Eigenvalue distribution of large weighted multipartite random sparse graphs

Abstract

We investigate the distribution of eigenvalues of weighted adjacency matrices from a specific ensemble of random graphs. We distribute $N$ vertices across a fixed number $\hat{I}º$ of components, with asymptotically $\hat{I}{\pm }_j\dot{N}$ vertices in each component, where the vector $(\hat{I}{\pm }_1,\hat{I}{\pm }_2,\dots ,\hat{I}{\pm }_{\hat{I}}º)$ is fixed. Consider a connected graph $\hat{I}“$ with $\hat{I}º$ vertices. We construct a multipartite graph with $N$ vertices, in which all vertices in the $i$-th component are connected to all vertices in the $j$-th component if if $\hat{I}{“}_{ij}=1$. Conversely, if $\hat{I}{“}_{ij}=0$, no edge connects the $i$-th and $j$-th components. In the resulting graph, we independently retain each edge with a probability of $p/N$, where $p$ is a fixed parameter. To each remaining edge, we assign an independent weight with a fixed distribution, that possesses all finite moments. We establish the weak convergence in probability of a random counting measure to a non-random probability measure. Furthermore, the moments of the limiting measure can be derived from a system of recurrence relations.

11 pages. arXiv admin note: substantial text overlap with arXiv:1312.0423