K-stars LDP: A Novel Framework for (p, q)-clique Enumeration under Local Differential Privacy

(p,q)-clique enumeration on a bipartite graph is critical for calculating clustering coefficient and detecting densest subgraph. It is necessary to carry out subgraph enumeration while protecting users' privacy from any potential attacker as the count of subgraph may contain sensitive information. Most recent studies focus on the privacy protection algorithms based on edge LDP (Local Differential Privacy). However, these algorithms suffer a large estimation error due to the great amount of required noise. In this paper, we propose a novel idea of k-stars LDP and a novel k-stars LDP algorithm for (p, q)-clique enumeration with a small estimation error, where a k-stars is a star-shaped graph with k nodes connecting to one node. The effectiveness of edge LDP relies on its capacity to obfuscate the existence of an edge between the user and his one-hop neighbors. This is based on the premise that a user should be aware of the existence of his one-hop neighbors. Similarly, we can apply this premise to k-stars as well, where an edge is a specific genre of 1-stars. Based on this fact, we first propose the k-stars neighboring list to enable our algorithm to obfuscate the existence of k-stars with Warner' s RR. Then, we propose the absolute value correction technique and the k-stars sampling technique to further reduce the estimation error. Finally, with the two-round user-collector interaction mechanism, we propose our k-stars LDP algorithm to count the number of (p, q)-clique while successfully protecting users' privacy. Both the theoretical analysis and experiments have showed the superiority of our algorithm over the algorithms based on edge LDP.