  In an era of unprecedented deluge of (mostly unstructured) data, graphs are
proving more and more useful, across the sciences, as a flexible abstraction to
capture complex relationships between complex objects. One of the main
challenges arising in the study of such networks is the inference of
macroscopic, large-scale properties affecting a large number of objects, based
solely on the microscopic interactions between their elementary constituents.
Statistical physics, precisely created to recover the macroscopic laws of
thermodynamics from an idealized model of interacting particles, provides
significant insight to tackle such complex networks.
  In this dissertation, we use methods derived from the statistical physics of
disordered systems to design and study new algorithms for inference on graphs.
Our focus is on spectral methods, based on certain eigenvectors of carefully
chosen matrices, and sparse graphs, containing only a small amount of
information. We develop an original theory of spectral inference based on a
relaxation of various mean-field free energy optimizations. Our approach is
therefore fully probabilistic, and contrasts with more traditional motivations
based on the optimization of a cost function. We illustrate the efficiency of
our approach on various problems, including community detection, randomized
similarity-based clustering, and matrix completion.
