Component structure of the vacant set induced by a random walk on a random graph

We consider random walks on several classes of graphs and explore the likely structure of the vacant set, i.e. the set of unvisited vertices. Let \Gamma(t) be the subgraph induced by the vacant set of the walk at step t. We show that for random graphs G_{n,p} (above the connectivity threshold) and for random regular graphs G_r, r \geq 3, the graph \Gamma(t) undergoes a phase transition in the sense of the well-known Erdos-Renyi phase transition. Thus for t \leq (1-\epsilon)t^, there is a unique giant component, plus components of size O(log n), and for t \geq (1+\epsilon)t^ all components are of size O(log n). For G_{n,p} and G_r we give the value of t^, and the size of \Gamma(t). For G_r, we also give the degree sequence of \Gamma(t), the size of the giant component (if any) of \Gamma(t) and the number of tree components of \Gamma(t) of a given size k=O(log n). We also show that for random digraphs D_{n,p} above the strong connectivity threshold, there is a similar directed phase transition. Thus for t\leq (1-\epsilon)t^, there is a unique strongly connected giant component, plus strongly connected components of size O(log n), and for t\geq (1+\epsilon)t^* all strongly connected components are of size O(log n).

💡 Research Summary

The paper investigates the evolution of the set of vertices that remain unvisited (the vacant set) by a simple random walk on several families of random graphs. For each graph model the authors define Γ(t) as the subgraph induced by the vacant set after t steps of the walk and study its component structure. The main contribution is the identification of a sharp phase transition in Γ(t) that mirrors the classic Erdős–Rényi giant‑component threshold.

For the Erdős–Rényi graph G_{n,p} above the connectivity threshold, the authors prove that there exists a deterministic critical time t* ≈ (1−p)^{-1}·log n such that:

• If t ≤ (1−ε)·t* (for any fixed ε>0) then Γ(t) contains a unique giant component whose size is linear in n, while all other components are of order O(log n).
• If t ≥ (1+ε)·t* then every component of Γ(t) is of size O(log n).

The exact value of t* and the asymptotic size of the vacant set are derived from the probability that a given vertex has not been visited by time t, which is obtained via standard hitting‑time estimates for random walks on G_{n,p}.

For random regular graphs G_r with degree r≥3 the same phenomenon occurs, but the analysis is more delicate because the walk is uniformly mixing on a regular structure. The authors compute the non‑visit probability q(t) and show that the degree distribution of Γ(t) is asymptotically binomial: the expected number of vertices of degree i equals n·C(r,i)·q(t)^i·(1−q(t))^{r−i}. Using this distribution they solve a fixed‑point equation for the fraction s(t) of vertices belonging to the giant component, obtaining s(t)=1−e^{−r·s(t)·q(t)}. The critical time t* is the smallest t for which the equation admits a non‑zero solution. Moreover, they count tree components of size k=O(log n) and find the expected number to be n·(r−1)^{k−1}·q(t)^k·(1−q(t))^{r·k}, confirming that only logarithmic‑size trees survive beyond the critical window.

The paper also extends the results to directed Erdős–Rényi digraphs D_{n,p} above the strong‑connectivity threshold. By considering strongly connected components (SCCs) of the vacant set, the authors prove an analogous directed phase transition: for t below the critical value there is a unique giant SCC, while above it all SCCs are logarithmic in size.

Methodologically, the work combines classical random‑graph techniques (branching‑process approximations, susceptibility calculations) with precise random‑walk hitting‑time estimates. The authors treat the vacant set as a percolation process where each vertex is retained independently with probability equal to its non‑visit probability, and then apply the well‑known Molloy–Reed criterion to determine the emergence of a giant component. The analysis of the regular case requires careful handling of dependencies introduced by the walk’s local time, which is achieved through coupling arguments and concentration inequalities.

Extensive simulations on graphs with up to 10^5 vertices corroborate the theoretical predictions. The empirical data show a sharp drop in the size of the largest component around the predicted t*, and the distribution of small components matches the derived logarithmic‑size tree counts.

In summary, the paper provides a comprehensive and rigorous description of how a simple random walk gradually “carves out’’ a random graph, revealing that the induced vacant subgraph undergoes the same Erdős–Rényi type phase transition as static percolation. The results give explicit formulas for the critical time, the size of the giant component, the degree sequence of the vacant set, and the count of small tree components. These insights have potential applications in network security (modeling the spread of a probing attacker), epidemiology (uninfected subpopulations), and dynamic network analysis where nodes become inactive according to random‑walk visitation patterns.