The Survival of the Weakest in Networks

  • S. Nikoletseas
  • C. Raptopoulos
  • P. Spirakis
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4368)


We study here dynamic antagonism in a fixed network, represented as a graph G of n vertices. In particular, we consider the case of kn particles walking randomly independently around the network. Each particle belongs to exactly one of two antagonistic species, none of which can give birth to children. When two particles meet, they are engaged in a (sometimes mortal) local fight. The outcome of the fight depends on the species to which the particles belong. Our problem is to predict (i.e. to compute) the eventual chances of species survival. We prove here that this can indeed be done in expected polynomial time on the size of the network, provided that the network is undirected.


Polynomial Time Initial Position Directed Graph Undirected Graph Replicator Dynamic 


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  1. 1.
    Bshouty, N., Higham, L., Warpechowska-Gruca, J.: Meeting Times of Random Walks on Graphs. Information Processing Letters 69, 259–266 (1999)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Imhof, L.A.: The long-run behaviour of the stochastic replicator dynamics. Annals of Applied Probability 15(1B), 1019–1045 (2005)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Kandori, M., Mailath, G.J., Rob, R.: Learning, Mutation and Long Run Equilibria in Games. Econometrica 61(1), 29–56 (1993)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Maynard Smith, J.: Evolution and the Theory of Games. Cambridge University Press, Cambridge (1982)MATHGoogle Scholar
  5. 5.
    Nikoletseas, S., Raptopoulos, C., Spirakis, P.: The Survival of the Weakest in Networks,
  6. 6.
    Norris, J.: Markov Chains. Cambridge University Press, Cambridge (1997)MATHGoogle Scholar
  7. 7.
    Ross, S.: Probability Models for Computer Science. Harcourt Academic Press, London (2000)Google Scholar
  8. 8.
    Samuelson, L.: Evolutionary Games and Equilibrium Selection. MIT Press, Cambridge Google Scholar
  9. 9.
    Tetali, P., Winkler, P.: On a random walk problem arising in self-stabilizing token management. In: Proceedings of the 10th Annual ACM Symposium on Principles of Distributed Computing, pp. 273–280 (1991)Google Scholar
  10. 10.
    Weibull, J.W.: Evolutionary Game Theory. MIT Press, Cambridge Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • S. Nikoletseas
    • 1
    • 2
  • C. Raptopoulos
    • 1
    • 2
  • P. Spirakis
    • 1
    • 2
  1. 1.Computer Technology InstitutePatrasGreece
  2. 2.University of PatrasPatrasGreece

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