- Liran Katzir
- Clara Shikhelman
- Eylon Yogev

## Abstract

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 $\eps>0$. The verifier performs $\widetilde{O}(1/\eps^2 \cdot \MixingTime \cdot \AverageDegree)$ queries to the graph, where $\MixingTime$ is the mixing time of the graph and $\AverageDegree$ 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.

## Research Areas

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