Charles Bordenave studied at Ecole Polytechnique in France, and received his PhD in 2006 under the supervision of François Baccelli. He was a postdoc at UC Berkeley before joining University of Toulouse as a CNRS researcher. Since 2018, he has been CNRS researcher at Aix-Marseille University. His research interests are in random matrices, random graphs, combinatorial optimization, stochastic geometry and more recently mixing times of Markov chains. He serves on the editorial boards of the Annals of Applied Probability, Annals of Probability and Bernoulli. He is the recipient of the 2017 Marc Yor prize from the French Academy of Sciences.
Charles Bordenave’s Medallion Lecture will be given at the INFORMS-APS 2019 meeting, July 3–5, 2019, in Brisbane, Australia [note that these are the correct dates, not July 2–4 as previously advertised]. The meeting website is http://informs-aps.smp.uq.edu.au/
Non-backtracking spectrum of random matrices.
A fruitful line of thought in the study of discrete combinatorial structures, such as graphs, is to look for natural matrices or operators whose spectrum will contain valuable and accessible information on the underlying combinatorial structure.
The adjacency matrix of a graph is the matrix acting on the vertices of the graph whose entries are zero or one depending on the presence or absence of an edge between a given pair of vertices. The entries of powers of the adjacency matrix count paths along the edges of the graph.
The non-backtracking matrix of a graph is a matrix acting on pairs of vertices sharing an edge. The entries of powers of the non-backtracking matrix count non-backtracking paths along the edges of the graph, that is, paths which do not successively visit the same edge twice. A non-backtracking path may be interpreted as a discrete geodesic.
This matrix was introduced by Hashimoto in 1988 in the context of the Ihara zeta function of a graph, which is an analog in a discrete setting of the Selberg zeta function of a Riemannian manifold. In recent years, due to its strong geometric flavor, this non-backtracking matrix has been promoted as a powerful tool to analyze the subtle interplay between the geometry of a graph and its spectrum.
It has found a wide range of applications. For example, in 2013, Krzakala et al. have used this matrix in the design of a spectral algorithm to detect communities in social networks which bypass some of the limitations of classical spectral algorithms. Before that, Friedman (2008) used this matrix to prove the celebrated Alon’s second eigenvalue conjecture which asserts that for any integer d, as n goes to infinity (with n×d even), almost all d-regular graphs on n vertices have the largest possible spectral gap at first order.
Non-backtracking matrices enjoy beautiful algebraic identities among which a spectral correspondence with matrices such as adjacency matrices. In this talk, we will explain how these spectral mappings can be used to translate problems on extremal eigenvalues of adjacency matrices into problems on extremal eigenvalues of weighted non-backtracking matrices.
We will then focus the talk on the study of the extremal eigenvalues of non-backtracking matrices of classical random graph ensembles: uniform regular graphs, Erdős-Rényi random graphs, stochastic block models and random n-lifts of a base graph. We will notably explain how these results can be used in community detection problems and in the theory of expander graphs, classical and quantum.
This is based on joint work with Benoit Collins, Marc Lelarge and Laurent Massoulié.