Differential Privacy from Locally Adjustable Graph Algorithms: k-Core Decomposition, Low Out-Degree Ordering, and Densest Subgraphs

Differentially private algorithms allow large-scale data analytics while preserving user privacy. Designing such algorithms for graph data is gaining importance with the growth of large networks that model various (sensitive) relationships between individuals. While there exists a rich history of important literature in this space, to the best of our knowledge, no results formalize a relationship between certain parallel and distributed graph algorithms and differentially private graph analysis. In this paper, we define locally adjustable graph algorithms and show that algorithms of this type can be transformed into differentially private algorithms.

Our formalization is motivated by a set of results that we present in the central and local models of differential privacy for a number of problems, including k-core decomposition, low out-degree ordering, and densest subgraphs. First, we design an ε-edge differentially private (DP) algorithm that returns a subset of nodes that induce a subgraph of density at least D * 1+η -O (poly(log n)/ε) , where D * is the density of the densest subgraph in the input graph (for any constant η > 0). This algorithm achieves a two-fold improvement on the multiplicative approximation factor of the previously best-known private densest subgraph algorithms while maintaining a near-linear runtime.

Then, we present an ε-locally edge differentially private (LEDP) algorithm for k-core decompositions. Our LEDP algorithm provides approximates the core numbers (for any constant η > 0) with (2 + η) multiplicative and O (poly (log n) /ε) additive error. This is the first differentially private algorithm that outputs private k-core decomposition statistics. We also modify our algorithm to return a differentially private low out-degree ordering of the nodes, where orienting the edges from nodes earlier in the ordering to nodes later in the ordering results in out-degree at most O (d + poly (log n) /ε) (where d is the degeneracy of the graph). A small modification to the algorithm also yields a ε-LEDP algorithm for (4 + η, O (poly (log n) /ε))-approximate densest subgraph (which returns both the set of nodes in the subgraph and its density). Our algorithm uses O(log 2 n) rounds of communication between the curator and individual nodes.

Whereas an algorithm that outputs the exact k-core decomposition does not satisfy the definition of DP (or LDP), we obtain an LDP algorithm for approximate k-core decomposition which gives (2+η, O(log 3 n/ε))approximate core numbers for any constant η > 0. We define the related concept of an approximate low out-degree ordering based on the definition of degeneracy .

An undirected graph G = (V, E) is d-degenerate if every induced subgraph of G has a node with degree at most d. The degeneracy of G is the smallest value of d for which G is d-degenerate.

It is well known that degeneracy d = max v∈V k(v).

The ordering D is an (φ, ζ)-approximate low out-degree ordering if orienting edges from earlier nodes to later nodes in D produces outdegree at most φ

When defining an approximate densest subgraph, we remove the condition on maximality of the subgraph and require the density to be within the specified approximation factors.

We consider two models of differential privacy: central [DMNS06] and local [KLN + 11]. In the central model, there is a trusted curator that has direct access to the input, whereas in the local model, the curator is not trusted and gets access only to outputs of private algorithms, called randomizers. Both notions of privacy require a definition of neighboring inputs. We focus on edge-neighboring graphs, defined next. Definition 2.6 (Edge-Neighboring [NRS07]). Graphs G 1 = (V 1 , E 1 ) and G 2 = (V 2 , E 2 ) are edge-neighboring if they differ in one edge, namely, if V 1 = V 2 and the size of the symmetric difference of E 1 and E 2 is 1. Definition 2.7 (ε-Edge Differential Privacy [NRS07]). Algorithm A(G), that takes as input a graph G and outputs some value in Range(A) 2 , is ε-edge differentially private (ε-edge DP) if for all S ⊆ Range(A) and all edge-neighboring graphs G and G ′ ,