We study the appearance of the giant component in random subgraphs of a given finite graph G = (V, E) in which each edge is present independently with probability p. We show that if G is an expander with vertices of bounded degree, then for any c ∈]0, 1[, the property that the random subgraph contains a giant component of size c|V | has a sharp threshold.

We analyse the hash function family based on walks in LPS Ramanujan graphs recently introduced by Charles et al. We present an algorithm for finding collisions that runs in quasi-linear time in the length of the hashed value. A concrete instance of the hash function is considered, based on a 100-digit prime. A short collision is given, together with implementation details.

We examine how a flexible process structure might be designed to allow the production system to better cope with fluctuating supply and demand, and to match supply with demand in a more effective manner. We argue that good flexible process structures are essentially highly connected graphs, and use the concept of graph expansion (a measure of graph connectivity) to achieve various insights into...

Let be an integer. Kouider and Lonc proved that the vertex set of every graph G with vertices minimum degree at least can covered by cycles. Our main result states for , same conclusion holds graphs are sparse in sense In particular, this allows us to determine local resilience random pseudorandom respect having a cover fixed number The proof uses version absorbing method expander graphs.

We present direct constructions of pseudorandom function (PRF) families based on Goldreich’s one-way function. Roughly speaking, we assume that non-trivial local mappings f : {0, 1} → {0, 1} whose input-output dependencies graph form an expander are hard to invert. We show that this one-wayness assumption yields PRFs with relatively low complexity. This includes weak PRFs which can be computed ...

We show that for integers k ≥ 2 and n ≥ 3, the diameter of the Cayley graph of SLn(Z/kZ) associated to a standard two-element generating set, is at most a constant times n 2 ln k. This answers a question of A. Lubotzky concerning SLn(Fp) and is unexpected because these Cayley graphs do not form an expander family. Our proof amounts to a quick algorithm for finding short words representing eleme...

We show that the Cayley graph of the fundamental group of a closed aspherical manifold with the hypereuclidean universal cover cannot contain an expander. This rules out for recent Gromov’s examples of exotic groups an approach to the Novikov Conjecture via the hypereuclideanness developed by Connes, Gromov and Moscovici [G],[CGM] and in a different languages by Ferry and Weinberger [FW],[DF], ...

We analyze classical Maker-Breaker games played on the edge set of a randomly generated graph G. We consider the random graph process and analyze, for each of the properties “being spanning k-vertex-connected” , “admitting a perfect matching”, and “being Hamiltonian”, the first time when Maker starts having a winning strategy for building a graph possessing the target property (the so called hi...

We show that for any constant ∆ ≥ 2, there exists a graph Γ with O(n) vertices which contains every n-vertex graph with maximum degree ∆ as an induced subgraph. For odd ∆ this significantly improves the best-known earlier bound of Esperet et al. and is optimal up to a constant factor, as it is known that any such graph must have at least Ω(n) vertices. Our proof builds on the approach of Alon a...

Deep Neural Networks, while being unreasonably effective for several vision tasks, have their usage limited by the computational and memory requirements, both during training and inference stages. Analyzing and improving the connectivity patterns between layers of a network has resulted in several compact architectures like GoogleNet, ResNet and DenseNet-BC. In this work, we utilize results fro...