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A graphon approach to spectral analysis of random matrices

Yizhe Zhu, University of Washington
Monday, October 16, 2017 - 2:30pm to 3:30pm
MEB 246
Graphons were introduced in 2006 by Lovász and Szegedy as limits of graph sequences. The set of finite graphs with the cut metric gives rise to a metric space, and the completion of such space is the space of graphons. These objects can be represented as symmetric measurable functions from \$[0, 1]^2\$ to \$[0, 1]\$.
I will present a graphon theory approach to the limiting spectral distribution of general Wigner-type matrices. A sufficient condition of the existence, a formula to calculate moments, and the Stieltjes transform of the distribution are obtained in terms of the homomorphism densities from trees into a graphon. As applications, we are able to decide the limiting spectral distributions for new random graph models: sparse \$W\$-random graphs and stochastic block models with a growing number of blocks.