Locally Differentially Private k-Triangle Counting in Real-Time Social Graph

Social networks are changing in real-time, and $ k $-triangle counting as a fundamental task in graph analysis is useful for finding meaningful connection patterns in these dynamical graphs. However, the dynamical graphs and the number of $ k $-triangles therein both are sensitive and imply users’ private information. The lasted works applied Local Differential Privacy (LDP) to the snapshot graph, counting simple statistical information. Nevertheless, these LDP-based works ignored the real-time development of social graphs and did not count more complex statistics, $ k $-triangle. To this end, we propose a locally differentially private k-triangles counting in real-time social graph, namely $\mathtt{LAPKE}$ that is the first work to provide edge LDP and guarantee a lower upper bound on the estimation error of the number of $ k $-triangles for dynamical graph models. Thereafter, to further reduce the estimation error, we propose $\mathtt{LAPKE^{+}}$ and its extension $\mathtt{LAPKE^{++}}$ that do not require a large number of users’ efforts and synchronization. The main intuition of $\mathtt{LAPKE}$, $\mathtt{LAPKE^{+}}$ and $\mathtt{LAPKE^{++}}$ is sampling $ k $ or 2 disjoint users in real-time social graph to construct the key substructure $ k $-wedge of $ k $-triangle, which minimizes the estimation error while preserving users’ private information. Moreover, we theoretically prove the lower upper bound on the estimation error of $ k $-triangle for dynamical graphs $\mathtt{LAPKE}$, $\mathtt{LAPKE^{+}}$ and $\mathtt{LAPKE^{++}}$ provide. Finally, the extensive experiment results on three real-world datasets and one synthetic dataset validate the superior performance of the proposed three algorithms compared with three latest existing methods.