Better bounds for coalescing-branching random walks

Abstract: Coalescing-branching random walks, or {\em 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 independently 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 $O(n \cdot polylog(n))$. We show that the cover time for a 2-cobra walk on an $n$-vertex $d$-regular graph with conductance $\phi_G$ is $O(\phi_G^{-2} \log^2 n)$, 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 $\Theta(n^3)$ for the worst-case cover time for random walks can be beaten using 2-cobra walks.