CONTACT PROCESSES ON RANDOM GRAPHS WITH POWER LAW DEGREE DISTRIBUTIONS HAVE CRITICAL VALUE 0
CONTACT PROCESSES ON RANDOM GRAPHS WITH POWER LAW DEGREE DISTRIBUTIONS HAVE CRITICAL VALUE 0
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DOI：
10.1214/09-aop471
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发表时间：
2009-11-01
影响因子：
2.3
通讯作者：
Durrett, Rick
中科院分区：
文献类型：
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作者：
Chatterjee, Shirshendu;Durrett, Rick
如果我们考虑一个具有幂律度分布的n阶随机图上传染率为λ的接触过程，平均场计算表明，当幂α> 3时，传染率的临界值λ（c）为正。物理学家似乎认为这是一个既定的事实，因为该结果最近已被推广到二分图戈麦斯Gardefies等人。Acad. Sci. USA 105（2008）1399-1404]。在这里，我们证明了对于任何alpha > 3的值，临界值lambda（c）都是零，并且从所有被感染的顶点开始的接触过程，随着n ->无穷大，概率趋于1，对于任何delta > 0，至少在exp（n（1-delta））的时间内保持正的感染点密度。使用最后的结果，再加上接触过程对偶，我们可以建立一个准平稳分布的存在性，其中一个随机选择的顶点被占据的概率ρ（λ）。预期p（lambda）类似于C lambda（beta），因为lambda -> 0。在这里我们展示了阿尔法-13。因此，即使图是局部树状的，beta也不取平均场临界值beta = 1。
If we consider the contact process with infection rate lambda on a random graph on n vertices with power law degree distributions, mean field calculations suggest that the critical value lambda(c) of the infection rate is positive if the power alpha > 3. Physicists seem to regard this as an established fact, since the result has recently been generalized to bipartite graphs by Gomez-Gardefies et al. [Proc. Nad. Acad. Sci. USA 105 (2008) 1399-1404]. Here, we show that the critical value lambda(c) is zero for any value of alpha > 3, and the contact process starting from all vertices infected, with a probability tending to 1 as n -> infinity, maintains a positive density of infected sites for time at least exp(n(1-delta)) for any delta > 0. Using the last result, together with the contact process duality, we can establish the existence of a quasi-stationary distribution in which a randomly chosen vertex is occupied with probability rho(lambda). It is expected that p(lambda) similar to C lambda(beta) as lambda -> 0. Here we show that alpha - 1 3. Thus even though the graph is locally tree-like, beta does not take the mean field critical value beta = 1.