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The presence of correlation is known to make privacy protection moredifficult. We investigate the privacy of socially contagious attributes on anetwork of individuals, where each individual possessing that attribute mayinfluence a number of others into adopting it. We show that for contagionsfollowing the Independent Cascade model there exists a giant connectedcomponent of infected nodes, containing a constant fraction of all the nodeswho all receive the contagion from the same set of sources. We further showthat it is extremely hard to hide the existence of this giant connectedcomponent if we want to obtain an estimate of the activated users at anacceptable level. Moreover, an adversary possessing this knowledge can predictthe real status ("active" or "inactive") with decent probability for many ofthe individuals regardless of the privacy (perturbation) mechanism used. As acase study, we show that the Wasserstein mechanism, a state-of-the-art privacymechanism designed specifically for correlated data, introduces a noise withmagnitude of order $\Omega(n)$ in the count estimation in our setting. Weprovide theoretical guarantees for two classes of random networks: Erdos Renyigraphs and Chung-Lu power-law graphs under the Independent Cascade model.Experiments demonstrate that a giant connected component of infected nodes canand does appear in real-world networks and that a simple inference attack canreveal the status of a good fraction of nodes.

Influencers and the Giant Component: the Fundamental Hardness in Privacy Protection for Socially Contagious Attributes