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

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, i.e. accuracy, speed and scalability. The present paper aims at addressing these problems by comparing the performance of three algorithms (i.e. 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 (e.g. 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.

/pdf/fast-and-scalable-likelihood-maximization-for-exponential-5ppn1s9j0c.pdf

Fast and scalable likelihood maximization for Exponential Random Graph Models with local constraints

Non-equivalence between the canonical and the microcanonical ensemble has been shown to arise for models defined by an extensive number of constraints (e.g. the Configuration Model). Here, we focus on the framework induced by entropy maximization and study the extent to which ensemble non-equivalence affects the description length of binary, canonical, and microcanonical models. Specifically, we consider the description length induced by the Normalized Maximum Likelihood (NML), which consists of two terms, i.e. a model log-likelihood and its complexity: while the effects of ensemble non-equivalence on the log-likelihood term are well understood, its effects on the complexity term have not been systematically studied yet. Here, we find that i) microcanonical models are always more complex than their canonical counterparts and ii) the difference between the canonical and the microcanonical description length is strongly influenced by the degree of non-equivalence, a result suggesting that non-equivalence should be taken into account when selecting models. Finally, we compare the NML-based approach to model selection with the Bayesian one induced by Jeffreys prior, showing that the two cannot be reconciled when non-equivalence holds.

Description length of canonical and microcanonical models

In the study of economic networks, econometric approaches interpret the traditional Gravity Model specification as the expected link weight coming from a probability distribution whose functional form can be chosen arbitrarily, while statistical-physics approaches construct maximum-entropy distributions of weighted graphs, constrained to satisfy a given set of measurable network properties. In a recent, companion paper, we integrated the two approaches and applied them to the World Trade Web, i.e. the network of international trade among world countries. While the companion paper dealt only with discrete-valued link weights, the present paper extends the theoretical framework to continuous-valued link weights. In particular, we construct two broad classes of maximum-entropy models, namely the integrated and the conditional ones, defined by different criteria to derive and combine the probabilistic rules for placing links and loading them with weights. In the integrated models, both rules follow from a single, constrained optimization of the continuous Kullback-Leibler divergence; in the conditional models, the two rules are disentangled and the functional form of the weight distribution follows from a conditional, optimization procedure. After deriving the general functional form of the two classes, we turn each of them into a proper family of econometric models via a suitable identification of the econometric function relating the corresponding, expected link weights to macroeconomic factors. After testing the two classes of models on World Trade Web data, we discuss their strengths and weaknesses. 

Reconciling econometrics with continuous maximum-entropy network models

In this chapter we describe the core method that will be used throughout the rest of the book, i.e. the construction of a constrained maximum-entropy ensemble of networks. This procedure requires the definition of the entropy of a network ensemble, the specification of structural properties to be enforced as constraints, the calculation of the resulting maximum-entropy probability of network configurations, and the maximization of the likelihood, given the empirical values of the enforced constraints. We describe this procedure explicitly, after giving some general motivations. In particular, we discuss the crucial importance of enforcing local constraints that preserve the (empirical) heterogeneity of node properties. The maximum-entropy method not only generates the exact probabilities of occurrence of any graph in the ensemble, but also the expectation values and the higher moments of any quantity of interest. Moreover, unlike most alternative approaches, it is applicable to networks that are either binary or weighted, either undirected or directed, either sparse or dense, either tree-like or clustered, either small or large. We also discuss various likelihood-based statistical criteria to rank competing models resulting from different choices of the constraints. These criteria are useful to assess the informativeness of different network properties.

Maximum-Entropy Ensembles of Graphs

One problem in the study of the Italian electric energy supply scenario is determining the ability of
photovoltaic production to provide a constant and stable energy background over space and time. Knowing
how the photovoltaic energy produced in a given node diffuses on the power grid is of crucial importance. A
smart grid able to face peaks of load must be designed. Approached here from a complex systems point of
view, the network of energy supply might be represented by a graph in which nodes are Italian municipalities
and edges cross the administrative boundaries from a municipality to its first neighbours. Using datasets
from ISTAT, GSE and ENEA, the node production and attraction of photovoltaic energy have been estimated
with high accuracy. The attraction index was built using demographic data, in accordance with medium per
capita energy consumption data. Moreover, the energy produced in each node could be determined using
data on the installed photovoltaic power and on local solar radiation. The available energy on each node was
calculated by running a distributive model assuming that the energy produced in one node which diffuses to
its first neighbours is proportional to the attraction index of the latter. Therefore the available energy at each
node is the sum of many contributions, coming from topological paths involving all the other nodes across
the network. The availability of cross temporal data on the photovoltaic power installed on the Italian territory
also make it possible to understand the evolution of the available photovoltaic energy landscape over time.

Tiziano Squartini

Papers

Fast and scalable likelihood maximization for Exponential Random Graph Models with local constraints

Description length of canonical and microcanonical models

Reconciling econometrics with continuous maximum-entropy network models

Maximum-Entropy Ensembles of Graphs

Complex Networks Approach to the Italian Photovoltaic Energy Distribution System