Robotic Surveillance and Markov Chains With Minimal Weighted Kemeny Constant

Patel, Rushabh; Agharkar, Pushkarini; Bullo, Francesco

doi:10.1109/tac.2015.2426317

Cited by 50 publications

(28 citation statements)

References 30 publications

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“…Second, we provided results for multiple Markov chains in which travel times are homogeneous. A clear extension is to consider the case of heterogeneous travel times similar to what was done in [32]. Finally, given the maximum pairwise hitting time of a Markov chain, there exist bounds on the cover time for multiple copies of that Markov chain running in parallel [11,2].…”

Section: Discussionmentioning

confidence: 99%

The Hitting Time of Multiple Random Walks

Patel¹,

Carron²,

Bullo³

2016

SIAM J. Matrix Anal. & Appl.

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This work provides generalized notions and analysis methods for the hitting time of random walks on graphs. The hitting time, also known as the Kemeny constant or the mean first passage time, of a random walk is widely studied; however, only limited work is available for the multiple random walker scenario. In this work we provide a novel method for calculating the hitting time for a single random walker as well as the first analytic expression for calculating the hitting time for multiple random walkers, which we denote as the group hitting time. We also provide a closed form solution for calculating the hitting time between specified nodes for both the single and multiple random walker cases. Our results allow for the multiple random walks to be different and, moreover, for the random walks to operate on different subgraphs. Finally, using sequential quadratic programming, we show that the combination of transition matrices that generate the minimal group hitting time for various graph topologies is often different.

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Section: Discussionmentioning

confidence: 99%

The Hitting Time of Multiple Random Walks

Patel¹,

Carron²,

Bullo³

2016

SIAM J. Matrix Anal. & Appl.

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“…Mean hitting time has found many applications in different areas [53]. For example, it can be applied to measure the efficiency of user navigation through the World Wide Web [52], as well as the efficiency of robotic surveillance in network environments [54].…”

Section: Mean Hitting Timementioning

confidence: 99%

Extended Corona Product as an Exactly Tractable Model for Weighted Heterogeneous Networks

Zhang

2017

The Computer Journal

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Various graph products and operations have been widely used to construct complex networks with common properties of real-life systems. However, current works mainly focus on designing models of binary networks, in spite of the fact that many real networks can be better mimicked by heterogeneous weighted networks. In this paper, we develop a corona product of two weighted graphs, based on which and an observed updating mechanism of edge weight in real networks, we propose a minimal generative model for inhomogeneous weighted networks. We derive analytically relevant properties of the weighted network model, including strength, weight and degree distributions, clustering coefficient, degree correlations and diameter. These properties are in good agreement with those observed in diverse real-world weighted networks. We then determine all the eigenvalues and their corresponding multiplicities of the transition probability matrix for random walks on the weighted networks. Finally, we apply the obtained spectra to derive explicit expressions for mean hitting time of random walks and weighted counting of spanning trees on the weighted networks. Our model is an exactly solvable one, allowing to analytically treat its structural and dynamical properties, which is thus a good test-bed and an ideal substrate network for studying different dynamical processes, in order to explore the impacts of heterogeneous weight distribution on these processes.

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“…For example, it is used to evaluate and improve user navigation efficiency through web graphs [20], and also for search and discovery in peer-to-peer networks by interpreting the Kemeny's constant as the search time of a random search method [7,26,30]. It is recently used as an indicator of network robustness to design surveillance policies for the quickest detection of intruders and anomalies in the network [25].…”

Section: Introductionmentioning

confidence: 99%

An Efficient and Scalable Algorithm for Estimating Kemeny's Constant of a Markov Chain on Large Graphs

Xin

Lee

2021

Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery &Amp; Data Mining

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The mean hitting time of a Markov chain on a graph from an arbitrary node to a target node randomly chosen according to its stationary distribution is called Kemeny's constant, which is an important metric for network analysis and has a wide range of applications. It is, however, still computationally expensive to evaluate the Kemeny's constant, especially when it comes to a large graph, since it requires the computation of the spectrum of the corresponding transition matrix or its normalized Laplacian matrix. In this paper, we propose a simple yet computationally efficient Monte Carlo algorithm to approximate the Kemeny's constant, which is equipped with an ( , )-approximation estimator. Thanks to its inherent algorithmic parallelism, we are able to develop its parallel implementation on a GPU to speed up the computation. We provide extensive experiment results on 13 real-world graphs to demonstrate the computational efficiency and scalability of our algorithm, which achieves up to 500× speed-up over the state-ofthe-art algorithm. We further present its practical enhancements to make our algorithm ready for practical use in real-world settings.

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Robotic Surveillance and Markov Chains With Minimal Weighted Kemeny Constant

Cited by 50 publications

References 30 publications

The Hitting Time of Multiple Random Walks

The Hitting Time of Multiple Random Walks

Extended Corona Product as an Exactly Tractable Model for Weighted Heterogeneous Networks

An Efficient and Scalable Algorithm for Estimating Kemeny's Constant of a Markov Chain on Large Graphs

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