[Paper Review] Improved Upper and Lower Bound Heuristics for Degree Anonymization in Social Networks

Sepp Hartung, Clemens Hoffmann+1 more|arXiv (Cornell University)|Feb 25, 2014

Privacy-Preserving Technologies in Data4 references6 citations

TL;DR

This paper presents an improved two-phase heuristic for degree anonymization in social networks, enhancing Liu and Terzi's approach with a new dynamic programming algorithm for k-anonymizing degree sequences and a refined local exchange heuristic for edge addition. It achieves provably optimal solutions on 26% of real-world instances and significantly reduces solution size compared to prior heuristics, with upper bounds within 3.6% of lower bounds on random graphs.

ABSTRACT

Motivated by a strongly growing interest in anonymizing social network data, we investigate the NP-hard Degree Anonymization problem: given an undirected graph, the task is to add a minimum number of edges such that the graph becomes k-anonymous. That is, for each vertex there have to be at least k-1 other vertices of exactly the same degree. The model of degree anonymization has been introduced by Liu and Terzi [ACM SIGMOD'08], who also proposed and evaluated a two-phase heuristic. We present an enhancement of this heuristic, including new algorithms for each phase which significantly improve on the previously known theoretical and practical running times. Moreover, our algorithms are optimized for large-scale social networks and provide upper and lower bounds for the optimal solution. Notably, on about 26 % of the real-world data we provide (provably) optimal solutions; whereas in the other cases our upper bounds significantly improve on known heuristic solutions.

Motivation & Objective

• To address the NP-hard Degree Anonymization problem, which requires minimizing edge additions to make a graph k-anonymous.
• To improve the theoretical and practical efficiency of the two-phase heuristic framework proposed by Liu and Terzi.
• To provide tighter lower bounds via an advanced Erdős-Gallai test and better upper bounds through an optimized local exchange heuristic.
• To evaluate the method on large-scale real-world and synthetic social networks with power-law degree distributions.
• To close the gap between lower and upper bounds by refining the degree sequence anonymization and realization processes.

Proposed method

• Proposes a new dynamic programming algorithm for k-anonymizing degree sequences with time complexity O(Δk²s), significantly improving over the prior O(nk) bound.
• Introduces a data reduction rule that accelerates the dynamic programming phase by pruning redundant degree values.
• Employs an advanced Erdős-Gallai test to compute tighter lower bounds on the minimum number of edge additions required.
• Uses a local exchange heuristic in the second phase to realize the anonymized degree sequence as a supergraph of the original, minimizing added edges.
• Applies the heuristic to both real-world and synthetic networks generated via the Barabási–Albert preferential attachment model.
• Employs linear regression and statistical analysis to compare solution quality and running times against the clustering-heuristic baseline.

Experimental results

Research questions

• RQ1 Can the dynamic programming approach for k-anonymizing degree sequences be optimized to reduce time complexity from O(nk) to O(Δk²s)?
• RQ2 To what extent can the advanced Erdős-Gallai test improve the lower bound on the minimum number of edge additions?
• RQ3 How does the performance of the new local exchange heuristic compare to the clustering-heuristic in terms of solution size and running time?
• RQ4 What proportion of real-world and synthetic networks can be solved optimally using the proposed framework?
• RQ5 Can the gap between lower and upper bounds be reduced by refining the realization phase or improving the lower bound computation?

Key findings

• The proposed dynamic programming algorithm achieves a significant speedup, with running time dependent on maximum degree Δ rather than number of vertices n.
• On real-world networks, the method provides provably optimal solutions for 26% of instances, outperforming previous heuristics.
• For the remaining instances, the upper bounds are on average 3.6% larger than the lower bounds on random graphs, indicating a tight gap.
• The new heuristic reduces solution size by up to 72% compared to the clustering-heuristic, with 74% of instances solved within one minute.
• For k=2 and k=3, only 14% of synthetic instances were solved optimally, highlighting the need for improved lower bounds.
• The running time increases exponentially with k, with only 18% of instances processed for k=150, indicating scalability challenges at high k.