RANDOM WALK MEETING TIMES on GRAPHS Contents 1. Introduction

RANDOM WALK MEETING TIMES ON GRAPHS ETHAN NAEGELE Abstract. We discuss the bounds on expected meeting times for a game involving two tokens which pass from vertex to vertex randomly on an arbitrary n-vertex graph. At each unit of time, the player chooses which token to move, and the game ends when the two tokens collide. We find the coefficients of the approximate bounds for the meeting times, which are of order n3, and the player strategies which give rise to such extremal meeting times. Contents 1. Introduction 1 2. Preliminaries 2 3. Strategies 4 4. Bounds 9 Acknowledgments 14 References 14 1. Introduction We investigate a game played on some connected, undirected graph G. The game to be discussed is played as follows: two tokens are placed at initial vertices (x; y) on G. At each unit of time (say, each second), the player chooses which of the two tokens is to be moved. The selected token moves according to the uniform distribution on its neighbors. When both tokens meet at the same vertex, the game ends. In this paper, we investigate the expected meeting times for each of the three main modes of play: Angel, Demon, and Random. Angel plays to minimize the meeting time; Demon plays to maximize it; Random flips a fair coin to select which token to move. The motivation for studying such games is well summarized in [1]. These games are useful in understanding \token-management" schemes, which pass a \token," some abstract object between nodes, or \processors" in a network. In the desired state, there is generally one token in the network. Were the system to enter some undesired state in which the network contains multiple tokens, the way to return to the desired state would be to remove the tokens by colliding them together at the same processor. It becomes of our interest, then, to study the maximum amount of time we should expect to wait in the worst case scenario, described by the Demon player. We would also like to find how much shorter we should expect to wait in the worst case scenario when we are trying to end the undesired state as soon as possible, i.e., the Angel player. Naturally, we also ask what the strategies are which result in these extremal Demon and Angel games, so that we may implement them, 1 2 ETHAN NAEGELE and so that we can understand under what conditions these meeting times arise. In response to these questions, we synthesize the main results from [4], [2], and [7]. We first build motivation for understanding how this game works by finding strategies for different graphs. We find the existence of pure optimal strategies for both Angel and Demon on any graph G, as well as finding an explicit strategy for the Demon game played on a path. We also deduce a general pure optimal strategy which utilizes a remote vertex for Demon and a central vertex for Angel. After developing an intuition for the game, a natural question arises: how long is the game expected to last? To answer this, we look for the graph which maximizes the hitting time of a random walk across all n-vertex graphs. The graph which achieves this is known as the lollipop graph (see Figure 1). We then calculate that 4 3 the maximal hitting time on such a graph is of order 27 n , and we develop this calculation explicitly. We use this calculation to obtain the Demon's bound, then again in obtaining the lower bound for the Angel by employing his pure optimal strategy. Then, since the random game must have bounds which lie between these two, we obtain the bounds on each of the three modes of play. 2. Preliminaries We assume basic knowledge of probability, random walks, and graph theory. It should be noted that we use the notation o(1) to denote some function f : N ! R f(n) such that limn!1 1 = 0. We again emphasize that G is required to be undirected and connected, so that the notion of maximum meeting time is always well-defined, and when we refer to \any graph G," we imply that G is also undirected and connected. The following definitions are required: Definition 2.1. The hitting time HG(x; y) is the expected time for a random walk starting at vertex x on graph G to reach vertex y. We may also define a special quantity involving hitting time as M HG (x; y) := HG(x; y) + M · e(G); where M is a positive real number, and e(G) is the number of edges on graph G. We denote V (G) as the set of vertices on G. Now let us define for fixed M the M M quantity H (n) := maxfHG (x; y): jV (G)j = n and x; y 2 V (G)g: Definition 2.2. An n-vertex graph G is (n; M)-extremal if there exists a pair of M M vertices x; y such that HG (x; y) = H (n). We also have some special vertices which are useful for later results. A remote vertex is a vertex r such that HG(r; x) ≤ HG(x; r) for all x of G. Similarly, a central vertex is a vertex c such that HG(x; c) ≤ HG(c; x) for all x. It is noted in [7] that for any G, some c and r both exist. Applying these definitions, we define the \potential function" Z(z) = HG(c; z) − HG(z; c): Intuitively, we can think of the potential of z as a measure of how far z is from being central; c has a potential of 0, whereas r has the maximum potential, which is easy to see from the aforementioned hitting time inequalities. RANDOM WALK MEETING TIMES ON GRAPHS 3 We also define the Demon's harmonic function as φ(x; y) = HG(x; y) + HG(y; r) − HG(r; y) 1 = (C(x; y) − Z(x) − Z(y)) + Z(r); 2 where C(x; y) = HG(x; y) + HG(y; x). We can think of C(x; y) as the \round trip time," or the expected amount of time it takes for a random walk to travel from x to y, then back to x. We define the Angel's harmonic function as (x; y) = HG(x; y) + HG(c; r) − HG(c; y) 1 = (C(x; y) − Z(x) − Z(y)): 2 In our investigation of which n-vertex graphs produce the maximal hitting time, it is useful to define the lollipop graph. m Definition 2.3. A lollipop graph Ln is an n-vertex graph containing a clique of m vertices which is connected to a path of t := n − m vertices (see Figure 1). In the setup that follows, we will have the starting point x of our random walk on m Ln be some vertex which is in the clique, z is the vertex in the clique which is also an endpoint of the path (it is the point where the path and the clique intersect), and y is the vertex which is the endpoint of the path that is farthest from the clique. 6 An L9 lollipop graph is shown in Figure 1 below, with the points x; y; and z as defined above. Note that this particular graph, as we will see, is part of a class of lollipop graphs which maximize HG(x; y), because the number of vertices in the 2 clique of the graph is 3 the total number of vertices: x z y 6 Figure 1. The lollipop graph L9. From Figure 1, we develop an intuition for why HG(u; v) 6= HG(v; u) for any pair u; v of vertices. In particular, if we consider the graph from Figure 1, we see that H(x; z) = 6−1 = 5; since deg u = 5 for any vertex u 6= z in the clique. However, we have H(z; x) > 5, since the random walk starting on z can not only reach any other vertex in the clique but may also retreat further down the path with probability p > 0, thereby extending the expected time to reach x. If w is a neighboring vertex of x, then we denote this as w ∼ x. As a convention used for a general function f on vertices of graphs, we write f(¯x; y) to denote the average value of f(w; y) over all the neighbors w of x and with some second, fixed input y. A \strategy" in the context of the token game is a set of criteria used by the player to assign some probability p such that with tokens on vertices x and y, the player moves the token on x with probability p. For example, the Random player 1 implements a strategy which states that at every move, p = 2 is assigned to either token. 4 ETHAN NAEGELE The goal of the token game depends upon the player. Demon wishes to avoid token collision as long as possible; Angel wishes to collide the tokens as soon as possible; Random is a \neutral" player which has no ultimate goal. Functions with a subscript or superscript of S, such as χS, denote the value of the function when the player follows strategy S. The function MG(x; y) denotes the maximum expected meeting time over all strategies S with starting vertices x; y. Lastly, some proofs require a notion of a \distance" metric on a graph G. The \distance" between two vertices x and y on G is the number of edges on the shortest path from x to y.

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