The maximum and minimum degrees of random bipartite multigraphs

Discussiones Mathematicae Graph Theory, 1996

The asymptotic distributions of the number of vertices of a given degree in random graphs, where the probabilities of edges may not be the same, are given. Using the method of Poisson convergence, distributions in a general and particular cases (complete, almost regular and bipartite graphs) are obtained.

2012

Theory of Computing Systems, 2014

Consider a random bipartite multigraph G with n left nodes and m ≥ n ≥ 2 right nodes. Each left node x has dx ≥ 1 random right neighbors. The average left degree ∆ is fixed, ∆ ≥ 2. We ask whether for the probability that G has a left-perfect matching it is advantageous not to fix dx for each left node x but rather choose it at random according to some (cleverly chosen) distribution. We show the following, provided that the degrees of the left nodes are independent: If ∆ is an integer then it is optimal to use a fixed degree of ∆ for all left nodes. If ∆ is non-integral then an optimal degree-distribution has the property that each left node x has two possible degrees, ∆ and ∆ , with probability px and 1 -px, respectively, where px is from the closed interval [0, 1] and the average over all px equals ∆ -∆. Furthermore, if n = c • m and ∆ > 2 is constant, then each distribution of the left degrees that meets the conditions above determines the same threshold c * (∆) that has the following property as n goes to infinity: If c < c * (∆) then there exists a left-perfect matching with high probability. If c > c * (∆) then there exists no left-perfect matching with high probability. The threshold c * (∆) is the same as the known threshold for offline k-ary cuckoo hashing for integral or non-integral k = ∆.

Arxiv preprint math.PR/0502580

In this paper we derive results concerning the connected components and the diameter of random graphs with an arbitrary iid degree sequence. We study these properties primarily, but not exclusively, when the tail of the degree ...

Discrete Mathematics, 2006

In this paper we consider the degree of a typical vertex in two models of random intersection graphs introduced in [E. Godehardt, J. Jaworski, Two models of random intersection graphs for classification, in: M. Schwaiger, O. ], the active and passive models. The active models are those for which vertices are assigned a random subset of a list of objects and two vertices are made adjacent when their subsets intersect. We prove sufficient conditions for vertex degree to be asymptotically Poisson as well as closely related necessary conditions. We also consider the passive model of intersection graphs, in which objects are vertices and two objects are made adjacent if there is at least one vertex in the corresponding active model "containing" both objects. We prove a necessary condition for vertex degree to be asymptotically Poisson for passive intersection graphs.

Physical Review E, 2018

Bipartite (two-mode) networks are important in the analysis of social and economic systems as they explicitly show conceptual links between different types of entities. However, applications of such networks often work with a projected (one-mode) version of the original bipartite network. The topology of the projected network, and the dynamics that take place on it, are highly dependent on the degree distributions of the two different node types from the original bipartite structure. To date, the interaction between the degree distributions of bipartite networks and their one-mode projections is well understood for only a few cases, or for networks that satisfy a restrictive set of assumptions. Here we show a broader analysis in order to fill the gap left by previous studies. We use the formalism of generating functions to prove that the degree distributions of both node types in the original bipartite network affect the degree distribution in the projected version. To support our analysis, we simulate several types of synthetic bipartite networks using a configuration model where node degrees are assigned from specific probability distributions, ranging from peaked to heavy-tailed distributions. Our findings show that when projecting a bipartite network onto a particular set of nodes, the degree distribution for the resulting one-mode network follows the distribution of the nodes being projected on to, but only so long as the degree distribution for the opposite set of nodes does not have a heavier tail. Furthermore, we show that bipartite degree distributions are not the only feature driving topology formation of projected networks, in contrast to what is commonly described in the literature.

Extremes, 2005

We study random graphs with an i.i.d. degree sequence of which the tail of the distribution function F is regularly varying with exponent τ ∈ (1, 2). Thus, the degrees have infinite mean. Such random graphs can serve as models for complex networks where degree power laws are observed.

The inductive dimension dim(G) of a finite undirected graph G is a rational number defined inductively as 1 plus the arithmetic mean of the dimensions of the unit spheres dim(S(x)) at vertices x primed by the requirement that the empty graph has dimension −1. We look at the distribution of the random variable dim on the Erdös-Rényi probability space G(n, p), where each of the n(n − 1)/2 edges appears independently with probability 0 ≤ p ≤ 1. We show that the average dimension dn(p) = Ep,n[dim] is a computable polynomial of degree n(n − 1)/2 in p. The explicit formulas allow experimentally to explore limiting laws for the dimension of large graphs. In this context of random graph geometry, we mention explicit formulas for the expectation Ep,n[χ] of the Euler characteristic χ, considered as a random variable on G(n, p). We look experimentally at the statistics of curvature K(v) and local dimension dim(v) = 1 + dim(S(v)) which satisfy χ(G) = v∈V K(v) and dim(G) = 1 |V | v∈V dim(v). We also look at the signature functions f (p) = Ep[dim], g(p) = Ep[χ] and matrix values functions Av,w(p) = Covp[dim(v), dim(w)], Bv,w(p) = Cov[K(v), K(w)] on the probability space G(p) of all subgraphs of a host graph G = (V, E) with the same vertex set V , where each edge is turned on with probability p.

2011

A two-rowed array α n = a 1 a 2 . . . a n b 1 b 2 . . . b n is said to be in lexicographic order if a k ≤ a k+1

2015

This thesis provides an introduction to the fundamentals of random graph theory. The study starts introduces the two fundamental building blocks of random graph theory, namely discrete probability and graph theory. The study starts by introducing relevant concepts probability commonly used in random graph theory-these include concentration inequalities such as Chebyshev's inequality and Chernoff's inequality. Moreover we proceed by introducing central concepts in graph theory, which will underpin the later discussion. In particular we provide results such as Mycielski's construction of a family of triangle-free graphs with high chromatic number and results in Ramsey theory. Next we introduce the concept of a random graph and present two of the most famous proofs in graph theory using the theory random graphs. These include the proof of the fact that there are graphs with arbitrarily high girth and chromatic number, and a bound on the Ramsey number R(k, k). Finally we conclude by introducing the notion of a threshold function for a monotone graph property and we present proofs for the threhold functions of certain properties.