k-Wise Independent Random Graphs

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 connectivitynumber, 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.

Invertible bloom lookup tables

“…Assuming that the cells associated with a key are chosen uniformly at random, we use known results on 2-cores of random hypergraphs. In particular, tight thresholds are known; when the number of hash values k of each is at least 2, there are constants c k > 1 such that if m > (c k + )n for any constant > 0, LISTENTRIES succeeds with high probability, that is with probability 1 − o (1). Similarly, if m < (c k − )n for any constant > 0, LISTENTRIES succeeds with probability o(1).…”

Generating k-Independent Variables in Constant Time

“…Fix the choice of finite field F. By Lemma 2 there exists an explicit m-generator in F for m ≤ |F| with period |F| that uses time O(m log 3 m) to emit m values. Fix some constant c > 1 and let (Γ i ) t i=1 denote an explicit sequence of constant degree expanders with the properties given by (4). The average number of operations per k-independent value output by g t when performing batch evaluation is given by…”

“…Sums of k-independent variables have their jth moment identical to fully random variables for j ≤ k which preserves many properties of full randomness. For output length n, Θ(log n)independence yields Chernoff-Hoeffding bounds [3] and random graph properties [4], while Θ(poly log n)-independence suffices to fool AC 0 circuits [5].…”

Fast and powerful hashing using tabulation

“…One reason for this is the Chernoff bounds of [43] for kindependent events, whose probability bounds differ from the full-independence Chernoff bound by 2 −Ω(k) . Another reason is that random graphs with O(lg n)-independent edges [3] share many of the properties of truly random graphs.…”