**Interactive Proofs for Social Graphs**

*Liran Katzir and Clara Shikhelman and 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 $\epsilon>0$. 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.

**Category / Keywords: **foundations / interactive proofs; social graphs; succinct arguments

**Date: **received 15 Nov 2020

**Contact author: **eylony at gmail com

**Available format(s): **PDF | BibTeX Citation

**Version: **20201115:121551 (All versions of this report)

**Short URL: **ia.cr/2020/1433

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