Random Walks, Interacting Particles, Dynamic Networks: Randomness Can Be Helpful

The aim of this article is to discuss some applications of random processes in searching and reaching consensus on finite graphs. The topics covered are: Why random walks?, Speeding up random walks, Random and deterministic walks, Interacting particles and voting, Searching changing graphs. As an introductory example consider the self-stabilizing mutual exclusion algorithm of Israeli and Jalfon [35], based on the random walk of tokens on a graph G. Initially each vertex emits a token which makes a random walk on G. On meeting at a vertex, tokens coalesce. Provided the graph is connected, and not bipartite, eventually only one token will remain, and the vertex with the token has exclusive access to some resource. The token makes a random walk on G, so in the long run it will visit all vertices of G. Typical questions are: how long before only one token remains (coalescence time), how long before every vertex has been visited by the token (cover time), what proportion of the time does each vertex have the token in the long run (stationary distribution of the random walk), how long before the walk approaches the stationary distribution (mixing time), how long before the number of visits to each vertex approximates the frequency given by the stationary distribution (blanket time). If these quantities can be understood and manipulated, then we can tune the random walk to perform efficiently on a given network. For example, in Fair Circulation of a Token [33], the transition probability of the walk is modified, so that each vertex has the same probability of holding the token in the long run, i.e. the stationary distribution is uniform. The questions asked in [33] are, how can this modification be achieved, and what