Generating graphs randomly

Abstract

Graphs are used in many disciplines to model the relationships that exist between objects in a complex discrete system. Researchers may wish to compare a network of interest to a "typical" graph from a family (or ensemble) of graphs which are similar in some way. One way to do this is to take a sample of several random graphs from the family, to gather information about what is "typical". Hence there is a need for algorithms which can generate graphs uniformly (or approximately uniformly) at random from the given family. Since a large sample may be required, the algorithm should also be computationally efficient.Rigorous analysis of such algorithms is often challenging, involving both combinatorial and probabilistic arguments. We will focus mainly on the set of all simple graphs with a particular degree sequence, and describe several different algorithms for sampling graphs from this family uniformly, or almost uniformly.

Modularity of the ABCD Random Graph Model with Community Structure

“…vol(C) = O(n ζ (log n) 4 ). Finally, by performing the same computation as in (16) we get that w.e.p.…”

Sequential construction of spatial networks with arbitrary degree sequence and edge length distribution

“…Constructing a general spatial network with a given degree distribution is a computationally difficult task. Even a simpler problem of constructing a non-spatial network that has a given degree sequence is already a known NP-hard problem with a long history of research [14]. That being said, randomised algorithms achieve such construction in linear time for undirected [15] and directed [16] graphs when such networks are sufficiently sparse.…”