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Consider an Erdos-Renyi random graph in which each edge is present independently with probability 1/2, except for a subset $sC_N$ of the vertices that form a clique (a completely connected subgraph). We consider the problem of identifying the clique, given a realization of such a random graph. The best known algorithm provably finds the clique in linear time with high probability, provided $|sC_N|ge 1.261sqrt{N}$ cite{dekel2011finding}. Spectral methods can be shown to fail on cliques smaller than $sqrt{N}$. In this paper we describe a nearly linear time algorithm that succeeds with high probability for $|sC_N|ge (1+eps)sqrt{N/e}$ for any $eps>0$. This is the first algorithm that provably improves over spectral methods. We further generalize the hidden clique problem to other background graphs (the standard case corresponding to the complete graph on $N$ vertices). For large girth regular graphs of degree $(Delta+1)$ we prove that `local algorithms succeed if $|sC_N|ge (1+eps)N/sqrt{eDelta}$ and fail if $|sC_N|le(1-eps)N/sqrt{eDelta}$.
In this paper, new techniques that allow conditional entropy to estimate the combinatorics of symbols are applied to animal communication studies to estimate the communications repertoire size. By using the conditional entropy estimates at multiple o
Let ${X_n}_{n=0}^{infty}$ be a stationary real-valued time series with unknown distribution. Our goal is to estimate the conditional expectation of $X_{n+1}$ based on the observations $X_i$, $0le ile n$ in a strongly consistent way. Bailey and Ryabko
Transfer entropy (TE) was introduced by Schreiber in 2000 as a measurement of the predictive capacity of one stochastic process with respect to another. Originally stated for discrete time processes, we expand the theory in line with recent work of S
Graph sparsification has been studied extensively over the past two decades, culminating in spectral sparsifiers of optimal size (up to constant factors). Spectral hypergraph sparsification is a natural analogue of this problem, for which optimal bou
We prove large (and moderate) deviations for a class of linear combinations of spacings generated by i.i.d. exponentially distributed random variables. We allow a wide class of coefficients which can be expressed in terms of continuous functions defi