The surviving rate of an infected network

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 snk(v) denote the maximum number of vertices in G the network administrator can save when a virus infects v. The k-surviving rate ρk(G) of G is defined to be the average value ∑v∈V(G)snk(v)/n2. In particular, we write ρ(G)=ρ1(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) for a planar network G of girth at least 9; (2) for a series–parallel network G; and (3) for a d-degenerate network G.