Consider an algorithm performing a computation on a huge random object (for example a random graph or a "long" random walk). Is it necessary to generate the entire object prior to the computation, or is it possible to provide query access to the object and sample it incrementally "onthefly" (as requested by the algorithm)? Such an implementation should emulate the random object by answering queries in a manner consistent with an instance of the random object sampled from the true distribution (or close to it). This paradigm is useful when the algorithm is sublinear and thus, sampling the entire object up front would ruin its efficiency. Our first set of results focus on undirected graphs with independent edge probabilities, i.e. each edge is chosen as an independent Bernoulli random variable. We provide a general implementation for this model under certain assumptions. Then, we use this to obtain the first efficient local implementations for the ErdösRényi G(n,p) model for all values of p, and the Stochastic Block model. As in previous localaccess implementations for random graphs, we support VertexPair and NextNeighbor queries. In addition, we introduce a new RandomNeighbor query. Next, we give the first localaccess implementation for AllNeighbors queries inmore »
Enhanced diffusivity in perturbed senile reinforced random walk models
We consider diffusivity of random walks with transition probabilities depending on the number of consecutive traversals of the last traversed edge, the so called senile reinforced random walk (SeRW). In one dimension, the walk is known to be subdiffusive with identity reinforcement function. We perturb the model by introducing a small probability δ of escaping the last traversed edge at each step. The perturbed SeRW model is diffusive for any δ > 0 , with enhanced diffusivity (≫ O ( δ^2 ) ) in the small δ regime. We further study stochastically perturbed SeRW models by having the last edge escape probability of the form δ ξ n with ξ n ’s being independent random variables. Enhanced diffusivity in such models are logarithmically close to the so called residual diffusivity (positive in the zero δ limit), with diffusivity between O ( 1/ log δ  ) and O ( 1/ log  log δ  ) . Finally, we generalize our results to higher dimensions where the unperturbed model is already diffusive. The enhanced diffusivity can be as much as O ( 1/log ^2 δ )
 Award ID(s):
 1632935
 Publication Date:
 NSFPAR ID:
 10158977
 Journal Name:
 Asymptotic analysis
 ISSN:
 09217134
 Sponsoring Org:
 National Science Foundation
More Like this


We present an algorithm that, with high probability, generates a random spanning tree from an edgeweighted undirected graph in \Otil(n^{5/3 }m^{1/3}) time\footnote{The \Otil(\cdot) notation hides \poly(\log n) factors}. The tree is sampled from a distribution where the probability of each tree is proportional to the product of its edge weights. This improves upon the previous best algorithm due to Colbourn et al. that runs in matrix multiplication time, O(n^\omega). For the special case of unweighted graphs, this improves upon the best previously known running time of \tilde{O}(\min\{n^{\omega},m\sqrt{n},m^{4/3}\}) for m >> n^{7/4} (Colbourn et al. '96, KelnerMadry '09, Madry et al. '15). The effective resistance metric is essential to our algorithm, as in the work of Madry et al., but we eschew determinantbased and random walkbased techniques used by previous algorithms. Instead, our algorithm is based on Gaussian elimination, and the fact that effective resistance is preserved in the graph resulting from eliminating a subset of vertices (called a Schur complement). As part of our algorithm, we show how to compute \epsapproximate effective resistances for a set SS of vertex pairs via approximate Schur complements in \Otil(m+(n + S)\eps^{2}) time, without using the JohnsonLindenstrauss lemma which requires \Otil( \min\{(m + S)\eps^{2},more »

Abstract Community detection is considered for a stochastic block model graph of n vertices, with K vertices in the planted community, edge probability p for pairs of vertices both in the community, and edge probability q for other pairs of vertices. The main focus of the paper is on weak recovery of the community based on the graph G , with o ( K ) misclassified vertices on average, in the sublinear regime n 1 o (1) ≤ K ≤ o ( n ). A critical parameter is the effective signaltonoise ratio λ = K 2 ( p  q ) 2 / (( n  K ) q ), with λ = 1 corresponding to the Kesten–Stigum threshold. We show that a belief propagation (BP) algorithm achieves weak recovery if λ > 1 / e, beyond the Kesten–Stigum threshold by a factor of 1 / e. The BP algorithm only needs to run for log * n + O (1) iterations, with the total time complexity O ( E log * n ), where log * n is the iterated logarithm of n . Conversely, if λ ≤ 1 / e, no local algorithm can asymptotically outperform trivial randommore »

Winnertakeall (WTA) refers to the neural operation that selects a (typically small) group of neurons from a large neuron pool. It is conjectured to underlie many of the brain’s fundamental computational abilities. However, not much is known about the robustness of a spikebased WTA network to the inherent randomness of the input spike trains.In this work, we consider a spikebased k–WTA model wherein n randomly generated input spike trains compete with each other based on their underlying firing rates and k winners are supposed to be selected. We slot the time evenly with each time slot of length 1 ms and model then input spike trains as n independent Bernoulli processes. We analytically characterize the minimum waiting time needed so that a target minimax decision accuracy (success probability) can be reached. We first derive an informationtheoretic lower bound on the waiting time. We show that to guarantee a (minimax) decision error≤δ(whereδ∈(0,1)), the waiting time of any WTA circuit is at least ((1−δ)log(k(n−k)+1)−1)TR,where R⊆(0,1)is a finite set of rates and TR is a difficulty parameter of a WTA task with respect to set R for independent input spike trains. Additionally,TR is independent of δ,n,and k. We then design a simple WTAmore »

We present new constructions of pseudorandom generators (PRGs) for two of the most widely studied nonuniform circuit classes in complexity theory. Our main result is a construction of the first nontrivial PRG for linear threshold (LTF) circuits of arbitrary constant depth and superlinear size. This PRG fools circuits with depth d∈N and n1+δ wires, where δ=2−O(d) , using seed length O(n1−δ) and with error 2−nδ . This tightly matches the best known lower bounds for this circuit class. As a consequence of our result, all the known hardness for LTF circuits has now effectively been translated into pseudorandomness. This brings the extensive effort in the last decade to construct PRGs and deterministic circuitanalysis algorithms for this class to the point where any subsequent improvement would yield breakthrough lower bounds. Our second contribution is a PRG for De Morgan formulas of size s whose seed length is s1/3+o(1)⋅polylog(1/ϵ) for error ϵ . In particular, our PRG can fool formulas of subcubic size s=n3−Ω(1) with an exponentially small error ϵ=exp(−nΩ(1)) . This significantly improves the inversepolynomial error of the previous stateoftheart for such formulas by Impagliazzo, Meka, and Zuckerman (FOCS 2012, JACM 2019), and again tightly matches the best currentlyknown lower boundsmore »