Interactive Proofs for Social Graphs

We consider interactive proofs for social graphs, where the verifier has only oracle access to the graph and can query for the $i^{th}$ neighbor of a vertex $v$, given $i$ and $v$. In this model, we construct a doubly-efficient public-coin two-message interactive protocol for estimating the size of the graph to within a multiplicative factor $\epsilon>0$. The verifier performs $\tilde{O}(1/\epsilon^2 \cdot \tau_{mix} \cdot \Delta)$ queries to the graph, where $\tau_{mix}$ is the mixing time of the graph and $\Delta$ is the average degree of the graph. The prover runs in quasi-linear time in the number of nodes in the graph. Furthermore, we develop a framework for computing the quantiles of essentially any (reasonable) function $f$ of vertices/edges of the graph. Using this framework, we can estimate many health measures of social graphs such as the clustering coefficients and the average degree, where the verifier performs only a small number of queries to the graph. Using the Fiat-Shamir paradigm, we are able to transform the above protocols to a non-interactive argument in the random oracle model. The result is that social media companies (e.g., Facebook, Twitter, etc.) can publish, once and for all, a short proof for the size or health of their social network. This proof can be publicly verified by any single user using a small number of queries to the graph

Estimating Sizes of Social Networks via Biased Sampling

Online social networks have become very popular in recent years and their number of users is already measured in many hundreds of millions. For various commercial and sociological purposes, an independent estimate of their sizes is important. In this work, algorithms for estimating the number of users in such networks are considered. The proposed schemes are also applicable for estimating the sizes of networks’ sub-populations. The suggested algorithms interact with the social networks via their public APIs only, and rely on no other external information. Due to obvious traffic and privacy concerns, the number of such interactions is severely limited. We therefore focus on minimizing the number of API interactions needed for producing good size estimates. We adopt the abstraction of social networks as undirected graphs and use random node sampling. By counting the number of collisions or non-unique nodes in the sample, we produce a size estimate. Then, we show analytically that the estimate error vanishes with high probability for smaller number of samples than those required by prior-art algorithms. Moreover, although our algorithms are provably correct for any graph, theyexcelwhenappliedtosocial network-likegraphs. The proposed algorithms were evaluated on syntheticas well real social networks such as Facebook, IMDB, and DBLP. Our experiments corroborated the theoretical results, and demonstrated the effectiveness of the algorithms