1 Introduction
Multiple random walks is a model for movement of several independent random walkers on a graph, and it is applied to various graph algorithms because of some advantages such as ease of analysis and light weight of processing. In order to design an efficient graph algorithm using multiple random walks, it is essential to study theoretical considerations for deeply understanding the characteristics of graph algorithms, including (a) a search algorithm for finding a particular node on the graph [Lv02:MRW, Gkantsidis:04:RW], (b) an algorithm for spreading information across networks by exchanging information between nodes [Dutta15:Coalescing], (c) a rendezvous algorithm for encountering a plurality of agent at the same node [Metivier00:Rendezvous].
Theare are some important metrics (e.g., first hitting time, recurrent time, cover time, reencountering time, and first meeting time) for the multiple random walks. The first hitting time is the time it takes for any random walker to arrive at a given node, and is important to measure the performance of the search algorithm. The recurrent time is the time required to return any one of random walkers to the starting node, and so it is the particular case of the first hitting time. The cover time is the time it takes for any random walker to reach all nodes, and corresponds to the maximum value of first hitting times. The cover time strongly affects the information dissemination speed in the graph. In addition, the reencountering time and the first meeting time are the times it takes for multiple random walkers to meet on a same node. The reencountering time considers a random walkers starting from the same node, but the first meeting time considers random walkers starting from a different node. In particular, first meeting time is closely related to the rendezvous problem, which is are a fundamental problem in the field of computer science. The rendezvous problem appears common to various engineering problems (e.g., selfstabilizing token management system problem [Israeli90:SelfStabilizing, Tetali91:SelfStabilizingRW] and server problem [Coppersmith93:OnLineAlgo]). In order to construct efficient algorithms for solving the rendezvous problem, the characteristics of first meeting time should be clarified.
The First meeting time of multiple random walk has been analyzed in [Aldous91:Meeting, Bshouty99:Meeting, Cooper09:MultiRW, Zhang14:Meeting, George16:Meeting]. However, many of these previous works focus on regular graphs. In [George16:Meeting], George et al provide a pioneering work for multiple random walks, and derive a closedform formula for calculating the expected value of the first meeting time in arbitrary graph. However, the effect of graph structures on the expected first meeting time has not been clarified. In order to design effective algorithms using multiple random walks for realistic graphs (e.g., social networks and communication networks), it is desirable to understand the effect of graph structures on the expected first meeting time. Since it is difficult to numerically clarify the effect using closedform formula derived in [George16:Meeting], the effect should be examined using the analysis of multiple random walks.
In this paper, we analyze first meeting time of multiple random walks in arbitrary graph, and clarify the effect of graph structures on its expected value. First, we derive the spectral formula for the expected first meeting time of two random walkers using the spectral graph theory that is used to analyze the charactristics of graphs. Then, the principal component of the expected first meeeting time are examined using the derived spectral formula. The resulting principal component reveals that (a) the expected first meeting time is almostly dominated by and (b) the expected first meeting time is independent of the beginning nodes of multiple random walks where is the number of nodes. and are the mean and standard deviation of the weighted degree of each node, respectively. and , and are related to the statistics of graph structures. In addition, we conform the validity of the analysis results through numerical examples.
The contributions in this paper are as follow:

to extend the analysis of a single random walk using the spectral graph theory to multiple random walks,

to derive the spectral formula for the expected first meeting time,

to clarify the principal component of the expected first meeting time,

to confirm the varidity of the derived spectral formula and the principal component for networks with various scales and several structures, and

to clarify the effect of graph structures on the expected first meeting time.
Acknowledgment
This work was supported by JSPS KAKENHI Grant Number 19K11927.
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