Largest sparse subgraphs of random graphs (original) (raw)

Subgraphs of Dense Random Graphs with Specified Degrees

Combinatorics, Probability and Computing, 2011

Let d = (d 1 , d 2 , . . . , d n ) be a vector of non-negative integers with even sum. We prove some basic facts about the structure of a random graph with degree sequence d, including the probability of a given subgraph or induced subgraph.

The Probability of Non-Existence of a Subgraph in a Moderately Sparse Random Graph

Combinatorics, Probability and Computing

We develop a general procedure that finds recursions for statistics counting isomorphic copies of a graph G0 in the common random graph models calG{\cal G}calG(n,m) and calG{\cal G}calG(n,p). Our results apply when the average degrees of the random graphs are below the threshold at which each edge is included in a copy of G0. This extends an argument given earlier by the second author for G0=K3 with a more restricted range of average degree. For all strictly balanced subgraphs G0, our results give much information on the distribution of the number of copies of G0 that are not in large ‘clusters’ of copies. The probability that a random graph in calG{\cal G}calG(n,p) has no copies of G0 is shown to be given asymptotically by the exponential of a power series in n and p, over a fairly wide range of p. A corresponding result is also given for calG{\cal G}calG(n,m), which gives an asymptotic formula for the number of graphs with n vertices, m edges and no copies of G0, for the applicable range of m. An example...

On graphs with subgraphs having large independence numbers

Journal of Graph Theory, 2007

Let G be a graph on n vertices in which every induced subgraph on s = log 3 n vertices has an independent set of size at least t = log n. What is the largest q = q(n) so that every such G must contain an independent set of size at least q ? This is one of several related questions raised by Erdős and Hajnal. We show that q(n) = Θ(log 2 n/ log log n), investigate the more general problem obtained by changing the parameters s and t, and discuss the connection to a related Ramsey-type problem.

Solitary subgraphs of random graphs

Discrete Mathematics, 2000

Let G be a graph and let G(n; p) be the binomial random graph with n vertices and edge probability p. We consider copies of G in G(n; p), vertex disjoint from all other such copies. For a strictly balanced graph G, initially, every copy of G in G(n; p) is solitary. Suen (Random Struct. Algorithms 1 (1990) 231-242) established a second (disappearance) threshold for a subclass of strictly balanced graphs. In this paper we extend his result to a more general case.

Random subgraphs of finite graphs: I. The scaling window under the triangle condition

Random Structures and Algorithms, 2005

We study random subgraphs of an arbitrary finite connected transitive graph G obtained by independently deleting edges with probability 1 − p. Let V be the number of vertices in G, and let Ω be their degree. We define the critical threshold p c = p c (G, λ) to be the value of p for which the expected cluster size of a fixed vertex attains the value λV 1/3 , where λ is fixed and positive. We show that for any such model, there is a phase transition at p c analogous to the phase transition for the random graph, provided that a quantity called the triangle diagram is sufficiently small at the threshold p c . In particular, we show that the largest cluster inside a scaling window of size |p − p c | = Θ(Ω −1 V −1/3 ) is of size Θ(V 2/3 ), while below this scaling window, it is much smaller, of order O( −2 log(V 3 )), with = Ω(p c − p). We also obtain an upper bound O(Ω(p − p c )V ) for the expected size of the largest cluster above the window. In addition, we define and analyze the percolation probability above the window and show that it is of order Θ(Ω(p − p c )). Among the models for which the triangle diagram is small enough to allow us to draw these conclusions are the random graph, the n-cube and certain Hamming cubes, as well as the spread-out n-dimensional torus for n > 6.

On the number of induced subgraphs of a random graph

Discrete Mathematics, 1987

Let G,, be a graph selected at random from the set of all labeled graphs of order n. We show that for many a type, the number of induced subgraphs of that type in Gn is asymptotically normally distributed as n tends to infinity.

The importance sampling technique for understanding rare events in Erdős–Rényi random graphs

Electronic Journal of Probability, 2015

In dense Erdős-Rényi random graphs, we are interested in the events where large numbers of a given subgraphs occur. The mean behaviour of subgraph counts is known, and only recently were the related large deviations results discovered. Consequently, it is natural to ask, what is the probability of an Erdős-Rényi graph containing an excessively large number of a given subgraph? Using the large deviation principle, we study an importance sampling scheme as a method to numerically compute the small probabilities of large triangle counts occurring within Erdős-Rényi graphs. The exponential tilt used in the importance sampling scheme comes from a generalized class of exponential random graphs. Asymptotic optimality, a measure of the efficiency of the importance sampling scheme, is achieved by the special choice of exponential random graph that is indistinguishable from the Erdős-Rényi graph conditioned to have many triangles. We show how this choice can be made for the conditioned Erdős-Rényi graphs both in the replica symmetric phase and also in parts of the replica breaking phase. Equally interestingly, we also show that the exponential tilt suggested directly by the large deviation principle does not always yield an optimal scheme.

A critical point for random graphs with a given degree sequence

Random Structures & Algorithms, 1995

Given a sequence of nonnegative real numbers λ0, λ1… which sum to 1, we consider random graphs having approximately λi n vertices of degree i. Essentially, we show that if Σ i(i ‐ 2)λi > 0, then such graphs almost surely have a giant component, while if Σ i(i ‐2)λ. < 0, then almost surely all components in such graphs are small. We can apply these results to Gn,p,Gn.M, and other well‐known models of random graphs. There are also applications related to the chromatic number of sparse random graphs.

Concentration and regularization of random graphs

Random Structures & Algorithms, 2017

This paper studies how close random graphs are typically to their expectations. We interpret this question through the concentration of the adjacency and Laplacian matrices in the spectral norm. We study inhomogeneous Erdös-Rényi random graphs on n vertices, where edges form independently and possibly with different probabilities pij. Sparse random graphs whose expected degrees are o(log n) fail to concentrate; the obstruction is caused by vertices with abnormally high and low degrees. We show that concentration can be restored if we regularize the degrees of such vertices, and one can do this in various ways. As an example, let us reweight or remove enough edges to make all degrees bounded above by O(d) where d = max npij. Then we show that the resulting adjacency matrix A concentrates with the optimal rate: A −E A = O(√ d). Similarly, if we make all degrees bounded below by d by adding weight d/n to all edges, then the resulting Laplacian concentrates with the optimal rate: L(A) − L(E A) = O(1/ √ d). Our approach is based on Grothendieck-Pietsch factorization, using which we construct a new decomposition of random graphs. We illustrate the concentration results with an application to the community detection problem in the analysis of networks.