Goltsev A V, Dorogovtsev S N, Mendes J F F
Departamento de Física da Universidade de Aveiro, Portugal.
Phys Rev E Stat Nonlin Soft Matter Phys. 2006 May;73(5 Pt 2):056101. doi: 10.1103/PhysRevE.73.056101. Epub 2006 May 1.
We develop the theory of the -core (bootstrap) percolation on uncorrelated random networks with arbitrary degree distributions. We show that the -core percolation is an unusual, hybrid phase transition with a jump emergence of the k-core as at a first order phase transition but also with a critical singularity as at a continuous transition. We describe the properties of the -core, explain the meaning of the order parameter for the k-core percolation, and reveal the origin of the specific critical phenomena. We demonstrate that a so-called "corona" of the k-core plays a crucial role (corona is a subset of vertices in the k-core which have exactly neighbors in the -core). It turns out that the k-core percolation threshold is at the same time the percolation threshold of finite corona clusters. The mean separation of vertices in corona clusters plays the role of the correlation length and diverges at the critical point. We show that a random removal of even one vertex from the k-core may result in the collapse of a vast region of the k-core around the removed vertex. The mean size of this region diverges at the critical point. We find an exact mapping of the k-core percolation to a model of cooperative relaxation. This model undergoes critical relaxation with a divergent rate at some critical moment.
我们发展了关于具有任意度分布的非相关随机网络上的(k)核（自展）渗流理论。我们表明，(k)核渗流是一种不寻常的混合相变，(k)核会像在一阶相变时那样突然出现，同时也具有像连续相变时那样的临界奇点。我们描述了(k)核的性质，解释了(k)核渗流序参量的意义，并揭示了特定临界现象的起源。我们证明了(k)核的所谓“日冕”起着关键作用（日冕是(k)核中在(k)核内恰好有(k)个邻居的顶点子集）。结果表明，(k)核渗流阈值同时也是有限日冕簇的渗流阈值。日冕簇中顶点的平均间距起着关联长度的作用，并在临界点发散。我们表明，从(k)核中随机移除哪怕一个顶点都可能导致围绕被移除顶点的(k)核的大片区域崩溃。该区域的平均大小在临界点发散。我们找到了(k)核渗流到一个协同弛豫模型的精确映射。这个模型在某个临界时刻以发散速率经历临界弛豫。
