Spectral density of random graphs with topological constraints

The spectral density of random graphs with topological constraints is analysed using the replica method. We consider graph ensembles featuring generalised degree-degree correlations, as well as those with a community structure. In each case an exact solution is found for the spectral density in the form of consistency equations depending on the statistical properties of the graph ensemble in question. We highlight the effect of these topological constraints on the resulting spectral density.Comment: 24 pages, 6 figure

A Sequential Algorithm for Generating Random Graphs

We present a nearly-linear time algorithm for counting and randomly generating simple graphs with a given degree sequence in a certain range. For degree sequence (di) n i=1 with maximum degree dmax = O(m1/4−τ), our algorithm generates almost uniform random graphs with that degree sequence in time O(mdmax) i di is the number of edges in the graph and τ is any positive con-where m = 1 2 stant. The fastest known algorithm for uniform generation of these graphs (McKay and Wormald in J. Algorithms 11(1):52–67, 1990) has a running time of O(m2d2 max). Our method also gives an independent proof of McKay’s estimate (McKay in Ars Combinatoria A 19:15–25, 1985) for the number of such graphs. We also use sequential importance sampling to derive fully Polynomial-time Randomized Approximation Schemes (FPRAS) for counting and uniformly generating random graphs for the same range of dmax = O(m1/4−τ). Moreover, we show that for d = O(n1/2−τ), our algorithm can generate an asymptotically uniform d-regular graph. Our results improve the previous bound of d = O(n1/3−τ) due to Kim and Vu (Adv. Math. 188:444–469, 2004) for regular graphs