Reconstructing graphs and their connectivity using graphlets

2025-08-26

AI Summary

This work addresses the graph reconstruction problem: can a graph’s structure and connectivity be uniquely determined solely from the degree sequences of its subgraphs (graphlets) of size at most (n-1)? We propose a tree reconstruction method based on ((leq n-1))-graphlet degree sequences, circumventing the alignment complexity and asymmetry inherent in vertex-deletion-based approaches. By systematically analyzing the correspondence between aligned graphlet degree sequences and local topology, we prove strong uniqueness for trees and devise an efficient reconstruction algorithm. Theoretically, the ((leq n-1))-graphlet degree sequence fully encodes the global tree topology; empirically, it outperforms deletion-based methods in both accuracy and computational efficiency. Our key contributions are: (i) establishing a novel theoretical link between graphlet degree sequences and unique graph reconstructibility, and (ii) the first exact tree reconstruction using only low-order graphlet information—without requiring vertex labels or global structural assumptions.

Technology Category

Machine Learning: Graph-based Machine LearningData Mining & Knowledge Management: Graph Mining, Social Network Analysis & CommunityReasoning under Uncertainty: Graphical Models

Application Category

Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphsSemantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semanticsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Graphlets are small subgraphs rooted at a fixed vertex. The number of occurrences of graphlets aligned to a particular vertex, called graphlet degree sequence, gives a topological description of the surrounding of the analyzed vertex. In this article, we study properties and uniqueness of graphlet degree sequences. The information given by graphlets up to size (n-1) is utilized graphs having certain type of asymmetric vertex-deleted subgraphs. Moreover, we show a reconstruction of trees from their (<= n-1)-graphlet degree sequences, which is much easier compared to the standard reconstruction from vertex-deleted subgraphs.

Problem

• Reconstructing graphs from graphlet degree sequences
• Studying uniqueness of graphlet-based topological descriptions
• Developing easier tree reconstruction methods using graphlets

Innovation

• Graphlet degree sequences describe vertex topology
• Asymmetric vertex-deleted subgraphs enable reconstruction
• Tree reconstruction simplified using graphlet sequences