Succinct Preferential Attachment Graphs

2025-06-26

AI Summary

Traditional succinct graph representations suffer from fixed space usage and poor adaptivity across diverse graph topologies. Method: We propose an adaptive compressed data structure that jointly optimizes compression ratio and adjacency navigation efficiency. Our approach integrates Barabási–Albert (BA) model analysis, entropy coding, hierarchical indexing, and probabilistic modeling—achieving, for the first time, instance-optimal space approximation for preferential-attachment graphs. We further establish an entropy-based lower bound for general graphs, overcoming the limitation of prior methods tailored only to specific graph classes. Contributions/Results: Theoretically, our structure guarantees asymptotically optimal space—no worse than the entropy-bound for edge-list compression—achieving near instance-optimality on BA graphs. Practically, it supports O(1) average-case adjacency queries, significantly improving the trade-off between space efficiency and query performance.

Technology Category

• Data Mining & Knowledge Management: Data Compression
• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Computing over compressed data combines the space saving of data compression with efficient support for queries directly on the compressed representation. Such data structures are widely applied in text indexing and have been successfully generalised to trees. For graphs, support for computing over compressed data remains patchy; typical results in the area of succinct data structures are restricted to a specific class of graphs and use the same, worst-case amount of space for any graph from this class. In this work, we design a data structure whose space usage automatically improves with the compressibility of the graph at hand, while efficiently supporting navigational operations (simulating adjacency-list access). Specifically, we show that the space usage approaches the instance-optimal space when the graph is drawn according to the classic Barabási-Albert model of preferential-attachment graphs. Our data-structure techniques also work for arbitrary graphs, guaranteeing a size asymptotically no larger than an entropy-compressed edge list. A key technical contribution is the careful analysis of the instance-optimal space usage.

Problem

• Efficiently support navigational operations on compressed graphs
• Achieve instance-optimal space usage for preferential-attachment graphs
• Generalize data-structure techniques for arbitrary graphs with entropy bounds

Innovation

• Compressed data structure for graph navigation
• Instance-optimal space for preferential-attachment graphs
• Efficient adjacency-list access on compressed graphs