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We improve the estimates of the subgraph probabilities in a random regular graph. Using the improved results, we further improve the limiting distribution of the number of triangles in random regular graphs.

High-dimensional data analysis typically focuses on low-dimensional structure, often to aid interpretation and computational efficiency. Graphical models provide a powerful methodology for learning the conditional independence structure in…

We develop a tensor network-based method for calculating disorder-averaged expectation values in random spin chains without having to explicitly sample over disorder configurations. The algorithm exploits statistical translation invariance…

The problem of efficiently sampling from a set of (undirected, or directed) graphs with a given degree sequence has many applications. One approach to this problem uses a simple Markov chain, which we call the switch chain, to perform the…

We introduce the concept of distance ideals of graphs, which can be regarded as a generalization of the Smith normal form and the spectra of the distance matrix of a graph. We obtain a classification of the graphs with at most one trivial…

In this work, we propose an end-to-end graph network that learns forward and inverse models of particle-based physics using interpretable inductive biases. Physics-informed neural networks are often engineered to solve specific problems…

It appeared recently that the classical random graph model used to represent real-world complex networks does not capture their main properties. Since then, various attempts have been made to provide accurate models. We study here a model…

Choosing a uniformly sampled simple directed graph realization of a degree sequence has many applications, in particular in social networks where self-loops are commonly not allowed. It has been shown in the past that one can perform a…

An undirected graphical model is a joint probability distribution defined on an undirected graph G*, where the vertices in the graph index a collection of random variables and the edges encode conditional independence relationships among…

Biological processes underlying the basic functions of a cell involve complex interactions between genes. From a technical point of view, these interactions can be represented through a graph where genes and their connections are,…

Consider a random graph model with $n$ vertices where each vertex has a vertex-type drawn from some discrete distribution. Suppose that the number of arcs to be placed between each pair of vertex-types is known, and that each arc is placed…

A directed diameter of a directed graph is the maximum possible distance between a pair of vertices, where paths must respect edge orientations, while undirected diameter is the diameter of the undirected graph obtained by symmetrizing the…

This paper studies how to capture dependency graph structures from real data which may not be Gaussian. Starting from marginal loss functions not necessarily derived from probability distributions, we utilize an additive…

Embedding static graphs in low-dimensional vector spaces plays a key role in network analytics and inference, supporting applications like node classification, link prediction, and graph visualization. However, many real-world networks…

In machine learning, graph embedding algorithms seek low-dimensional representations of the input network data, thereby allowing for downstream tasks on compressed encodings. Recently, within the framework of network renormalization,…

This is an overview of the basic tools of nonsmooth analysis which are grounded on nonstandard models of set theory. By way of illustration we give a criterion for an infinitesimally optimal path of a general discrete dynamic system.

The galaxies of nonstandard enlargements of conventionally infinite as well as of transfinite graphs are defined, analyzed, and illustrated by some examples. It is then shown that any such enlargement either has exactly one galaxy, its…

We develop a statistical mechanics approach for random networks with uncorrelated vertices. We construct equilibrium statistical ensembles of such networks and obtain their partition functions and main characteristics. We find simple…

Random graph models play a central role in network analysis. The Chung-Lu model, which connects nodes based on their expected degrees is of particular interest. It is widely used to generate null-graph models with expected degree sequences…

In this paper we consider higher isoperimetric numbers of a (finite directed) graph. In this regard we focus on the $n$th mean isoperimetric constant of a directed graph as the minimum of the mean outgoing normalized flows from a given set…