Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution

Even though power-law or close-to-power-law degree distributions are ubiquitously observed in a great variety of large real networks, the mathematically satisfactory treatment of random power-law graphs satisfying basic statistical requirements of realism is still lacking. These requirements are: sparsity, exchangeability, projectivity, and unbiasedness. The last requirement states that entropy of the graph ensemble must be maximized under the degree distribution constraints. Here we prove that the hypersoft configuration model (HSCM), belonging to the class of random graphs with latent hyperparameters, also known as inhomogeneous random graphs or $W$-random graphs, is an ensemble of random power-law graphs that are sparse, unbiased, and either exchangeable or projective. The proof of their unbiasedness relies on generalized graphons, and on mapping the problem of maximization of the normalized Gibbs entropy of a random graph ensemble, to the graphon entropy maximization problem, showing that the two entropies converge to each other in the large-graph limit

Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution

Even though power-law or close-to-power-law degree distributions are ubiquitously observed in a great variety of large real networks, the mathematically satisfactory treatment of random power-law graphs satisfying basic statistical requirements of realism is still lacking. These requirements are: sparsity, exchangeability, projectivity, and unbiasedness. The last requirement states that entropy of the graph ensemble must be maximized under the degree distribution constraints. Here we prove that the hypersoft configuration model (HSCM), belonging to the class of random graphs with latent hyperparameters, also known as inhomogeneous random graphs or $W$-random graphs, is an ensemble of random power-law graphs that are sparse, unbiased, and either exchangeable or projective. The proof of their unbiasedness relies on generalized graphons, and on mapping the problem of maximization of the normalized Gibbs entropy of a random graph ensemble, to the graphon entropy maximization problem, showing that the two entropies converge to each other in the large-graph limit

Low-temperature behaviour of social and economic networks

Real-world social and economic networks typically display a number of particular topological properties, such as a giant connected component, a broad degree distribution, the small-world property and the presence of communities of densely interconnected nodes. Several models, including ensembles of networks also known in social science as Exponential Random Graphs, have been proposed with the aim of reproducing each of these properties in isolation. Here we define a generalized ensemble of graphs by introducing the concept of graph temperature, controlling the degree of topological optimization of a network. We consider the temperature-dependent version of both existing and novel models and show that all the aforementioned topological properties can be simultaneously understood as the natural outcomes of an optimized, low-temperature topology. We also show that seemingly different graph models, as well as techniques used to extract information from real networks, are all found to be particular low-temperature cases of the same generalized formalism. One such technique allows us to extend our approach to real weighted networks. Our results suggest that a low graph temperature might be an ubiquitous property of real socio-economic networks, placing conditions on the diffusion of information across these systems

High-Dimensional Gaussian Graphical Model Selection: Walk Summability and Local Separation Criterion

We consider the problem of high-dimensional Gaussian graphical model selection. We identify a set of graphs for which an efficient estimation algorithm exists, and this algorithm is based on thresholding of empirical conditional covariances. Under a set of transparent conditions, we establish structural consistency (or sparsistency) for the proposed algorithm, when the number of samples n=omega(J_{min}^{-2} log p), where p is the number of variables and J_{min} is the minimum (absolute) edge potential of the graphical model. The sufficient conditions for sparsistency are based on the notion of walk-summability of the model and the presence of sparse local vertex separators in the underlying graph. We also derive novel non-asymptotic necessary conditions on the number of samples required for sparsistency

The statistical mechanics of networks

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.Comment: 15 pages, 4 figure