Talks

Abstract It is well known that sparse random graphs (where the number of edges is of the order of the number of vertices) contain only a small number of triangles. On the other hand, many networks observed in real life are sparse but contain a large number of triangles. Motivated by this discrepancy we ask the following two questions: How (un)likely is it that a sparse random graph contains a large number of triangles? What does the graph look like when it contains a large number of triangles? We also ask a related question: What is the probability that in a sparse random graph a large number of vertices are part of some triangle? We discuss these questions, as well as some applications to exponential random graph models.

Abstract   In many domains, we are interested in estimating the total causal treatment effect in the presence of network interference, where the outcome of one individual or unit is affected by the treatment assignment of those in its local network. Additional challenges arise when complex cluster randomized designs are not feasible to implement, or the network is unknown and costly to estimate. We propose a new measure of model complexity that characterizes the difficulty of estimating the total treatment effect under the standard A/B testing setup. We provide a class of unbiased estimators whose variance is optimal with respect to the population size and the treatment budget. Furthermore, we show that when the network is completely unknown, we can still estimate the total treatment effect under a richer yet simple staggered rollout experimental design. The proposed design principles, and related estimator, work with a broad class of outcome models. Our solution and statistical guarantees do not rely on restrictive network properties, allowing for highly connected networks that may not be easily clustered. This is joint work with Edoardo Airoldi, Christian Borgs, Jennifer Chayes, Mayleen Cortez, and Matthew Eichhorn

Abstract A random structure exhibits symmetry if its law remains invariant under a group of transformations. Exchangeability (of graphs, sequences, etc) and stationarity are examples. Under suitable conditions, the transformation group can be used to define an estimator that averages over an instance of the structure, and such estimators turn out to satisfy a law of large numbers, a central limit theorem, and further convergence results. Loosely speaking: The large-sample theory of i.i.d averages still holds if the i.i.d assumption is substituted by a suitable symmetry assumption.  Joint work with Morgane Austern.

Within the AdS/CFT correspondence, the entanglement properties of the CFT are related to wormholes in the dual gravity theory. This gives rise to questions about the factorisation properties of the Hilbert spaces on both sides of the correspondence.  We  show how the Berry phase, a geometrical phase encoding information about topology, may be used to reveal the Hilbert space structure. Wormholes are characterized by a non-exact symplectic form that gives rise to the Berry phase. For wormholes connecting two spacelike regions in AdS3 spacetimes, we find that the non-exactness is linked to a variable appearing in the phase space of the boundary CFTs. Mathematical concepts such as coadjoint orbits and geometric actions play an important role in this analysis.  We classify Berry phases according to the type of dual bulk diffeomorphism involved, distinguishing between Virasoro, gauge and modular Berry phases.

In addition to its relevance for quantum gravity, the approach presented also suggests how to experimentally realize the Berry phase and its relation to entanglement in table-top experiments involving photons or electrons. This provides a new example for relations between very different branches of physics that follow from the AdS/CFT correspondence and its generalizations. Based on 2202.11717 and 2109.06190.

Zoom link: https://pitp.zoom.us/j/96113910200?pwd=YXJnSWxiMHRIb21xdGFnNFM0cFFvUT09

Abstract Dense graph limit theory is mainly concerned with law-of large-number type of results. We propose a corresponding central limit theorem - or rather fluctuation theory - based on Janson's theory of Gaussian Hilbert Spaces and generalised U-statistics from the 1990s. Our approach provides rates and allows for proper statistical inference based on subgraph counts.

Abstract   The problem of completing a large low rank matrix using a subset of revealed entries has received much attention in the last ten years. I will give a necessary and sufficient condition, stated in the language of graph limit theory, for a sequence of matrix completion problems with arbitrary missing patterns to be asymptotically solvable. I will also present an algorithm that is able to approximately recover the matrix whenever recovery is possible.

Abstract Ramsey's Theorem states that for every graph H, there is an integer R(H) such that every 2-edge-coloring of R(H)-vertex complete graph contains a monochromatic copy of H. In this talk, we focus on a natural quantitative extension: how many monochromatic copies of H can we find in every 2-edge-coloring of K_N, and what graphs H are so-called common, i.e., the number of monochromatic copies of H is asymptotically minimized by a random 2-edge-coloring. A classical result of Goodman from 1959 states that the triangle is a common graph. On the other hand, Thomason proved in 1989 that no clique of order at least four is common, and the existence of a common graph with chromatic number larger than three was open until 2012, when Hatami, Hladky, Kral, Norin and Razborov proved that the 5-wheel is common. In this talk, we show that for every k>4 there exists a common graph with chromatic number k. This is a joint work with D. Kral and F. Wei

Information about the late-time Universe is imprinted on the small scale CMB as photons travel to us from the surface of last scattering. Several processes are at play and small scale fluctuations are very rich and non-Gaussian in nature. I will review some of the most important effects and I will focus on the Sunyaev-Zel'dovich (SZ) effect and gravitational lensing. I will discuss how a combination of measurements can probe velocity fields at cosmological distances and inform us on cluster energetics. I will also show recent measurements of weak lensing of the CMB and how they can help us interpret intriguing discrepancies in cosmological parameters between the high and low redshift Universe.

Zoom link: https://pitp.zoom.us/j/94451033605?pwd=Tkx4dHZTblMxUFJlZENyblJQVFo2dz09

Abstract Motifs (patterns of subgraphs), such as edges and triangles, encode important structural information about the geometry of a network. Consequently, counting motifs in a large network is an important statistical and computational problem. In this talk we will consider the problem of estimating motif densities and fluctuations of subgraph counts in an inhomogeneous random graph sampled from a graphon. We will show that the limiting distributions of subgraph counts can be Gaussian or non-Gaussian, depending on a notion of regularity of subgraphs with respect to the graphon. Using these results and a novel multiplier bootstrap for graphons, we will construct joint confidence sets for the motif densities. Finally, we will discuss various structure theorems and open questions about degeneracies of the limiting distribution. Joint work with Anirban Chatterjee, Soham Dan, and Svante Janson.

Abstract We initiate a study of large deviations for block model random graphs in the dense regime. Following Chatterjee and Varadhan (2011), we establish an LDP for dense block models, viewed as random graphons. As an application of our result, we study upper tail large deviations for homomorphism densities of regular graphs. We identify the existence of a "symmetric'' phase, where the graph, conditioned on the rare event, looks like a block model with the same block sizes as the generating graphon. In specific examples, we also identify the existence of a "symmetry breaking'' regime, where the conditional structure is not a block model with compatible dimensions. This identifies a "reentrant phase transition'' phenomenon for this problem---analogous to one established for Erdős–Rényi random graphs (Chatterjee and Dey (2010), Charrerjee and Varadhan (2011)). Finally, extending the analysis of Lubetzky and Zhao (2015), we identify the precise boundary between the symmetry and symmetry breaking regimes for homomorphism densities of regular graphs and the operator norm on Erdős–Rényi bipartite graphs. Joint work with Christian Borgs, Jennifer Chayes, Subhabrata Sen, and Samantha Petti