Dear Colleagues,

We would like to invite you to the following SPASS seminar, jointly organized by UniPi, SNS, UniFi and UniSi: *Local and 'almost' local structure of random graphs* by *Remco van der Hofstad* (*Eindhoven University of Technology*)

The seminar will take place on TUE, 07.05.2024 at 14:00 CET in Aula 102, Dipartimento di Matematica e Informatica "Ulisse Dini", Università degli Studi di Firenze, and streamed online at this link https://meet.google.com/gji-phwo-vbg.

The organizers, A. Agazzi, G. Bet, A. Caraceni, F. Grotto, G. Zanco https://sites.google.com/unipi.it/spass -------------------------------------------- *Abstract:*

* Ever since its invention by Benjamini and Schramm, and independently by Aldous and Steele, local convergence has become an indispensable tool in random graph theory. Many properties, such as subgraph counts, clustering, degree distributions and degree-degree dependencies, but also the number of spanning trees on a graph, and, under mild extra conditions, the free energy of the Ising model, are asymptotically determined by the local limit. Even more properties are not local, yet their behavior in many random graph models is still dictated by the local limit. An example is the size of the maximal connected component, which was recently shown to be `almost' local, in the sense that the proportion of vertices in it equals the survival probability of the local limit subject to one natural necessary and sufficient condition. In this lecture, we give an overview of local convergence and the local and `almost local’ structure of random graphs. We discuss that most real-world networks are sparse, in that their average degrees are bounded, while they can be quite inhomogeneous in that vertices of very high degree may exist. For such settings, local convergence is the natural notion of graph convergence for graph sequences with diverging sizes. It describes how the proportion of vertices whose neighborhoods have a certain shape converges to a limiting value, which can be seen as the neighborhood of a vertex in a limiting infinite graph, which can be random even when the finite graph is deterministic. We then discuss consequences of local convergence for random graph sequences.*