estimating the resilience of these systems to events such as financial shocks, crises and cascade failures. Here we present an innovative method to reconstruct the structure of such partially-accessible systems, based on the knowledge of intrinsic node-specific properties and of the number of connections of only a limited subset of nodes. This information is used to calibrate an inference procedure based on fundamental concepts derived from statistical physics, which allows to generate ensembles of directed weighted networks intended to represent the real system—so that the real network properties can be estimated as their average values within the ensemble. We test the method both on synthetic and empirical networks, focusing on the properties that are commonly used to measure systemic risk. Indeed, the method shows a remarkable robustness with respect to the limitedness of the information available, thus representing a valuable tool for gaining insights on privacy-protected economic and financial systems.

Common asset holding by financial institutions (portfolio overlap) is nowadays regarded as an important channel for financial contagion with the potential to trigger fire sales and severe losses at the systemic level. We propose a method to assess the statistical significance of the overlap between heterogeneously diversified portfolios, which we use to build a validated network of financial institutions where links indicate potential contagion channels. The method is implemented on a historical database of institutional holdings ranging from 1999 to the end of 2013, but can be applied to any bipartite network. We find that the proportion of validated links (i.e. of significant overlaps) increased steadily before the 2007–2008 financial crisis and reached a maximum when the crisis occurred. We argue that the nature of this measure implies that systemic risk from fire sales liquidation was maximal at that time. After a sharp drop in 2008, systemic risk resumed its growth in 2009, with a notable acceleration in 2013. We finally show that market trends tend to be amplified in the portfolios identified by the algorithm, such that it is possible to have an informative signal about institutions that are about to suffer (enjoy) the most significant losses (gains).