Better Bounds for Coalescing-Branching Random Walks

Abstract

Coalescing-branching random walks, or cobra walks for short, are a natural variant of random walks on graphs that can model the spread of disease through contacts or the spread of information in networks. In a k-cobra walk, at each time step a subset of the vertices are active; each active vertex chooses k random neighbors (sampled indpendently and uniformly with replacement) that become active at the next step, and these are the only active vertices at the next step. A natural quantity to study for cobra walks is the cover time, which corresponds to the expected time when all nodes have become infected or received the disseminated information.In this work, we extend previous results for cobra walks in multiple ways. We show that the cover time for the 2-cobra walk on [0, n] d is O(n) (where the order notation hides constant factors that depend on d); previous work had shown the cover time was O(n·polylog(n)). We show that the cover time for a 2-cobra walk on an n-vertex d-regular graph with conductance φG is O(φ −2 G log 2 n), significantly generalizing a previous result that held only for expander graphs with sufficiently high expansion. And finally we show that the cover time for a 2-cobra walk on a graph with n vertices is always O(n 11/4 log n); this is the first result showing that the bound of Θ(n 3 ) for the worst-case cover time for random walks can be beaten using 2-cobra walks.

Tight Bounds for Coalescing-Branching Random Walks on Regular Graphs

“…This result improves the previous bounds of O n 2−1/d log n for d-regular graphs [27], and the recent O n 2 log n bound which however applies also to non-regular graphs [9]. Our bound is tight, as demonstrated by the example of an n-node graph obtained by two cliques of size n/2, by removing one of the edges in each clique, and adding two new edges between the cliques such that the graph remains regular.…”

“…bound in [27], the O (1 − λ) −3 · log n bound in [8], [9]. Note that φ is an upper bound on 1 − λ, but the latter can be much smaller than φ.…”

“…They also provided tight bounds for the complete graph, trees, and the d-dimensional grid. Mitzenmacher et al [27] gave a w.h.p. bound of O d 4 · φ −2 · log 2 n for d-regular graphs with conductance φ, and general bounds of O n 2−1/d · log n for d-regular graphs and O n 11/4 · log n for non-regular graphs.…”

Improved Cover Time Bounds for the Coalescing-Branching Random Walk on Graphs

“…indicates the presence of a poly-log n term. Improved bounds were shown later in [8]: an O((r 4 /ϕ 2 ) log 2 n) bound for r -regular graphs with conductance ϕ, an O(D 2 n 1/D ) bound for D-dimensional grids, and an O(n 11/4 log n) bound for general graphs.…”

“…In this paper we show two new bounds on the cover time of the COBRA process for branching factor b = 2. For arbitrary connected graphs, we improve the O(n 11/4 log n) bound given in [8] to O(m + (d max ) 2 log n) = O(n 2 log n), where d max is the maximum degreeof a vertex. For r -regular connected graphs, we show a bound of O (r 2 + r /(1 − λ)) log n , which improves the O((1/(1 − λ) 3 log n) bound given in [4] for the case when 1 − λ = o(1/ √ r ).…”