Yizhe Zhu, University of Washington

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.