Eigenvalue distribution of large weighted multipartite random sparse graphs

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 $\kappa$ of components, with asymptotically $\alpha_j \dot N$ vertices in each component, where the vector $(\alpha_1,\alpha_2, \ldots, \alpha_{\kappa})$ is fixed. Consider a connected graph $\Gamma$ with $\kappa$ 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 $\Gamma_{ij}=1$. Conversely, if $\Gamma_{ij}=0$, 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.