You Can't See Me: Anonymizing Graphs Using the Szemerédi Regularity Lemma

Abstract

Complex networks gathered from our online interactions provide a rich source of information that can be used to try to model and predict our behavior. While this has very tangible benefits that we have all grown accustomed to, there is a concrete privacy risk in sharing potentially sensitive data about ourselves and the people we interact with, especially when this data is publicly available online and unprotected from malicious attacks. k-anonymity is a technique aimed at reducing this risk by obfuscating the topological information of a graph that can be used to infer the nodes' identity. In this paper we propose a novel algorithm to enforce k-anonymity based on a well-known result in extremal graph theory, the Szemerédi regularity lemma. Given a graph, we start by computing a regular partition of its nodes. The Szemerédi regularity lemma ensures that such a partition exists and that the edges between the sets of nodes behave almost randomly. With this partition, we anonymize the graph by randomizing the edges within each set, obtaining a graph that is structurally similar to the original one yet the nodes within each set are structurally indistinguishable. We test the proposed approach on real-world networks extracted from Facebook. Our experimental results show that the proposed approach is able to anonymize a graph while retaining most of its structural information.

k-Anonymity on Graphs Using the Szemerédi Regularity Lemma

“…The two heuristics differ in the way the groups are split: 1) the degree based heuristic groups together nodes with similar degree, while 2) the indeg guided heuristic splits a sparse (dense) partition into two sparse (dense) partitions. We use the latter heuristic in our experiments as it has been shown to achieve better results [11]. Note that due to the nature of the algorithm the cardinality of the final partition is a power of 2.…”

“…Finally, Qian et al [26] and Ma et al [23] looked at the complementary problem of de-anonymizing a graph in the case where the attacker has access to richer features as well as structural information. While most of the previous k-anonymity approaches assume that the attacker has access only to a certain level of structural information (from the degree of a node, to its immediate neighborhood or even the whole graph), Foffano et al [11] have recently proposed a k-anonymization framework where the resulting graph is not susceptible to any particular structure-based attack. Their approach is based on the Szemerédi regularity lemma [7], a well-known result of extremal graph theory.…”

“…Crucially, the lemma was later extended to account for sparse graphs as well [15]. The idea underpinning the work of Foffano et al [11] is that the groups of nodes that form regular pairs can be anonymized by rewiring the intra-group edges according to an Erdös-Rényi process [8]. Thanks to the theoretical guarantees of the Szemerédi regularity lemma, this has minimal effect on the overall graph structure and, together with the random-like behavior of the inter-groups connections, ensures that the each group of nodes is anonymous.…”

“…While the seminal work of Foffano et al [11] provides a limited theoretical and experimental analysis of their method, in this paper we propose to go one step further by making the following contributions:…”

A Review of Anonymization for Healthcare Data

“…Then they randomly selected edges from the removed edges and assigns them as the edge of the cluster. In order to anonymize graphs while maintaining structural similarity, Foffano et al [26] applied Szemerédi regularity lemma in partitioning each node into clusters and ensuring that the intra-partition edges behave almost randomly. By this, the interpartition edges are left unaltered, making it semantically similar to the original graph.…”