Inverse clustering of Gibbs Partitions via independent fragmentation and dual dependent coagulation operators

Gibbs partitions of the integers generated by stable subordinators of index $\alpha\in(0,1)$ form remarkable classes of random partitions where in principle much is known about their properties, including practically effortless obtainment of otherwise complex asymptotic results potentially relevant to applications in general combinatorial stochastic processes, random tree/graph growth models and Bayesian statistics. This class includes the well-known models based on the two-parameter Poisson-Dirichlet distribution which forms the bulk of explicit applications. This work continues efforts to provide interpretations for a larger classes of Gibbs partitions by embedding important operations within this framework. Here we address the formidable problem of extending the dual, infinite-block, coagulation/fragmentation results of Jim Pitman (1999, Annals of Probability), where in terms of coagulation they are based on independent two-parameter Poisson-Dirichlet distributions, to all such Gibbs (stable Poisson-Kingman) models. Our results create nested families of Gibbs partitions, and corresponding mass partitions, over any $0<\beta<\alpha<1.$ We primarily focus on the fragmentation operations, which remain independent in this setting, and corresponding remarkable calculations for Gibbs partitions derived from that operation. We also present definitive results for the dual coagulation operations, now based on our construction of dependent processes, and demonstrate its relatively simple application in terms of Mittag-Leffler and generalized gamma models. The latter demonstrates another approach to recover the duality results in Pitman (1999)

Consensus clustering in complex networks

The community structure of complex networks reveals both their organization and hidden relationships among their constituents. Most community detection methods currently available are not deterministic, and their results typically depend on the specific random seeds, initial conditions and tie-break rules adopted for their execution. Consensus clustering is used in data analysis to generate stable results out of a set of partitions delivered by stochastic methods. Here we show that consensus clustering can be combined with any existing method in a self-consistent way, enhancing considerably both the stability and the accuracy of the resulting partitions. This framework is also particularly suitable to monitor the evolution of community structure in temporal networks. An application of consensus clustering to a large citation network of physics papers demonstrates its capability to keep track of the birth, death and diversification of topics.Comment: 11 pages, 12 figures. Published in Scientific Report