We investigate the heterogeneity of outcomes of repeated instances of percolation experiments in complex networks using a message passing approach to evaluate heterogeneous, node dependent probabilities of belonging to the giant or percolating cluster, i.e. the set of mutually connected nodes whose size scales linearly with the size of the system. We evaluate these both for large finite single instances, and for synthetic networks in the configuration model class in the thermodynamic limit. For the latter, we consider both Erdos-Renyi and scale free networks as examples of networks with narrow and broad degree distributions respectively. For real-world networks we use an undirected version of a Gnutella peer-to-peer file-sharing network with $N=62,568$ nodes as an example. We derive the theory for multiple instances of both uncorrelated and correlated percolation processes. For the uncorrelated case, we also obtain a closed form approximation for the large mean degree limit of Erdos-Renyi networks.
Cluster-size heterogeneity in the two-dimensional Ising model
我们数值研究了二维伊辛模型中团簇大小的不均匀性，并验证了最近在渗流问题[Phys. Rev. E 84，010101（R）（2011）]背景下提出的标度形式。通过有限尺寸标度分析得到的标度指数与理论值的分形维数$d_f$和Fisher指数$τ$的集群分布是一致的。我们还指出，强大的有限尺寸效应存在由于集群大小的异质性的几何性质。
Fluctuating micro-heterogeneity in water - tert-butyl alcohol mixtures and lambda-type divergence of the mean cluster size with phase transition-like multiple anomalies
$(k,n)$-core percolation on hypergraphs with anchor nodes
Advanced-Basic-Convergence Research Institute, Chungbuk National University（忠北国立大学先进基础融合研究院）；Department of Physics, Chungbuk National University（忠北国立大学物理系）；Centre for Complex Systems, School of Mathematical Sciences, Queen Mary University of London（伦敦大学玛丽女王学院数学科学学院复杂系统中心）
