Shotgun assembly of random regular graphs

TL;DR

<p>The paper resolves the shotgun assembly problem for random d-regular graphs by establishing sharp radii for unique graph recovery from rooted R-neighborhoods. It develops a cycle-structure framework and a sophisticated coupling of BFS explorations to overcome dependencies among overlapping neighborhoods, yielding tight upper and lower bounds around the threshold R ≈ (log n + log log n)/(2 log(d-1)). Key contributions include a reduction to distinct directed BFS explorations, a cycle-distance metric for comparing local structures, and a polynomial-time certifiable non-isomorphism test for independent graphs. The results have implications for graph reconstruction, automorphism certification, and the tractability of verifying non-isomorphism via local neighborhoods in random regular graphs.</p>

Abstract

Mossel and Ross (2019) introduce the shotgun assembly problem for random graphs: what radius $R$ ensures that the random graph $G$ can be uniquely recovered from its list of rooted $R$-neighborhoods, with high probability? Here we consider this question for random regular graphs of fixed degree $d\ge3$. A result of Bollobás (1982) implies efficient recovery at $R = (1 + ε) \frac12 \log_{d-1}n$ with high probability -- moreover, this recovery algorithm uses only a summary of the distances in each neighborhood. We show that using the full neighborhood structure gives a sharper bound \[ R = \frac{\log n + \log\log n}{2\log(d-1)} + O(1)\,, \] which we prove is tight up to the $O(1)$ term. One consequence of our proof is that if $G,H$ are independent graphs where $G$ follows the random regular law, then with high probability the graphs are non-isomorphic; furthermore, this can be efficiently certified by testing the $R$-neighborhood list of $H$ against the $R$-neighborhood of a single adversarially chosen vertex of $G$.

Figures (10)

• Figure 1: Crossing edges and neighborhoods in $B_R({\boldsymbol{u}}) \cup B_R({\boldsymbol{v}})$. While the number of the number of lower crossings can diverge, their associated crossing neighborhoods are small and rarely intersect. In fact, we will show that with high probability, all but $O(1)$ of them are non-intersecting trees.
• Figure 2: A sequence of cycle additions and deletions in a cycle structure with $|v({\boldsymbol{s}})| = 1$, where this vertex is marked. All the half-edges adjacent to the root have bit 1. Edges added in the add operations are marked in blue. Edges cut in delete operations are marked in red, and edges pruned thereafter are marked in orange. Note that in general, deletions can increase some vertices' distance to the root, as occurs in the first deletion in this sequence. In §\ref{['ss:cycle-deletion-lemma']} we describe certain sequences of deletions that do not increase vertices' distance to the root.
• Figure 3: Left: a possible edge-labeled graph $H_T$. Right: the corresponding labeled cycle structure.
• Figure 4: Case 1: solid triangles represent trees that do not intersect the rest of the graph up to depth $L_\circ + 1$, while dotted triangles represent descendant subgraphs that may not be trees, and may intersect the rest of the graph. If there exists ${\overline {\boldsymbol{w}}}_1 \subset {\overline {\boldsymbol{w}}}$ of size $|{\overline {\boldsymbol{w}}}_1| = d-2$ such that $B_{L_\circ}({\overline {\boldsymbol{w}}}_1)$ does not contain $u$, then $B_{L_\circ}({\boldsymbol{w}}_1) \cap B_{L_\circ}({\overline {\boldsymbol{w}}}_1) = \varnothing$, so the event ${\mathcal{E}}_1$ holds for ${\boldsymbol{w}}_1, {\overline {\boldsymbol{w}}}_1$. Thus, for ${\boldsymbol{w}}_1 = \varphi^{-1}({\overline {\boldsymbol{w}}}_1)$, $B_R({\boldsymbol{w}}_1) \ncong B_R({\overline {\boldsymbol{w}}}_1)$.
• Figure 5: Case 2: if $u\in B_{L_\circ}({\overline {\boldsymbol{w}}}_1)$ for all ${\overline {\boldsymbol{w}}}_1 \subseteq {\overline {\boldsymbol{w}}}$ of size $|{\overline {\boldsymbol{w}}}_1| = d-2$, then $B_{L_\circ}({\overline {\boldsymbol{w}}})$ is not a tree. Thus $B_{L_\circ}({\boldsymbol{w}}) \ncong B_{L_\circ}({\overline {\boldsymbol{w}}})$.

Theorems & Definitions (130)

• Theorem 1.1
• Definition 2.1: Breadth-first search
• Remark 2.2
• Definition 2.3: Ancestor path; descendant
• Definition 2.4: Oriented neighborhood
• Definition 2.5
• Remark 2.6
• Remark 2.7: Simulated BFS exploration
• proof
• Remark 2.9