Private Release of Graph Statistics using Ladder Functions

Abstract

Protecting the privacy of individuals in graph structured data while making accurate versions of the data available is one of the most challenging problems in data privacy. Most efforts to date to perform this data release end up mired in complexity, overwhelm the signal with noise, and are not effective for use in practice. In this paper, we introduce a new method which guarantees differential privacy. It specifies a probability distribution over possible outputs that is carefully defined to maximize the utility for the given input, while still providing the required privacy level. The distribution is designed to form a 'ladder', so that each output achieves the highest 'rung' (maximum probability) compared to less preferable outputs. We show how our ladder framework can be applied to problems of counting the number of occurrences of subgraphs, a vital objective in graph analysis, and give algorithms whose cost is comparable to that of computing the count exactly. Our experimental study confirms that our method outperforms existing methods for counting triangles and stars in terms of accuracy, and provides solutions for some problems for which no effective method was previously known. The results of our algorithms can be used to estimate the parameters of suitable graph models, allowing synthetic graphs to be sampled.

Publishing Attributed Social Graphs with Formal Privacy Guarantees

“…However, rather than using a transitive closure probability, we count the number of triangles in the input graph and then add transitive edges until the number of triangles in the resulting graph matches that of the input graph. The number of triangles in a graph can be accurately estimated under differential privacy using a recent technique [37] (Appendix C.3). Like TCL, for each transitive edge we add, we remove a seed edge to maintain the expected degree distribution; however, since the deleted edge may itself be part of one or more existing triangles, we reject proposed replacements that decrease the net triangle count.…”

“…Wang et al outlined a general, divide and conquer approach using smooth sensitivity for graph analysis tasks that have low local sensitivity, such as clustering coefficient [35]. Zhang et al [37] defined the Ladder framework for producing accurate DP estimates of subgraph counting queries, including triangles and k-stars. The Ladder framework combines the concept of "local sensitivity at distance t" [25] with the exponential mechanism [22].…”

“…Fortunately, real-world graphs typically do not contain such pathological structures, and tend to have local sensitivities that are much lower. Triangle counting has been extensively studied in the context of differential privacy (e.g., [25,37,13,5]). The most effective approaches take advantage of the above observation in some way to reduce the amount of noise required to satisfy differential privacy.…”

“…The state-of-the-art approach for differentially private triangle counting is based on the Ladder framework [37], which effectively combines the concept of "local sensitivity at distance t", from the smooth sensitivity framework [25], with the exponential mechanism [22]. In the context of triangle counting, it has been shown to provide better accuracy and has the added benefit that it provides pure differential privacy, rather than the weaker notion of (ε, δ )- Add independent Laplace noise to each coordinate of degree sequence S 6: Apply constrained inference (Algorithm 1 from [11]) to S to getS 7: ford i ∈S do 8:…”

“…. , n − 1} 9: n ∆ ← NOISESAMPLE( f n ∆ , E, ε ∆ ) Get a differentially private estimate of the triangle count using Algorithm 1 from [37] 10: return Θ M = {S, n ∆ } differential privacy provided by the smooth sensitivity framework. The local sensitivity of a function f at distance t (from the input graph) is denoted LS t f (G), and quantifies the maximum local sensitivity among all graphs that can be formed from G by adding or deleting up to t edges.…”