The surviving rate of an infected network

  • Authors:

  • Weifan Wang;Stephen Finbow;Ping Wang

  • Affiliations:

  • Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China;Department of Mathematics, Statistics and Computer Science, St. Francis Xavier University, Antigonish, Nova Scotia, Canada;Department of Mathematics, Statistics and Computer Science, St. Francis Xavier University, Antigonish, Nova Scotia, Canada

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2010

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Abstract

Let G be a connected network. Let k=1 be an integer. Suppose that a vertex v of G becomes infected. A program is then installed on k-nodes not yet infected. Afterwards, the virus spreads to all its unprotected neighbors in each time interval. The virus and the network administrator take turns until the virus can no longer spread further. Let sn"k(v) denote the maximum number of vertices in G the network administrator can save when a virus infects v. The k-surviving rate @r"k(G) of G is defined to be the average value @?"v"@?"V"("G")sn"k(v)/n^2. In particular, we write @r(G)=@r"1(G). In this paper, we first use a probabilistic method to show that almost all networks have k-surviving rate arbitrarily close to 0. Then, we prove the following results: (1) @r(G)=235 for a planar network G of girth at least 9; (2) @r"2(G)=116 for a series-parallel network G; and (3) @r"""2"""d"""-"""1(G)=25d for a d-degenerate network G.