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In this work we study the dynamics of systems composed of numerous interacting elements interconnected through a random weighted directed graph, such as models of random neural networks. We develop an original theoretical approach based on…
In this paper we study the one dimensional random geometric graph when the location of the nodes are independent and exponentially distributed. We derive exact results and the limit theorems for the connectivity and other properties…
We study regular graphs whose distance-$2$ graph or distance-$1$-or-$2$ graph is strongly regular. We provide a characterization of such graphs $\Gamma$ (among regular graphs with few distinct eigenvalues) in terms of the spectrum and the…
We define a growing model of random graphs. Given a sequence of nonnegative integers $\{d_n\}_{n=0}^\infty$ with the property that $d_i\leq i$, we construct a random graph on countably infinitely many vertices $v_0,v_1\ldots$ by the…
We study the $k$-core of a random (multi)graph on $n$ vertices with a given degree sequence. In our previous paper [Random Structures Algorithms 30 (2007) 50--62] we used properties of empirical distributions of independent random variables…
This paper studies the problem of detecting anomalous graphs using a machine learning model trained on only normal graphs, which has many applications in molecule, biology, and social network data analysis. We present a self-discriminative…
We study limits of the largest connected components (viewed as metric spaces) obtained by critical percolation on uniformly chosen graphs and configuration models with heavy-tailed degrees. For rank-one inhomogeneous random graphs, such…
The vertex-random graphs called proximity catch digraphs (PCDs) have been introduced recently and have applications in pattern recognition and spatial pattern analysis. A PCD is a random directed graph (i.e., digraph) which is constructed…
When studying networks using random graph models, one is sometimes faced with situations where the notion of adjacency between nodes reflects multiple constraints. Traditional random graph models are insufficient to handle such situations.…
We consider site (vertex) percolation on $d$-regular graphs, for both constant-degree and growing-degree cases. We give sufficient, and relatively tight, conditions for the emergence of the ``Erd\H{o}s-R\'enyi component phenomenon" in the…
We give sufficient conditions under which a random graph with a specified degree sequence is symmetric or asymmetric. In the case of bounded degree sequences, our characterisation captures the phase transition of the symmetry of the random…
Following recent work on the VC-dimension of subsets of various pseudorandom graphs, we study the VC-dimension of Hamming graphs, which have proved somewhat resistant to the standard techniques in the literature. Our methods are elementary,…
Let $d\geq 3$ be a fixed integer and $A$ be the adjacency matrix of a random $d$-regular directed or undirected graph on $n$ vertices. We show there exist constants $\mathfrak d>0$, \begin{align*} {\mathbb P}(\text{$A$ is singular in…
The purpose of the present paper is to establish moderate deviation principles for a rather general class of random variables fulfilling certain bounds of the cumulants. We apply a celebrated lemma of the theory of large deviations…
The theory of graphons is an important tool in understanding properties of large networks. We investigate a power-law random graph model and cast it in the graphon framework. The distinctively different structures of the limit graph are…
We introduce and study a novel semi-random multigraph process, described as follows. The process starts with an empty graph on $n$ vertices. In every round of the process, one vertex $v$ of the graph is picked uniformly at random and…
In this note we define and study graph invariants generalizing to higher dimension the maximum degree of a vertex and the vertex-connectivity (our $0$-dimensional cases). These are known to coincide almost surely in any regime for…
Inspired by the prospect of having discretized spaces emerge from random graphs, we construct a collection of simple and explicit exponential random graph models that enjoy, in an appropriate parameter regime, a roughly constant vertex…
In this paper we investigate phenomena of spontaneous emergence or purposeful formation of highly organized structures in networks of related agents. We show that the formation of large organized structures requires exponentially large, in…
The dissertation is related to combinatorial geometry with a strong probabilistic flavor. The main results can be split into three parts. The results of the first part guarantee that each "unit distance graph" in the plane has an induced…