Dear colleagues,

we are pleased to invite you to the following two seminars (on site) on Wednesday, November 17, 2021, at the University of Florence.

The room is the Aula Magna, building Morgagni 65 (ex Farmacologia), Viale Morgagni 65, 50134, Firenze.

Time: 11:45-12:45 Speaker: Elena Magnanini, WIAS Berlin Title: LIMIT THEOREMS FOR THE EDGE DENSITY IN EXPONENTIAL RANDOM GRAPHS Abstract: In the present talk we focus on the derivation of some asymptotic properties for the family of exponential random graphs. This model can be seen as the generalization of the dense Erdӧs-Rényi random graph and follows the statistical mechanics approach of defining a Hamiltonian to weight the probability measure on the space of graphs, assigning higher mass to graphs with “desirable” properties. In particular our analysis will be focused on the edge-triangle model, a two-parameter family of exponential random graphs where the Hamiltonian only includes edge and triangle densities. We borrow tools from statistical mechanics together with large deviations techniques to obtain limit theorems for the edge density in the so-called replica symmetric regime, where the limiting free energy of the model is known together with a complete characterization of the phase diagram. First, we determine the asymptotic distribution of the edge density, as the graph size n tends to infinity, in the entire replica symmetric regime. In particular, we obtain a strong law of large numbers when the parameters are chosen outside the critical curve and the convergence to a mixture of Dirac measures whenever working on the critical curve. We then study the fluctuations of the edge density around its average for all parameter values outside the critical curve and off the critical point and we formulate conjectures about the behavior at criticality based on the analysis of a mean-field approximation of the model. Joint work with Alessandra Bianchi and Francesca Collet.

Time: 14:15-15:15 Speaker: Tejas Iyer, WIAS Berlin Title: CONDENSATION IN PREFERENTIAL ATTACHMENT TREES WITH NEIGHBOURHOOD INFLUENCE Abstract: Motivated by the structure of complex networks such as the internet, we consider a growing model of preferential attachment trees with neighbourhood influence, where vertices arrive one at a time, are equipped with independent weights, and connect to existing vertices with probability proportional to their fitness function: a function of their own weight and the weights of their neighbours. In this model we prove almost sure limiting statements for the proportion of vertices with a given degree having weight belonging to a given measurable set, and the proportion of edges in the tree with endpoint belonging to a measurable set. We show that under certain conditions, the latter quantity demonstrates a condensation phenomenon, in which a positive proportion of edges in the network accumulate among those of weight that confers maximal reinforcement of fitness. Finally, we derive almost sure limiting statements for the log of the degree of a fixed vertex in this model, and prove that in this model the degree distribution behaves like a power law - a ubiquitous feature of many real-world complex networks.

According to the current regulations to prevent the spread of Coronavirus the Green pass is mandatory to access the venue and contact tracing will be done.

Best regards,

Luisa Andreis and Gianmarco Bet

Luisa Andreis ------------------------------------------------------- RTD-A Dipartimento di Matematica e Informatica “Ulisse Dini" Università degli Studi di Firenze Viale Morgagni 65, 50134, Firenze, IT Personal webpage: https://sites.google.com/view/luisaandreis/home Email: luisa.andreis@unifi.it