Percolation on sparse random graphs with given degree sequence

We study the two most common types of percolation process on a sparse random graph with a given degree sequence. Namely, we examine first a bond percolation process where the edges of the graph are retained with probability p and afterwards we focus on site percolation where the vertices are retained with probability p. We establish critical values for p above which a giant component emerges in both cases. Moreover, we show that in fact these coincide. As a special case, our results apply to power law random graphs. We obtain rigorous proofs for formulas derived by several physicists for such graphs.Comment: 20 page

Rumor Spreading on Random Regular Graphs and Expanders

Broadcasting algorithms are important building blocks of distributed systems. In this work we investigate the typical performance of the classical and well-studied push model. Assume that initially one node in a given network holds some piece of information. In each round, every one of the informed nodes chooses independently a neighbor uniformly at random and transmits the message to it. In this paper we consider random networks where each vertex has degree d, which is at least 3, i.e., the underlying graph is drawn uniformly at random from the set of all d-regular graphs with n vertices. We show that with probability 1 - o(1) the push model broadcasts the message to all nodes within (1 + o(1))C_d ln n rounds, where C_d = 1/ ln(2(1-1/d)) - 1/(d ln(1 - 1/d)). In particular, we can characterize precisely the effect of the node degree to the typical broadcast time of the push model. Moreover, we consider pseudo-random regular networks, where we assume that the degree of each node is very large. There we show that the broadcast time is (1+o(1))C ln n with probability 1 - o(1), where C= 1/ ln 2 + 1, is the limit of C_d as d grows.Comment: 18 page

On the Insertion Time of Cuckoo Hashing

Cuckoo hashing is an efficient technique for creating large hash tables with high space utilization and guaranteed constant access times. There, each item can be placed in a location given by any one out of k different hash functions. In this paper we investigate further the random walk heuristic for inserting in an online fashion new items into the hash table. Provided that k > 2 and that the number of items in the table is below (but arbitrarily close) to the theoretically achievable load threshold, we show a polylogarithmic bound for the maximum insertion time that holds with high probability.Comment: 27 pages, final version accepted by the SIAM Journal on Computin

The Multiple-orientability Thresholds for Random Hypergraphs

A $k$-uniform hypergraph $H = (V, E)$ is called $\ell$-orientable, if there is an assignment of each edge $e\in E$ to one of its vertices $v\in e$ such that no vertex is assigned more than $\ell$ edges. Let $H_{n,m,k}$ be a hypergraph, drawn uniformly at random from the set of all $k$-uniform hypergraphs with $n$ vertices and $m$ edges. In this paper we establish the threshold for the $\ell$-orientability of $H_{n,m,k}$ for all $k\ge 3$ and $\ell \ge 2$, i.e., we determine a critical quantity $c_{k, \ell}^*$ such that with probability $1-o(1)$ the graph $H_{n,cn,k}$ has an $\ell$-orientation if $c c_{k, \ell}^*$. Our result has various applications including sharp load thresholds for cuckoo hashing, load balancing with guaranteed maximum load, and massive parallel access to hard disk arrays.Comment: An extended abstract appeared in the proceedings of SODA 201

Bootstrap percolation in inhomogeneous random graphs

A bootstrap percolation process on a graph G is an "infection" process which evolves in rounds. Initially, there is a subset of infected nodes and in each subsequent round every uninfected node which has at least r infected neighbours becomes infected and remains so forever. The parameter r > 1 is fixed. We consider this process in the case where the underlying graph is an inhomogeneous random graph whose kernel is of rank 1. Assuming that initially every vertex is infected independently with probability p > 0, we provide a law of large numbers for the number of vertices that will have been infected by the end of the process. We also focus on a special case of such random graphs which exhibit a power-law degree distribution with exponent in (2,3). The first two authors have shown the existence of a critical function a_c(n) such that a_c(n)=o(n) with the following property. Let n be the number of vertices of the underlying random graph and let a(n) be the number of the vertices that are initially infected. Assume that a set of a(n) vertices is chosen randomly and becomes externally infected. If a(n) << a_c(n), then the process does not evolve at all, with high probability as n grows, whereas if a(n)>> a_c(n), then with high probability the final set of infected vertices is linear. Using the techniques of the previous theorem, we give the precise asymptotic fraction of vertices which will be eventually infected when a(n) >> a_c (n) but a(n) = o(n). Note that this corresponds to the case where p approaches 0.Comment: 42 page