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

Based on the misleading expectation that weighted network properties always offer a more complete description than purely topological ones, current economic models of the International Trade Network (ITN) generally aim at explaining local weighted properties, not local binary ones. Here we complement our analysis of the binary projections of the ITN by considering its weighted representations. We show that, unlike the binary case, all possible weighted representations of the ITN (directed/undirected, aggregated/disaggregated) cannot be traced back to local country-specific properties, which are therefore of limited informativeness. Our two papers show that traditional macroeconomic approaches systematically fail to capture the key properties of the ITN. In the binary case, they do not focus on the degree sequence and hence cannot characterize or replicate higher-order properties. In the weighted case, they generally focus on the strength sequence, but the knowledge of the latter is not enough in order to understand or reproduce indirect effects.

Rewiring World Trade. Part II: A Weighted Network Analysis

Since 2007, several contributions have tried to identify early-warning signals of the financial crisis. However, the vast majority of analyses has focused on financial systems and little theoretical work has been done on the economic counterpart. In the present paper we fill this gap and employ the theoretical tools of network theory to shed light on the response of world trade to the financial crisis of 2007 and the economic recession of 2008-2009. We have explored the evolution of the bipartite World Trade Web (WTW) across the years 1995-2010, monitoring the behavior of the system both before and after 2007. Our analysis shows early structural changes in the WTW topology: since 2003, the WTW becomes increasingly compatible with the picture of a network where correlations between countries and products are progressively lost. Moreover, the WTW structural modification can be considered as concluded in 2010, after a seemingly stationary phase of three years. We have also refined our analysis by considering specific subsets of countries and products: the most statistically significant early-warning signals are provided by the most volatile macrosectors, especially when measured on developing countries, suggesting the emerging economies as being the most sensitive ones to the global economic cycles.

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

In all empirical-network studies, the observed properties of economic networks are informative only if compared with a well-defined null model that can quantitatively predict the behavior of such properties in constrained graphs. However, predictions of the available null-model methods can be derived analytically only under assumptions (e.g., sparseness of the network) that are unrealistic for most economic networks like the World Trade Web (WTW). In this paper we study the evolution of the WTW using a recently-proposed family of null network models. The method allows to analytically obtain the expected value of any network statistic across the ensemble of networks that preserve on average some local properties, and are otherwise fully random. We compare expected and observed properties of the WTW in the period 1950-2000, when either the expected number of trade partners or total country trade is kept fixed and equal to observed quantities. We show that, in the binary WTW, node-degree sequences are sufficient to explain higher-order network properties such as disassortativity and clustering-degree correlation, especially in the last part of the sample. Conversely, in the weighted WTW, the observed sequence of total country imports and exports are not sufficient to predict higher-order patterns of the WTW. We discuss some important implications of these findings for international-trade models.

Null Models of Economic Networks: The Case of the World Trade Web

This corrects the article DOI: 10.1103/PhysRevE.92.040802.

Erratum: Estimating topological properties of weighted networks from limited information [Phys. Rev. E 92, 040802(R) (2015)].

Introduction. Network reconstruction is an active field of research. Among the methods proposed so far, some assume that the binary and weighted constraints jointly determine the reconstruction output; others consider the weights estimation step as completely unrelated to the binary one. Amidst the former ones, a special mention is deserved by the Enhanced Configuration Model; the algorithms of the second group, instead, are those iteratively adjusting the link weights on top of some previously-determined topology.
Methods and Results. Here we develop a theoretical framework that provides an analytical, unbiased procedure to estimate the weighted structure of a network, once its topology has been determined, thus extending the Exponential Random Graph (ERG) recipe. Our approach treats the information about the topological structure as a priori; together with the proper weighted constraints, it represents the input of our generalized reconstruction procedure. The probability distribution describing link weights is, then, determined by maximizing the key quantity of our algorithm, i.e. the conditional entropy under a properly-defined set of constraints. This algorithm returns a conditional probability distribution depending on a vector of unknown parameters; in alignment with previous results, their estimation is carried out by considering a generalized likelihood function. In our work, we compare two possible specifications of this framework.
Conclusions. The knowledge of the structure of a financial network gives valuable information for estimating the systemic risk. However, since financial data are typically subject to confidentiality, network reconstruction techniques become necessary to infer both the presence of connections and their intensity. Recently, several "horse races" have been conducted to compare the performance of these methods. Here, we establish a generalised likelihood approach to rigorously define and compare methods for reconstructing weighted networks: the best one is obtained by "dressing" the best-performing, available binary method with an exponential distribution of weights.

Tiziano Squartini

Papers

Rewiring World Trade. Part II: A Weighted Network Analysis

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

Null Models of Economic Networks: The Case of the World Trade Web

Erratum: Estimating topological properties of weighted networks from limited information [Phys. Rev. E 92, 040802(R) (2015)].

Generalized inference for the efficient reconstruction of weighted networks