We consider two competing first passage percolation processes started from uniformly chosen subsets of a random regular graph on N vertices. The processes are allowed to spread with different rates, start from vertex subsets of different sizes or at different times. We obtain tight results regarding the sizes of the vertex sets occupied by each process, showing that in the generic situation one process will occupy Θ(1)N α vertices, for some 0 < α < 1. The value of α is calculated in terms of the relative rates of the processes, as well as the sizes of the initial vertex sets and the possible time advantage of one process.The motivation for this work comes from the study of viral marketing on social networks. The described processes can be viewed as two competing products spreading through a social network (random regular graph). Considering the processes which grow at different rates (corresponding to different attraction levels of the two products) or starting at different times (the first to market advantage) allows to model aspects of real competition. The results obtained can be interpreted as one of the two products taking the lion share of the market. We compare these results to the same process run on d dimensional grids where we show that in the generic situation the two products will have a linear fraction of the market each.

We are given a graph G with n vertices, where a random subset of k vertices has been made into a clique, and the remaining edges are chosen independently with probability 1 2 . This random graph model is denoted G(n, 1 2 , k). The hidden clique problem is to design an algorithm that finds the k-clique in polynomial time with high probability. An algorithm due to Alon, Krivelevich and Sudakov uses spectral techniques to find the hidden clique with high probability when k = c √ n for a sufficiently large constant c > 0. Recently, an algorithm that solves the same problem was proposed by Feige and Ron. It has the advantages of being simpler and more intuitive, and of an improved running time of O(n 2 ). However, the analysis in the paper gives success probability of only 2/3. In this paper we present a new algorithm for finding hidden cliques that both runs in time O(n 2 ), and has a failure probability that is less than polynomially small.

We initiate a systematic study of eigenvectors of random graphs. Whereas much is known about eigenvalues of graphs and how they reflect properties of the underlying graph, relatively little is known about the corresponding eigenvectors. Our main focus in this paper is on the nodal domains associated with the different eigenfunctions. In the analogous realm of Laplacians of Riemannian manifolds, nodal domains have been the subject of intensive research for well over a hundred years. Graphical nodal domains turn out to have interesting and unexpected properties. Our main theorem asserts that there is a constant c such that for almost every graph G, each eigenfunction of G has at most two large nodal domains, and in addition at most c exceptional vertices outside these primary domains. We also discuss variations of these questions and briefly report on some numerical experiments which, in particular, suggest that almost surely there are just two nodal domains and no exceptional vertices.

We initiate a systematic study of eigenvectors of random graphs. Whereas much is known about eigenvalues of graphs and how they reflect properties of the underlying graph, relatively little is known about the corresponding eigenvectors. Our main focus in this paper is on the nodal domains associated with the different eigenfunctions. In the analogous realm of Laplacians of Riemannian manifolds, nodal domains have been the subject of intensive research for well over a hundred years. Graphical nodal domains turn out to have interesting and unexpected properties. Our main theorem asserts that there is a constant c such that for almost every graph G, each eigenfunction of G has at most two large nodal domains, and in addition at most c exceptional vertices outside these primary domains. We also discuss variations of these questions and briefly report on some numerical experiments which, in particular, suggest that almost surely there are just two nodal domains and no exceptional vertices.

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