Shortest Paths and Distances with Differential Privacy

Abstract

We introduce a model for differentially private analysis of weighted graphs in which the graph topology (V, E) is assumed to be public and the private information consists only of the edge weights w : E → R + . This can express hiding congestion patterns in a known system of roads. Differential privacy requires that the output of an algorithm provides little advantage, measured by privacy parameters ǫ and δ, for distinguishing between neighboring inputs, which are thought of as inputs that differ on the contribution of one individual. In our model, two weight functions w, w ′ are considered to be neighboring if they have ℓ 1 distance at most one. We study the problems of privately releasing a short path between a pair of vertices and of privately releasing approximate distances between all pairs of vertices. We are concerned with the approximation error, the difference between the length of the released path or released distance and the length of the shortest path or actual distance.For the problem of privately releasing a short path between a pair of vertices, we prove a lower bound of Ω(|V|) on the additive approximation error for fixed privacy parameters ǫ, δ. We provide a differentially private algorithm that matches this error bound up to a logarithmic factor and releases paths between all pairs of vertices, not just a single pair. The approximation error achieved by our algorithm can be bounded by the number of edges on the shortest path, so we achieve better accuracy than the worst-case bound for pairs of vertices that are connected by a low-weight path consisting of o(|V|) vertices.For the problem of privately releasing all-pairs distances, we show that for trees we can release all-pairs distances with approximation error O(log 2.5 |V|) for fixed privacy parameters. For arbitrary bounded-weight graphs with edge weights in [0, M ] we can release all distances with approximation errorÕ( |V|M ).

Differentially-Private Multi-Party Sketching for Large-Scale Statistics

“…We may consider the sensitivity of the MST cost when perturbing a single edge weight of the graph, but keeping the topology of the graph fixed. This is similar to the privacy model considered by Sealfon [61].…”

Private Graph Data Release: A Survey

“…Instead, a more suitable setting is to define differential privacy in the context of neighboring weight functions for the given graph. This was first formally introduced in [122]. A use case of this is a navigation system that has access to a public map and road user-traffic data and is promised to keep user data private.…”

“…Difference Sequence [125] Degree sequence: [30,66,109] Subgraph Counting: [11,32,66] Edge Weight DP [122] Local DP [131,150] Graph Mining…”

SoK: Differential privacies

“…Many adaptations of DP simply change the neighborhood definition to protect different types of input data than datasets. DP was adopted to graph-structured data in [38,85,136,140,144,151,153], to streaming data in [51,55,56,64,102], to symbolic control systems in [93], to text vectors in [175], to set operations in [172], to images in [174], to genomic data in [147], to recommendation systems in [80], to machine learning in [119], to location data in [28], to outsourced database systems in [101], to bandit algorithms in [12,156], to RAMs in [3,21,158] and to Private Information Retrieval in [135,155]. We detail the corresponding definitions in the full version of this work.…”