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Quantifying the complexity of large graphs requires measures that extend beyond predefined structural features and scale efficiently with graph size. This work adopts a generative perspective, modeling large networks as exchangeable graphs…
The study of very large graphs is a prominent theme in modern-day mathematics. In this paper we develop a rigorous foundation for studying the space of finite labelled graphs and their limits. These limiting objects are naturally countable…
The paper systematically classifies rings based on the dominant metric dimensions (Ddim) of their associated CZDG, establishing consequential bounds for the Ddim of these compressed zero-divisor graphs. The authors investigate the interplay…
Graph signals are functions of the underlying graph. When the edge-weight between a pair of nodes is high, the corresponding signals generally have a higher correlation. As a result, the signals can be represented in terms of a graph-based…
We derive representation theorems for exchangeable distributions on finite and infinite graphs using elementary arguments based on geometric and graph-theoretic concepts. Our results elucidate some of the key differences, and their…
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…
We study the random graph G_{n,\lambda/n} conditioned on the event that all vertex degrees lie in some given subset S of the non-negative integers. Subject to a certain hypothesis on S, the empirical distribution of the vertex degrees is…
We develop a new robust technique to deduce variance principles for non-integrable discrete systems. To illustrate this technique, we show the existence of a variational principle for graph homomorphisms from $\Z^m$ to a $d$-regular tree.…
Random intersection graphs have received much interest and been used in diverse applications. They are naturally induced in modeling secure sensor networks under random key predistribution schemes, as well as in modeling the topologies of…
A rigidity theory is developed for the Euclidean and non-Euclidean placements of countably infinite simple graphs in R^d with respect to the classical l^p norms, for d>1 and 1<p<\infty. Generalisations are obtained for the Laman and…
Random walk on changing graphs is considered. For sequences of finite graphs increasing monotonically towards a limiting infinite graph, we establish transition probability upper bounds. It yields sufficient transience criteria for simple…
We introduce the matching measure of a finite graph as the uniform distribution on the roots of the matching polynomial of the graph. We analyze the asymptotic behavior of the matching measure for graph sequences with bounded degree. A…
We study characteristics which might distinguish two-graphs by introducing different numerical measures on the collection of graphs on $n$ vertices. Two conjectures are stated, one using these numerical measures and the other using the deck…
A paradigm that was successfully applied in the study of both pure and algorithmic problems in graph theory can be colloquially summarized as stating that "any graph is close to being the disjoint union of expanders". Our goal in this paper…
The hard core model in statistical physics is a probability distribution on independent sets in a graph in which the weight of any independent set I is proportional to lambda^(|I|), where lambda > 0 is the vertex activity. We show that…
Given a graph $G$ and $p\in [0,1]$, the random subgraph $G_p$ is obtained by retaining each edge of $G$ independently with probability $p$. We show that for every $\epsilon>0$, there exists a constant $C>0$ such that the following holds.…
A random algebraic graph is defined by a group $G$ with a uniform distribution over it and a connection $\sigma:G\longrightarrow[0,1]$ with expectation $p,$ satisfying $\sigma(g)=\sigma(g^{-1}).$ The random graph…
We investigate how to find generic and globally rigid realizations of graphs in $\mathbb{R}^d$ based on elementary geometric observations. Our arguments lead to new proofs of a combinatorial characterization of the global rigidity of graphs…
We generalize subgraph densities, arising in dense graph limit theory, to Markov spaces (symmetric measures on the square of a standard Borel space). More generally, we define an analogue of the set of homomorphisms in the form of a measure…
Let $G=(V,E)$ be a $d$-regular graph on $n$ vertices and let $\mu_0$ be a probability measure on $V$. The act of moving to a randomly chosen neighbor leads to a sequence of probability measures supported on $V$ given by $\mu_{k+1} = A…