Random Schreier graphs and expanders

Random Cayley Graphs and Expanders

For every 1 > δ > 0 there exists a c = c(δ) > 0 such that for every group G of order n, and for a set S of c(δ) log n random elements in the group, the expected value of the second largest eigenvalue of the normalized adjacency matrix of the Cayley graph X(G,S) is at most (1−δ). This implies that almost every such a graph is an ε(δ)-expander. For Abelian groups this is essentially tight, and ex...

. Expanders with Respect to Hadamard Spaces and Random Graphs

It is shown that there exists a sequence of 3-regular graphs {Gn}n=1 and a Hadamard space X such that {Gn}n=1 forms an expander sequence with respect to X, yet random regular graphs are not expanders with respect to X. This answers a question of [NS11]. {Gn}n=1 are also shown to be expanders with respect to random regular graphs, yielding a deterministic sublinear time constant factor approxima...

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 th...

Random Unitaries Give Quantum Expanders

Sharpness of KKL on Schreier graphs

Recently, the Kahn–Kalai–Linial (KKL) Theorem on influences of functions on {0, 1} was extended to the setting of functions on Schreier graphs. Specifically, it was shown that for an undirected Schreier graph Sch(G,X,U) with log-Sobolev constant ρ and generating set U closed under conjugation, if f : X → {0, 1} then E [f ] & log(1/M[f ]) · ρ ·Var[f ]. Here E [f ] denotes the average of f ’s inf...