非齐次随机图中Bootstrap过程演化的相变
A phase transition regarding the evolution of bootstrap processes in inhomogeneous random graphs
摘要：
感染阈值为$r\ge 1$的图上的自举渗流过程是一个按时间步长演化的传播过程。该过程从受感染顶点的子集开始，并且在每个后续步骤中，至少具有$r$受感染邻居的每个未受感染顶点都被感染并且永远保持感染。自举逾渗过程中的临界现象最初是由Aizenman和Lebowitz在20世纪80年代末观察到的，作为$\mathbb{Z}^d$中的有限体积相变，由感染顶点的小局部岛的积累引起。在稠密（齐次）随机图的情况下，Janson，Turova and Valier（2012）也观察到了它们。在本文中，我们考虑一类称为Chung-Lu模型的非齐次随机图：每个顶点都配备了一个正的权重，每对顶点作为一个边出现的概率与权重的乘积成正比。特别是，我们专注于稀疏制度，其中的边缘的数量是成比例的顶点的数量。本文的主要结果确定这些权重序列的一个关键现象发生：有一个临界密度的顶点，在开始的过程中被感染，一个小的（次线性）组以上的感染顶点创建一个雪崩的感染，从而导致爆发。我们表明，这基本上只发生在尾部的权重分布占主导地位的幂律指数为3，我们确定在这种情况下的临界密度。
英文摘要：
A bootstrap percolation process on a graph with infection threshold $r\ge 1$ is a dissemination process that evolves in time steps. The process begins with a subset of infected vertices and in each subsequent step every uninfected vertex that has at least $r$ infected neighbours becomes infected and remains so forever. Critical phenomena in bootstrap percolation processes were originally observed by Aizenman and Lebowitz in the late 1980s as finite-volume phase transitions in $\mathbb{Z}^d$ that are caused by the accumulation of small local islands of infected vertices. They were also observed in the case of dense (homogeneous) random graphs by Janson, Łuczak, Turova and Valier (2012). In this paper, we consider the class of inhomogeneous random graphs known as the Chung-Lu model: each vertex is equipped with a positive weight and each pair of vertices appears as an edge with probability proportional to the product of the weights. In particular, we focus on the sparse regime, where the number of edges is proportional to the number of vertices. The main results of this paper determine those weight sequences for which a critical phenomenon occurs: there is a critical density of vertices that are infected at the beginning of the process, above which a small (sublinear) set of infected vertices creates an avalanche of infections that in turn leads to an outbreak. We show that this occurs essentially only when the tail of the weight distribution dominates a power law with exponent 3 and we determine the critical density in this case.
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