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Local weak convergence, Zeta limits and random topology

Banff International Research Station for Mathematical Innovation and Discovery

Local weak convergence is a powerful framework for study of sparse graph limits and has been successfully applied in obtaining exact expectation asymptotics in probabilistic combinatorial optimization​, statistical physics and random graph theory. In particular, it can be used to show that sum of lifetime sum of $H_0$-persistent diagram on a mean field model (complete graph with i.i.d. weights) converges to $\zeta(3)$, where $\zeta$ is the Riemann-zeta function. Further, using this framework the minimum cost function on the complete bipartite graph with i.i.d. weights was shown to converge to $\zeta(2)$. In this talk, we shall look at some underlying ideas behind such results and wonder about the possibility of extensions to random topology. As is to be expected, when we move from random graphs to random complexes, there will be fewer answers and more questions.

Mathematics; Algebraic topology; Probability theory and stochastic processes; Data mining

video/mp4

Author affiliation: Indian Statistical Institute

2018-01-30

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/64488

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/