Spectral-radius asymptotic conjecture for directed Chung–Lu random graphs

Let $A$ be the adjacency matrix of a realization from the directed Chung–Lu model with expected in-degree and out-degree vectors $\mathbf{a}$ and $\mathbf{b}$, respectively, and let

$$S=\sum_i a_i=\sum_i b_i$$

be the expected number of edges. Write $\rho(A)$ for the absolute value of the dominating eigenvalue of $A$. Spectral-radius asymptotic conjecture. For a sequence of realizations in which $S$ tends to infinity,

$$\lim_{N\to\infty}\frac{\rho(A)}{\frac{\mathbf{a}\cdot\mathbf{b}}{S}}=1$$

almost surely. This conjecture would provide a rigorous justification for the asymptotic approximation to the spectral radius proposed for directed Chung–Lu random graphs and could support spectral methods for community detection. The supplied text does not establish whether the conjecture has been proved or disproved.