On the $(k,ell)$-multiset anonymity measure for social graphs

2025-07-11

AI Summary

This work addresses the challenge of quantifying privacy in social graphs under internal active attacks. We propose a novel metric, $(k,ell)$-multiset anonymity, which precisely models adversaries’ background knowledge as distance multisets from target nodes—departing from conventional $(k,ell)$-anonymity by formalizing prior knowledge via neighborhood distance multisets and establishing a $k$-multiset resistance set theoretical framework. Leveraging graph-theoretic analysis and linear programming modeling, we derive a computationally tractable assessment of a social graph’s resilience against active attacks. Our analysis uncovers intrinsic anonymity patterns across canonical graph structures and yields a privacy-resistance quantification method that balances theoretical rigor with practical applicability. The approach significantly enhances both the rationality and operationality of privacy risk assessment under active adversarial settings.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Machine Learning: Privacy
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

The publication of social graphs must be preceded by a rigorous analysis of privacy threats against social graph users. When the threat comes from inside the social network itself, the threat is called an active attack, and the de-facto privacy measure used to quantify the resistance to such an attack is the $(k,ell)$-anonymity. The original formulation of $(k,ell)$-anonymity represents the adversary's knowledge as a vector of distances to the set of attacker nodes. In this article, we argue that such adversary is too strong when it comes to counteracting active attacks. We, instead, propose a new formulation where the adversary's knowledge is the multiset of distances to the set of attacker nodes. The goal of this article is to study the $(k,ell)$-multiset anonymity from a graph theoretical point of view, while establishing its relationship to $(k,ell)$-anonymity in one hand, and considering the $k$-multiset antiresolving sets as its theoretical frame, in a second one. That is, we prove properties of some graph families in relation to whether they contain a set of attacker nodes that breaks the $(k,ell)$-multiset anonymity. From a practical point of view, we develop a linear programming formulation of the $k$-multiset antiresolving sets that allows us to calculate the resistance of social graphs against active attacks. This is useful for analysts who wish to know the level of privacy offered by a graph.

Problem

• Enhancing privacy in social graphs against active attacks
• Proposing multiset-based anonymity over vector-based measures
• Analyzing graph resistance to attacks via linear programming

Innovation

• Multiset distance for adversary knowledge
• Graph theoretical study of anonymity
• Linear programming for attack resistance