On the Renormalization of Random Network Models

This thesis studies the renormalization of random network models, with a focus on the topological and algebraic struc- tures that emerge when networks are coarse-grained or par- tially observed. It is divided into two parts. The first anal- yses the Multi-Scale Model (MSM), a random graph model that arises as a fixed point of a node-aggregation renormal- ization procedure. In the infinite-mean fitness regime, Chap- ter 2 characterizes the spectrum of the adjacency matrix: the leading eigenvalues scale as √ n, alternate in sign, and are as- sociated with a logarithmic-spiral structure and log-periodic eigenvectors, while the spectral bulk reaches the same scale. Chapter 3 proves that the same model generates non-trivial clustering without latent geometry and exhibits a breakdown of self-averaging for relevant network observables. The sec- ond part develops a renormalization framework for random network models with correlated edges. Chapter 4 shows that marginalization over unobserved variables can be formulated as a renormalization flow of effective Hamiltonians and that higher-order interactions are generically induced. Chapter 5 solves the resulting flow exactly for one-dimensional, maxi- mum coordination-two topologies and examines disordered extensions and applications to network dynamics and infer- ence. Chapter 6 provides a general solution of second-order random network models by reducing their partition functions to the diagonalization of a link-interaction adjacency matrix, revealing connections with algebraic graph theory and the phase structure of correlated network ensembles.