Statistical field theory of random graphs with prescribed degrees

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.

Figures (1)

• Figure 1: Solutions of the nonlinear system (\ref{['eqJ']}) for $\beta_j$, followed by reconstructing the eigenvalue densities (\ref{['pmixreg']}), plotted as solid black lines (left) for $70\%$ of degree 3 vertices and $30\%$ of degree 5 vertices, solved at $J=7$, $\gamma=2$ and (right) for $50\%$ of degree 4 vertices and $50\%$ of degree 12 vertices, solved at $J=7$, $\gamma=3/2$. As the distributions are reflection-symmetric, only the $\lambda>0$ parts of the analytic predictions (solid black lines) are plotted explicitly. For the initial seed in the root search algorithm, $\beta_0=1$ and $\beta_{j>0}=0$ is used at $\lambda=0$, and the previous solution is reused as the seed for each next value of $\lambda$. The grey shaded areas represent empirical eigenvalue density histograms obtained from a sample of 500 random graphs with 10000 vertices each and the corresponding degree proportions.