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Title: Central limit theorem for Betti numbers in stochastic block models for random graphs Authors:  Yang Di - University of Nottingham (United Kingdom) [presenting]
Andrew Wood - The University of Nottingham (United Kingdom)
Huiling Le - University of Nottingham (United Kingdom)
Karthik Bharath - University of Nottingham (United Kingdom)
Abstract: In recent years there has been growing interest within statistics in topological aspects of random objects, one important direction being topological data analysis and the key concept of persistent homology. One interesting theoretical contribution is a central limit theorem (CLT) for Betti numbers in random clique complexes in Erdos-Renyi (ER) random graphs. The clique complex of an undirected graph is the simplicial complex formed by the sets of vertices in the cliques of the graph, a clique being a subgraph in which all edges are present. For each dimension, there is a certain range of probabilities where the Betti number is non-zero asymptotically almost surely in ER models. We consider a generalisation of the previous result to a CLT for Betti numbers in stochastic block model random graphs. The content of the result will be explained and the main ideas in the proof will be briefly outlined.