## Abstract

There are hundreds of online social networks with altogether billions of users. Many such networks publicly release structural information, with all personal information removed. Empirical studies have shown, however, that this provides a false sense of privacy—it is possible to identify almost all users that appear in two such anonymized network as long as a few initial mappings are known. We analyze this problem theoretically by reconciling two versions of an artificial power-law network arising from independent subsampling of vertices and edges. We present a new algorithm that identifies most vertices and makes no wrong identifications with high probability. The number of vertices matched is shown to be asymptotically optimal. For an *n*-vertex graph, our algorithm uses \(n^\varepsilon \) seed nodes (for an arbitrarily small \(\varepsilon \)) and runs in quasilinear time. This improves previous theoretical results which need \(\Theta (n)\) seed nodes and have runtimes of order \(n^{1+\Omega (1)}\). Additionally, the applicability of our algorithm is studied experimentally on different networks.

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## Notes

As our graphs are given by adjacency lists, this means that for each graph \(G_i\) we pick a random permutation \(\pi _i\). Then the

*j*th adjacency list becomes the \(\pi _i(j)\)th adjacency list, each entry \(\ell \) of an adjacency list is replaced by \(\pi _i(\ell )\), and finally we sort each adjacency list to obtain a proper description of the permuted graph.Throughout the paper, we say that a bound holds

*with high probability*(w.h.p.) if it holds with probability at least \(1-n^{-c}\) for some \(c > 0\).In the whole paper \(\mathcal {O}(\cdot )\) and \(\Omega (\cdot )\) hide any dependency on the power law exponent \(\beta \) of

*G*. We always assume \(2<\beta < 3\).A realistic application of deanonymization algorithms such as ours could be to manually identify a few high degree nodes, corresponding to public figures, and to run the algorithm to identify the remaining nodes. For the manual step, one may exploit additional metadata—such high-profile vertices are typically public and share lots of information. The algorithm itself does not rely on any such information and only uses graph structure.

In this model, a bipartite graph of users and interests is constructed; and two users are connected if they share an interest. To create two subsampled graphs, each interest is deleted independently with probability 0.25 in both graphs.

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## Acknowledgements

We thank Silvio Lattanzi from Google Inc. for fruitful discussions, sharing their data sets, and sending us a preliminary version of [18] at the early stages of this project. Karl Bringmann is a recipient of the *Google Europe Fellowship in Randomized Algorithms*, and this research is supported in part by this Google Fellowship. Tobias Friedrich received funding from the German Research Foundation (DFG) under Grant Agreement No. FR 2988 (ADLON).

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A preliminary conference version [8] without most proofs appeared in the 22nd European Symposium on Algorithms (ESA 2014).

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Bringmann, K., Friedrich, T. & Krohmer, A. De-anonymization of Heterogeneous Random Graphs in Quasilinear Time.
*Algorithmica* **80**, 3397–3427 (2018). https://doi.org/10.1007/s00453-017-0395-0

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DOI: https://doi.org/10.1007/s00453-017-0395-0

### Keywords

- Social networks
- Locality-sensitive hashing
- Network privacy