Statistical field theory of random graphs with prescribed degrees
arXiv:2410.11191 · doi:10.1088/1742-5468/add0a4
Abstract
Statistical field theory methods have been very successful with a number of random graph and random matrix problems, but it is challenging to apply these methods to graphs with prescribed degree sequences due to the extensive number of constraints that enforce the desired degree at each vertex. Building on top of recent results where similar methods are applied to random regular graph counting, we develop an accurate statistical field theory description for the adjacency matrix spectra of graphs with prescribed degrees. For large graphs, the expectation values are dominated by functional saddle points satisfying explicit equations. For the case of regular graphs, this immediately leads to the known McKay distribution. We then consider mixed-regular graphs with N1 vertices of degree d1, N2 vertices of degree d2, etc, such that the ratios N_i/N are kept fixed as N goes to infinity. For such graphs, the eigenvalue densities are governed by nonlinear integral equations of the Hammerstein type. Solving these equations numerically reproduces with an excellent accuracy the empirical eigenvalue distributions.
v2: version accepted for publication
References in corpus (11)
- Cavity Approach to the Spectral Density of Sparse Symmetric Random Matrices
- Spectra of Sparse Random Matrices
- Spectra of random networks with arbitrary degrees
- Return times of random walk on generalized random graphs
- Analytic solution of the resolvent equations for heterogeneous random graphs: spectral and localization properties
- Random matrices with row constraints and eigenvalue distributions of graph Laplacians
- Entropy of microcanonical finite-graph ensembles
- Robust extended states in Anderson model on partially disordered random regular graphs
- A Gaussian integral that counts regular graphs
- Hammerstein equations for sparse random matrices
- First return times on sparse random graphs