Minimum stationary values of sparse random directed graphs

TL;DR

This study analyzes extremal stationary values for sparse directed graphs under bounded-degree assumptions, showing that the minimum stationary probability π_min decays polynomially with n at rate n^{-(1+hat{H}^-/φ(a_0))} whp in the uniqueness regime (δ^+ ≥ 2). The approach builds a bridge between local graph exploration and marked Bienaymé–Galton–Watson branching processes, leveraging large deviation theory to capture rare, highly non-typical trajectories that govern π_min. It further links these extremal values to hitting and cover times, establishing n^{1+hat{H}^-/φ(a_0)+o(1)} scaling, and provides explicit constants in special degree-sequence cases. Overall, the results complement prior ergodicity-regime bounds with precise polynomial exponents and actionable asymptotics for fundamental random-walk metrics on sparse directed networks.

Abstract

We consider the stationary distribution of the simple random walk on the directed configuration model with bounded degrees. Provided that the minimum out-degree is at least $2$, with high probability (whp) there is a unique stationary distribution. We show that the minimum positive stationary value is whp $n^{-(1+C+o(1))}$ for some constant $C \ge 0$ determined by the degree distribution. In particular, $C$ is the competing combination of two factors: (1) the contribution of atypically "thin" in-neighbourhoods, controlled by subcritical branching processes; and (2) the contribution of atypically "light" trajectories, controlled by large deviation rate functions. Additionally, our proof implies that whp the hitting and the cover time are both $n^{1+C+o(1)}$. Our results complement those of Caputo and Quattropani who showed that if the minimum in-degree is at least 2, stationary values have logarithmic fluctuations around $n^{-1}$.

Figures (4)

• Figure 1: Instance of a branching process $(X_r)_{t\geq r\geq 0}$ with $t=6$. Black individuals form the spine of $x$. Grey individuals correspond to the process $(X_r-X_r^{(t)})_{t\geq r\geq 0}$ (that is, individuals with no progeny in the $6$-th generation. There are three ramifications, i.e. $R(x)=3$, namely at indices $1$, $2$ and $5$. The colors indicate the spine decomposition of $(X_{r}^{(t)})_{t \ge r \ge 0}$.
• Figure 2: An ongoing exploration process at step $i=5$ and the associated tree $T_f^-(5)$. Active nodes are depicted as small gray squares and marks are assigned to paired nodes.
• Figure 3: Schematic drawing of the event ${\cal{Y}}_f$ for $h_1=3$, $h_2=4$ and $\omega=7$. In it, ${\cal{N}}^-_{\leq h_1+h_2}(f)$ is depicted and ${\cal{N}}^-_{\leq h_1}(f)$ is coloured in blue; we only add the out-degrees in the latter as the other ones are irrelevant for ${\cal{Y}}_f$. Note that $t_{\omega}^-(f)= 6$. Events $E^ f_0$ and $E^ f_2$ hold as $h_1<t_{\omega}^-(f)\leq h_1+h_2$. Using the out-degrees of the vertices in ${\cal{N}}^-_{\leq h_1}(f)$, one can compute the weight of each of the three heads at distance $h_1$ of $f$ to obtain $\Gamma_{h_1}^-(f)= \tfrac{1}{4\cdot 5 \cdot 4} + \tfrac{1}{6\cdot 5 \cdot 4} + \tfrac{1}{3\cdot 5 \cdot 4} = \tfrac{3}{80}$. So event $E^f_1$ will hold provided that $\gamma>\tfrac{3}{80}$. Event $E^f_3$ holds as ${\cal{N}}^-_{\leq h_1}(f)$ induces a tree.
• Figure 4: Plot of the function $\phi(a)$

Theorems & Definitions (43)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Remark 1.6: Minimum out-degree and uniqueness of $\pi$
• Remark 1.7: Maximum stationary value
• Remark 1.8: Explicit polynomial exponents for $\pi_{{\min}}$
• Remark 1.9
• Remark 1.10