Limiting Spectra of inhomogeneous random graphs

We consider sparse inhomogeneous Erd\H{o}s-R\'enyi random graph ensembles where edges are connected independently with probability $p_{ij}$. We assume that $p_{ij}= \varepsilon_N f(w_i, w_j)$ where $(w_i)_{i\ge 1}$ is a sequence of deterministic weights, $f$ is a bounded function and $N\varepsilon_N\to \lambda\in (0,\infty)$. We characterise the limiting moments in terms of graph homomorphisms and also classify the contributing partitions. We present an analytic way to determine the Stieltjes transform of the limiting measure. The convergence of the empirical distribution function follows from the theory of local weak convergence in many examples but we do not rely on this theory and exploit combinatorial and analytic techniques to derive some interesting properties of the limit. We extend the methods of Khorunzhy et al. (2004) and show that a fixed point equation determines the limiting measure. The limiting measure crucially depends on $\lambda$ and it is known that in the homogeneous case, if $\lambda\to\infty$, the measure converges weakly to the semicircular law (Jung and Lee (2018)). We extend this result of interpolating between the sparse and dense regimes to the inhomogeneous setting and show that as $\lambda\to \infty$, the measure converges weakly to a measure which is known as the operator-valued semicircular law.