k-Wise Independent Random Graphs

TL;DR

This paper investigates the properties of k-wise independent random graphs, establishing bounds on k needed to preserve key features of classical G(N,p) graphs, with implications for efficient graph generation.

Contribution

It provides bounds on the independence parameter k required to maintain properties like connectivity and clique number in k-wise independent graphs, extending understanding of their structure.

Findings

Properties like connectivity and Hamiltonicity are preserved with constant or polylogarithmic k.

Most graph properties are maintained with k=poly(log(N)) for various p.

Efficient graph generation with small seed size is possible using k-wise independence.

Abstract

We study the k-wise independent relaxation of the usual model G(N,p) of random graphs where, as in this model, N labeled vertices are fixed and each edge is drawn with probability p, however, it is only required that the distribution of any subset of k edges is independent. This relaxation can be relevant in modeling phenomena where only k-wise independence is assumed to hold, and is also useful when the relevant graphs are so huge that handling G(N,p) graphs becomes infeasible, and cheaper random-looking distributions (such as k-wise independent ones) must be used instead. Unfortunately, many well-known properties of random graphs in G(N,p) are global, and it is thus not clear if they are guaranteed to hold in the k-wise independent case. We explore the properties of k-wise independent graphs by providing upper-bounds and lower-bounds on the amount of independence, k, required for maintaining the main properties of G(N,p) graphs: connectivity, Hamiltonicity, the connectivity-number, clique-number and chromatic-number and the appearance of fixed subgraphs. Most of these properties are shown to be captured by either constant k or by some k= poly(log(N)) for a wide range of values of p, implying that random looking graphs on N vertices can be generated by a seed of size poly(log(N)). The proofs combine combinatorial, probabilistic and spectral techniques.