Index statistical properties of sparse random graphs

Index statistical properties of sparse random graphs.

Using the replica method, we develop an analytical approach to compute the characteristic function for the probability P(N)(K,λ) that a large N×N adjacency matrix of sparse random graphs has K eigenvalues below a threshold λ. The method allows to determine, in principle, all moments of P(N)(K,λ), from which the typical sample-to-sample fluctuations can be fully characterized. For random graph m...

Statistical Mechanics and Algorithms on Sparse and Random Graphs

Bisecting sparse random graphs

Consider partitions of the vertex set of a graph G into two sets with sizes differing by at most 1: the bisection width of G is the minimum over all such partitions of the number of ‘‘cross edges’’ between the parts. We are interested in sparse random graphs Ž . G with edge probability c n. We show that, if c ln 4, then the bisection width is n n, c n with high probability; while if c ln 4, the...

Sparse Quasi-Random Graphs

Quasi-random graph properties form a large equivalence class of graph properties which are all shared by random graphs. In recent years, various aspects of these properties have been treated by a number of authors (e.g., see [5]-[14], [16], [23]-[27]). Almost all of these results deal with dense graphs, that is, graphs on n vertices having cn edges for some c > 0 as n → ∞. In this paper, we ext...

Dismantling Sparse Random Graphs

We consider the number of vertices that must be removed from a graph G in order that the remaining subgraph has no component with more than k vertices. Our principal observation is that, if G is a sparse random graph or a random regular graph on n vertices with n→∞, then the number in question is essentially the same for all values of k that satisfy both k →∞ and k = o(n). The process of removi...