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Meeting times of random walks on graphs

Nader H. Bshouty, Lisa Higham, Jolanta Warpechowska-Gruca
1999 Information Processing Letters  
W e prove an upper bound on the meeting time of an arbitrary number of random walks in any connected undirected graph in terms of the meeting times of fewer random walks.  ...  W Concluding remarks and open problems Theorem 6 bounds the meeting time of several random walks on an undirected connected graph by a function of the meeting times of fewer random walks.  ...  The expected time until several random walks on a graph collapse into one characterizes the performance of this protocol. In the literature, however, one random walk on a graph is usually analyzed.  ... 
doi:10.1016/s0020-0190(99)00017-4 fatcat:5i4c4gyf25ghrfg4ed3uux2iwe

Exact computation for meeting times and infection times of random walks on graphs

Takuya Ohwa
2015 Pacific Journal of Mathematics for Industry  
We consider independent multiple random walks on graphs and study comparison results of meeting times and infection times between many conditions of the random walks by obtaining the exact density functions  ...  The author was supported by JST, ERATO, Kawarabayashi Large Graph Project.  ...  It is mentioned in [1] that meeting times of two random walks on some graphs can be regarded as a first hitting time of a single random walk. We give a generalization of this fact.  ... 
doi:10.1186/s40736-015-0016-2 fatcat:6yyovejhebdufkr23v25qqvw2e

MEETING TIMES OF RANDOM WALKS ON GRAPHS [article]

Jolanta Warpechowska-Gruca, University Of Calgary, University Of Calgary
2008
Both bounds express the meeting time of several random walks in terms of the meeting times of fewer random walks.  ...  These techniques are used to prove two upper bounds on the meeting time of an arbitrary number of random walks in any connected undirected graph.  ...  Concluding remarks and open problems Theorem 6 bounds the meeting time of several random walks on an undirected connected graph by a function of the meeting times of fewer random walks.  ... 
doi:10.11575/prism/31330 fatcat:b2s6nhrpbjclnojfemdeprc26y

An efficient similarity search framework for SimRank over large dynamic graphs

Yingxia Shao, Bin Cui, Lei Chen, Mingming Liu, Xing Xie
2015 Proceedings of the VLDB Endowment  
In the preprocessing stage, TSF samples a set of one-way graphs to index raw random walks in a novel manner within O(N Rg) time and space, where N is the number of vertices and Rg is the number of one-way  ...  In this paper, we propose a novel two-stage random-walk sampling framework (TSF) for SimRank-based similarity search (e.g., top-k search).  ...  In the tsM ap, each meeting vertex is associated with its all possible meeting times, like vertex 1 can be met at times 1 or 2 based on the two sampled random walks.  ... 
doi:10.14778/2757807.2757809 fatcat:kxltrqvpgramfbznz7b3ahcodm

Multiple Random Walks and Interacting Particle Systems [chapter]

Colin Cooper, Alan Frieze, Tomasz Radzik
2009 Lecture Notes in Computer Science  
For k independent walks on a random regular graph G, the cover time CG(k) is asymptotic to CG/k, where CG is the cover time of a single walk.  ...  We study properties of multiple random walks on a graph under various assumptions of interaction between the particles. To give precise results, we make our analysis for random regular graphs.  ...  To make our analysis, we reduce the multiple random walks to a single random walk on a suitably defined product graph, to which we apply the technique of [6] .  ... 
doi:10.1007/978-3-642-02930-1_33 fatcat:axxnxxnv2fdcbfz2bb5ehupw2i

Graph Degree Heterogeneity Facilitates Random Walker Meetings [article]

Yusuke Sakumoto, Hiroyuki Ohsaki
2020 arXiv   pre-print
The first meeting time is one of the important metrics for multiple random walks.  ...  The first meeting time of multiple random walks has been analyzed previously, but many of these analyses have focused on regular graphs.  ...  The first meeting time is one of the important metrics for multiple random walks.  ... 
arXiv:2005.11161v3 fatcat:nnfqyidte5gvvkiac4ulyemuwy

Coalescing random walks and voting on connected graphs [article]

Colin Cooper, Robert Elsasser, Hirotaka Ono, Tomasz Radzik
2016 arXiv   pre-print
In a coalescing random walk, a set of particles make independent random walks on a graph.  ...  Whenever one or more particles meet at a vertex, they unite to form a single particle, which then continues the random walk through the graph.  ...  Previous work on coalescing random walks We summarize some known results for coalescing random walks. There are two distinct models for the transition times of random walks on finite graphs.  ... 
arXiv:1204.4106v3 fatcat:rwq5xit4xne33kee76d6qstdbe

On coalescence time in graphs--When is coalescing as fast as meeting? [article]

Varun Kanade, Frederik Mallmann-Trenn, Thomas Sauerwald
2018 arXiv   pre-print
Coalescing random walks is a fundamental stochastic process, where a set of particles perform independent discrete-time random walks on an undirected graph.  ...  As a general result, we establish that for graphs whose meeting time is only marginally larger than the mixing time (a factor of log^2 n), the coalescence time of n random walks equals the meeting time  ...  the hitting time (Theorem 1.4) .  ... 
arXiv:1611.02460v4 fatcat:5av5epd5mbf7layr6i46g44t6y

Coalescing Random Walks and Voting on Connected Graphs

Colin Cooper, Robert Elsässer, Hirotaka Ono, Tomasz Radzik
2013 SIAM Journal on Discrete Mathematics  
In a coalescing random walk, a set of particles make independent discrete-time random walks on a graph.  ...  Whenever one or more particles meet at a vertex, they unite to form a single particle, which then continues a random walk through the graph.  ...  We summarize some known results for coalescing random walks. There are two distinct models for the transition times of random walks on finite graphs.  ... 
doi:10.1137/120900368 fatcat:rfptirzvvbgnpklh5zji2rwpna

Random Walks on Random Graphs [chapter]

Colin Cooper, Alan Frieze
2009 Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering  
Among them are the following: Mixing time of the random walk, cover time of a random graph, properties of multiple particle walks, random walks on graph processes, constructing random networks using random  ...  Random Graphs Various topics arise in the context of random walks on random graphs.  ... 
doi:10.1007/978-3-642-02427-6_18 fatcat:jsowgcrxznf6jgw4kvwggwkv4e

An experimental evaluation of simrank-based similarity search algorithms

Zhipeng Zhang, Yingxia Shao, Bin Cui, Ce Zhang
2017 Proceedings of the VLDB Endowment  
Our study reveals that none of these algorithms dominates the others: algorithms based on iterative method often have higher accuracy while algorithms based on random walk can be more scalable.  ...  Given a graph, SimRank is one of the most popular measures of the similarity between two vertices.  ...  An FPG organizes in a compact way one random walk with length T for each vertex together with the distance where each two random walks meet for the first time.  ... 
doi:10.14778/3055540.3055552 fatcat:qb6alvxakfcd5dtyxqzrvpc6sm

On the Convergence Time of Asynchronous Distributed Quantized Averaging Algorithms

Minghui Zhu, Sonia Martinez
2011 IEEE Transactions on Automatic Control  
By appealing to random walks on graphs, we derive polynomial bounds on the expected convergence time of the algorithms presented, as a function of the number of agents in the network.  ...  The focus of this technical note is on the study of the convergence time of the proposed quantized averaging algorithms.  ...  The Meeting Time of Two Natural Random Walks on a Fixed Graph G We first study a variation of the problem in [8] , namely, the meeting time of two natural random walks on a fixed graph G.  ... 
doi:10.1109/tac.2010.2093276 fatcat:ig6ybtimd5g6xcatdbfvoc7jri

An Upper Bound on the Convergence Time for Quantized Consensus of Arbitrary Static Graphs

Shang Shang, Paul Cuff, Pan Hui, Sanjeev Kulkarni
2015 IEEE Transactions on Automatic Control  
We use the theory of electric networks, random walks, and couplings of Markov chains to derive an O(N^3 N) upper bound for the expected convergence time on an arbitrary graph of size N, improving on the  ...  Our result is not dependent on graph topology. Example of complete graphs is given to show how to extend the analysis to graphs of given topology.  ...  We use the degree of nodes on the shortest path on the graph to improve the bound on the hitting time of the biased random walk. • The analysis for arbitrary graphs is extended to a tighter bound for certain  ... 
doi:10.1109/tac.2014.2342071 fatcat:daapmymcw5ht5nt7ryf3ixtixm

The infection time of graphs

Tassos Dimitriou, Sotiris Nikoletseas, Paul Spirakis
2006 Discrete Applied Mathematics  
We easily get that an upper bound on the expected value of T k is the worst case (over all initial positions) expected meeting time m * of two random walks multiplied by (log k).  ...  The infection time T k of infecting all the white particles with red color is then a random variable that depends on k, the initial position of the particles, the number of nodes and edges of the graph  ...  The machinery stated above can be used also for the meeting times: Definition 8. Let M i,j be the first time that two independent copies of a random walk on G meet given that they start from i, j .  ... 
doi:10.1016/j.dam.2006.04.026 fatcat:d6cmaplzlbhxzozct5sx3ihjau

Robotic Surveillance Based on the Meeting Time of Random Walks [article]

Xiaoming Duan, Mishel George, Rushabh Patel, Francesco Bullo
2019 arXiv   pre-print
First, by analyzing multiple random walks on a common graph as a single random walk on the Kronecker product graph, we provide the first closed-form expression for the expected meeting time in terms of  ...  This paper analyzes the meeting time between a pair of pursuer and evader performing random walks on digraphs.  ...  The meeting time of two Markov chains Consider the pursuer and evader performing random walks on a strongly connected graph G = (V, E) with the node set V = {1, . . . , n} and E ⊂ V × V .  ... 
arXiv:1912.02693v1 fatcat:fssldkrtwjfdzeh4prjiy64wpi
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