  Triangle counting in networks under LDP (Local Differential Privacy) is a
fundamental task for analyzing connection patterns or calculating a clustering
coefficient while strongly protecting sensitive friendships from a central
server. In particular, a recent study proposes an algorithm for this task that
uses two rounds of interaction between users and the server to significantly
reduce estimation error. However, this algorithm suffers from a prohibitively
high communication cost due to a large noisy graph each user needs to download.
  In this work, we propose triangle counting algorithms under LDP with a small
estimation error and communication cost. We first propose two-rounds algorithms
consisting of edge sampling and carefully selecting edges each user downloads
so that the estimation error is small. Then we propose a double clipping
technique, which clips the number of edges and then the number of noisy
triangles, to significantly reduce the sensitivity of each user's query.
Through comprehensive evaluation, we show that our algorithms dramatically
reduce the communication cost of the existing algorithm, e.g., from 6 hours to
8 seconds or less at a 20 Mbps download rate, while keeping a small estimation
error.
