4th International ICST Conference on Performance Evaluation Methodologies and Tools

Research Article

A random walk model for infection on graphs

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  • @INPROCEEDINGS{10.4108/ICST.VALUETOOLS2009.7447,
        author={A. Ganesh and M. Draief},
        title={A random walk model for infection on graphs},
        proceedings={4th International ICST Conference on Performance Evaluation Methodologies and Tools},
        publisher={ICST},
        proceedings_a={VALUETOOLS},
        year={2010},
        month={5},
        keywords={random graphs complex networks communication systems wireless networks.},
        doi={10.4108/ICST.VALUETOOLS2009.7447}
    }
    
  • A. Ganesh
    M. Draief
    Year: 2010
    A random walk model for infection on graphs
    VALUETOOLS
    ICST
    DOI: 10.4108/ICST.VALUETOOLS2009.7447
A. Ganesh1,*, M. Draief2,*
  • 1: Department of Mathematics, University of Bristol, University Walk, Clifton Bristol BS8 1TW, U.K.
  • 2: Department of Electrical and Electronic Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, U.K.
*Contact email: A.Ganesh@bristol.ac.uk, M.Draief@imperial.ac.uk

Abstract

We address the question of understanding the effect of the underlying network topology on the spread of a virus and the dissemination of information when users are mobile performing independent random walks on a graph. To this end we propose a simple model of infection that enables to study the coincidence time of two random walkers on an arbitrary graph. By studying the coincidence time of a susceptible and an infected individual both moving in the graph we obtain estimates of the infection probability. The main result of this paper is to pinpoint the impact of the network topology on the infection probability. More precisely, we prove that for homogeneous graph including regular graphs and the classical Erd¨os-R´enyi model, the coincidence time is inversely proportional to the number of nodes in the graph. We then study the model on power-law graphs, that exhibit heterogeneous connectivity patterns, and show the existence of a phase transition for the coincidence time depending on the parameter of the power-law of the degree distribution.