Models and Algorithms for Graph Watermarking

TL;DR

This paper develops theoretical models and algorithms for graph watermarking, addressing feasibility and security, and demonstrates their application on random graph models like Erdős-Rényi and power-law graphs.

Contribution

It introduces a formal framework for graph watermarking, including security definitions and feasibility conditions, with practical schemes for specific random graph models.

Findings

Feasibility characterized via keygen, marking, and identification functions.

Security proofs provided for watermarking schemes.

Applicable to Erdős-Rényi and power-law graph models.

Abstract

We introduce models and algorithmic foundations for graph watermarking. Our frameworks include security definitions and proofs, as well as characterizations when graph watermarking is algorithmically feasible, in spite of the fact that the general problem is NP-complete by simple reductions from the subgraph isomorphism or graph edit distance problems. In the digital watermarking of many types of files, an implicit step in the recovery of a watermark is the mapping of individual pieces of data, such as image pixels or movie frames, from one object to another. In graphs, this step corresponds to approximately matching vertices of one graph to another based on graph invariants such as vertex degree. Our approach is based on characterizing the feasibility of graph watermarking in terms of keygen, marking, and identification functions defined over graph families with known distributions. We demonstrate the strength of this approach with exemplary watermarking schemes for two random graph models, the classic Erd\H{o}s-R\'{e}nyi model and a random power-law graph model, both of which are used to model real-world networks.