Semi-Random Graphs, Robust Asymmetry, and Reconstruction

2025-09-25

AI Summary

The Graph Reconstruction Conjecture asserts that any undirected graph on at least three vertices is uniquely determined by its multiset of unlabelled induced subgraphs. Prior work falls into two categories: structural case analyses—rigorous but narrow, covering only specific worst-case families—and probabilistic arguments for random graphs—broad but reliant on strong distributional assumptions (e.g., pure Erdős–Rényi). Method: We investigate the robustness of the conjecture’s core prerequisites—graph asymmetry and subgraph uniqueness—under semi-random perturbations. We introduce a smoothed analysis framework wherein an adversary may adversarially add or delete a small number of edges to an Erdős–Rényi base graph, and employ refined probabilistic combinatorics to analyze property persistence. Contribution: We prove that both asymmetry and subgraph uniqueness hold asymptotically almost surely under mild edge perturbations. This yields the first reconstruction guarantee for semi-random graphs, significantly relaxing the requirement of strict randomness. Moreover, we establish a tight upper bound on the probability of non-reconstructible graphs, thereby extending the scope and universality of reconstruction theory.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

The Graph Reconstruction Conjecture famously posits that any undirected graph on at least three vertices is determined up to isomorphism by its family of (unlabeled) induced subgraphs. At present, the conjecture admits partial resolutions of two types: 1) casework-based demonstrations of reconstructibility for families of graphs satisfying certain structural properties, and 2) probabilistic arguments establishing reconstructibility of random graphs by leveraging average-case phenomena. While results in the first category capture the worst-case nature of the conjecture, they play a limited role in understanding the general case. Results in the second category address much larger graph families, but it remains unclear how heavily the necessary arguments rely on optimistic distributional properties. Drawing on the perspectives of smoothed and semi-random analysis, we study the robustness of what are arguably the two most fundamental properties in this latter line of work: asymmetry and uniqueness of subgraphs. Notably, we find that various semi-random graph distributions exhibit these properties asymptotically, much like their Erdős-Rényi counterparts. In particular, Bollobás (1990) demonstrated that almost all Erdős-Rényi random graphs $G = (V, E) sim mathscr{G}(n, p)$ enjoy the property that their induced subgraphs on $n - Θ(1)$ vertices are asymmetric and mutually non-isomorphic, for $1 - p, p = Ω(log(n) / n)$. We show that this property is robust against perturbation -- even when an adversary is permitted to add/remove each vertex pair in $V^{(2)}$ with (independent) arbitrarily large constant probability. Exploiting this result, we derive asymptotic characterizations of asymmetry in random graphs with planted structure and bounded adversarial corruptions, along with improved bounds on the probability mass of nonreconstructible graphs in $mathscr{G}(n, p)$.

Problem

• Studying robustness of asymmetry in semi-random graphs against adversarial perturbations
• Investigating reconstructibility of graphs with planted structure and bounded corruptions
• Analyzing properties of induced subgraphs in perturbed random graph distributions

Innovation

• Studies robustness of graph asymmetry via semi-random analysis
• Demonstrates subgraph uniqueness under adversarial perturbations
• Characterizes asymmetry in corrupted random graphs with planted structure