Complete Graphs and Bipartite Graphs in a Random Graph

Abstract

Random graphs, or more precisely the Erd&#x0151;s-R&#x00E9;nyi random graph model, is a major tool for modeling complex networks. The most distinctive property of a random graph is inarguably the threshold phenomenon. In this paper, we study the threshold phenomenon for the existence of a complete graph and distribution of complete bipartite graphs in random graphs using Markov&#x2019;s inequality and indicator functions.We review basic theorems in graph theory and random graphs. A graph is denoted by $G(W,E)$, where the elements of W are the vertices of the graph G and the elements of E are its edges. A random graph is a graph where vertices or edges or both are determined by some random procedure. In the 1980&#x2019;s, Bollob&#x00E1;s showed that every non-trivial monotone increasing property in a random graph has a threshold. Graphs of a size less than this threshold have a low probability to have the property, but graphs with a size larger than this threshold are almost guaranteed to have the property. This is known as a phase transition.For such random graphs denoted by $G(n, p)$, where n is the number of vertices of the graph G and p is the probability of an edge between any two vertices is present, we present a proof of the threshold probability that a random graph contains a complete graph, K<inf>d</inf>, which occurs at $p=n^{-\frac{2}{d-1}}$. A calculation of the probability distribution for a random graph to contain a complete bipartite graph $K_{r, s}$ as an induced subgraph is also presented which exhibits a global maximum at $p=\frac{2 r s}{r(r-1)+s(s-1)+2 r s}$.