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Maximum entropy null models of networks come in different flavors that depend on the type of constraints under which entropy is maximized. If the constraints are on degree sequences or distributions, we are dealing with configuration…
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…
It is known that after an appropriate rescaling the maximum degree of the binomial random graph converges in distribution to a Gumbel random variable. The same holds true for the maximum number of common neighbours of a $k$-vertex set, and…
Ensembles of networks arise in various fields where multiple independent networks are observed on the same set of nodes, for example, a collection of brain networks constructed on the same brain regions for different individuals. However,…
The volume of phase space may grow super-exponentially ("explosively") with the number of degrees of freedom for certain types of complex systems such as those encountered in biology and neuroscience, where components interact and create…
This paper presents the foundational ideas for a new way of modeling social aggregation. Traditional approaches have been using network theory, and the theory of random networks. Under that paradigm, every social agent is represented by a…
Graphs are widely used for describing systems made up of many interacting components and for understanding the structure of their interactions. Various statistical models exist, which describe this structure as the result of a combination…
We consider the joint density distribution of the elements of certain random matrix models which are example of globally correlated and asymptotically scale-invariant distributions. It is shown that in their cases, the nonadditive entropy…
A set of vertices in a graph forms a potential maximal clique if there exists a minimal chordal completion in which it is a maximal clique. Potential maximal cliques were first introduced as a key tool to obtain an efficient, though…
The exponential family of random graphs is among the most widely-studied network models. We show that any exponential random graph model may alternatively be viewed as a lattice gas model with a finite Banach space norm. The system may then…
We introduce a method for the theoretical analysis of exponential random graph models. The method is based on a large-deviations approximation to the normalizing constant shown to be consistent using theory developed by Chatterjee and…
A well-known result across information theory, machine learning, and statistical physics shows that the maximum entropy distribution under a mean constraint has an exponential form called the Gibbs-Boltzmann distribution. This is used for…
We apply the Principle of Maximum Entropy to the study of a general class of deterministic fractal sets. The scaling laws peculiar to these objects are accounted for by means of a constraint concerning the average content of information in…
The entropy of a hierarchical network topology in an ensemble of sparse random networks with "hidden variables" associated to its nodes, is the log-likelihood that a given network topology is present in the chosen ensemble.We obtain a…
In this article we introduce a simple tool to derive polynomial upper bounds for the probability of observing unusually large maximal components in some models of random graphs when considered at criticality. Specifically, we apply our…
Exponential-family random graph models (ERGMs) are probabilistic network models that are parametrized by sufficient statistics based on structural (i.e., graph-theoretic) properties. The ergm package for the R statistical computing system…
The evidence lower bound (ELBO) is one of the most central objectives for probabilistic unsupervised learning. For the ELBOs of several generative models and model classes, we here prove convergence to entropy sums. As one result, we…
A drawback of the classic approach for complexity analysis of distributed graph problems is that it mostly informs about the complexity of notorious classes of ``worst case'' graphs. Algorithms that are used to prove a tight (existential)…
Observational entropy -- a quantity that unifies Boltzmann's entropy, Gibbs' entropy, von Neumann's macroscopic entropy, and the diagonal entropy -- has recently been argued to play a key role in a modern formulation of statistical…
Understanding the structural complexity and predictability of complex networks is a central challenge in network science. Although recent studies have revealed a relationship between compression-based entropy and link prediction…