Let D(n,r) be a random r-out regular directed multigraph on the set of vertices 1,...,n . In this work, we establish that for every r � 2, there existsr > 0 such that diam(D(n,r)) = (1 + �r + o(1))log r n. Our techniques also allow us to bound some extremal quantities related to the stationary distribution of a simple random walk on D(n,r). In particular, we determine the asymptotic behaviour ofmax andmin, the maximum and the minimum values of the stationary distribution. We show that with
To conclude this section, we discuss the stationary distribution of a simple random walk in these other models. While in undirected graphs the stationary distribution (if it exists) is completely determined by the degrees of the vertices, this is not the case in directed graphs. Cooper and Frieze @cite_10 give a very precise description of the stationary distribution of @math when @math , for any constant @math , and use their result to compute the cover time of @math . It is worth noticing that for such values of @math , both the in-degrees and out-degrees are of logarithmic order and concentrated around their expected values, which turns to be very useful for the analysis. It seems harder to find an interesting question about the stationary distribution of @math when @math since like in random @math -in regular digraphs, typically there are vertices with no out-edges.
@cite_10
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We study properties of multiple random walks on a graph under various assumptions of interaction between the particles. To give precise results, we make the analysis for random regular graphs. The cover time of a random walk on a random @math -regular graph was studied in [C. Cooper and A. Frieze, SIAM J. Discrete Math., 18 (2005), pp. 728-740], where it was shown with high probability (whp) that for @math the cover time is asymptotic to @math , where @math . In this paper we prove the following (whp) results, arising from the study of multiple random walks on a random regular graph @math . For @math independent walks on @math , the cover time @math is asymptotic to @math , where @math is the cover time of a single walk. For most starting positions, the expected number of steps before any of the walks meet is @math . If the walks can communicate when meeting at a vertex, we show that, for most starting positions, the expected time for @math walks to broadcast a single piece of information to each other is asymptotic to @math as @math . We also establish properties of walks where there are two types of particles, predator and prey, or where particles interact when they meet at a vertex by coalescing or by annihilating each other. For example, the expected extinction time of @math explosive particles ( @math even) tends to @math as @math . The case of @math coalescing particles, where one particle is initially located at each vertex, corresponds to a voter model defined as follows: Initially each vertex has a distinct opinion, and at each step each vertex changes its opinion to that of a random neighbor. The expected time for a unique opinion to emerge is the same as the expected time for all the particles to coalesce, which is asymptotic to @math . Combining results from the predator-prey and multiple random walk models allows us to compare expected detection times of all prey in the following scenarios: Both the predator and the prey move randomly, the prey moves randomly and the predators stay fixed, and the predators move randomly and the prey stays fixed. In all cases, with @math predators and @math prey the expected detection time is @math , where @math is the @math th harmonic number. We study properties of a simple random walk on the random digraph D"n","p when np=dlogn, d>1. We prove that whp the value @p"v of the stationary distribution at vertex v is asymptotic to deg^-(v) m where deg^-(v) is the in-degree of v and m=n(n-1)p is the expected number of edges of D"n","p. If d=d(n)-> with n, the stationary distribution is asymptotically uniform whp. Using this result we prove that, for d>1, whp the cover time of D"n","p is asymptotic to dlog(d (d-1))nlogn. If d=d(n)-> with n, then the cover time is asymptotic to nlogn. Let D(n,r) be a random r-out regular directed multigraph on the set of vertices 1,...,n . In this work, we establish that for every r � 2, there existsr > 0 such that diam(D(n,r)) = (1 + �r + o(1))log r n. Our techniques also allow us to bound some extremal quantities related to the stationary distribution of a simple random walk on D(n,r). In particular, we determine the asymptotic behaviour ofmax andmin, the maximum and the minimum values of the stationary distribution. We show that with Given a discrete random walk on a finite graph @math , the vacant set and vacant net are, respectively, the sets of vertices and edges which remain unvisited by the walk at a given step @math . These sets induce subgraphs of the underlying graph. Let @math be the subgraph of @math induced by the vacant set of the walk at step @math . Similarly, let @math be the subgraph of @math induced by the edges of the vacant net. For random @math -regular graphs @math , it was previously established that for a simple random walk, the graph @math of the vacant set undergoes a phase transition in the sense of the phase transition on Erd os -Renyi graphs @math . Thus, for @math there is an explicit value @math of the walk, such that for @math , @math has a unique giant component, plus components of size @math , whereas for @math all the components of @math are of size @math . We establish the threshold value @math for a phase transition in the graph @math of the vacant net of a simple random walk on a random @math -regular graph. We obtain the corresponding threshold results for the vacant set and vacant net of two modified random walks. These are a non-backtracking random walk, and, for @math even, a random walk which chooses unvisited edges whenever available. This allows a direct comparison of thresholds between simple and modified walks on random @math -regular graphs. The main findings are the following: As @math increases the threshold for the vacant set converges to @math in all three walks. For the vacant net, the threshold converges to @math for both the simple random walk and non-backtracking random walk. When @math is even, the threshold for the vacant net of the unvisited edge process converges to @math , which is also the vertex cover time of the process.