Paper:
Inferring Networks From Random Walk-Based Node Similarities
Digital presence in the world of online social media entails significant privacy risks [31,56]. In this work we consider a privacy threat to a social network in which an attacker has access to a subset of random walk-based node similarities, such as effective resistances (i.e., commute times) or personalized PageRank scores. Using these similarities, the attacker seeks to infer as much information as possible about the network, including unknown pairwise node similarities and edges. For the effective resistance metric, we show that with just a small subset of measurements, one can learn a large fraction of edges in a social network. We also show that it is possible to learn a graph which accurately matches the underlying network on all other effective resistances. This second observation is interesting from a data mining perspective, since it can be expensive to compute all effective resistances or other random walk-based similarities. As an alternative, our graphs learned from just a subset of effective resistances can be used as surrogates in a range of applications that use effective resistances to probe graph structure, including for graph clustering, node centrality evaluation, and anomaly detection. We obtain our results by formalizing the graph learning objective mathematically, using two optimization problems. One formulation is convex and can be solved provably in polynomial time. The other is not, but we solve it efficiently with projected gradient and coordinate descent. We demonstrate the effectiveness of these methods on a number of social networks obtained from Facebook. We also discuss how our methods can be generalized to other random walk-based similarities, such as personalized PageRank scores. Our code is available at https://github.com/cnmusco/graph-similarity-learning.
In this work we focus on these random walk-based similarity metrics. We initiate the study of a fundamental question:
How much information about a network can be learned given access to a subset of potentially noisy estimates of pairwise node similarities?
This question is important from a privacy perspective. A common privacy breach is social link disclosure [6,56], in which an attacker attempts to learn potentially sensitive links between nodes in a network. Such attacks are very common; fake accounts are used to infiltrate social groups, potential employers may want to inspect a job candidate's social network, and advertisers may wish to probe a user's information to offer targeted ads. Thus, studying the ability of an attacker to reveal link information using node similarities is important in understanding the privacy implications of releasing similarities, or information that can be used to compute them.
There are many scenarios in which node similarities may be released, either directly or indirectly, with the potential to reveal private link information. For example, when searching for users on a social network platform, node similarity is indirectly revealed since similar users (in terms of social connections) are often displayed together in search results. As a second example, random walk-based graph embeddings (e.g. PageRank or DeepWalk embeddings) may be released publicly for research purposes since, naively, they appear to contain no identifying information.
From a data mining perspective, computing all pairwise node similarities can be infeasible for large networks since the number of similarities grows quadratically in the number of nodes. Additionally, when the network cannot be accessed in full but can only be probed via crawling [28], we may only have access to similarity estimates rather than their exact values. Thus, understanding what information can be learned from a partial, potentially noisy, set of node similarities is important when using these metrics in large scale graph mining.
Finally, in some scenarios, it may be possible to measure node similarities for an underlying graph, which we cannot directly access but wish to recover. For example, in evolutionary ecology, effective resistance distances in planar "environment graphs" have been shown to correlate with genetic differentiation in geographically distributed populations [36,37,41]. In this context, measurements of geographic genetic variation give incomplete and noisy measurements of effective resistances. Recovering an underlying graph from these measurements corresponds to recovering plausible limitations on migration and movement that could have led to the observed genetic variations.

Review:
Updates after reading author response and other reviews  Thanks for the clarity on the egonets. I agree that perhaps the social structure can be interesting and this could be a strength rather than weakness.  I didn't actually have a concern on practical use cases, as the response states. I just think it would be more interesting to consider noise that appears from the types of situations described, rather than Gaussian (and the authors acknowledged this point).  Overall, this was an interesting submission. I am upping my score to 7.  --------------------  This paper proposes to learn an unknown graph given a subset of the effective resistances of pairs of nodes. The proposed methods can substantially out-perform random guessing in network reconstruction on a couple synthetic networks and some Facebook egonets.  The main idea of the paper is interesting and original, and the proposed methods do well on the datasets considered. However, I think the choice of egonets as real-world datasets is limiting. The fact that all nodes share a common neighbor implies substantial social structure (and possibly substantial graph structure, if the ego is included --- it is not clear if the ego is included in these datasets). I would suggest to instead use a variety of social networks, or if the authors are keen on Facebook, use some of the Facebook100 graphs.  Another weakness in the experimental design is the following. The noise terms for the experiments are Gaussian, but I read the motivation for the noise terms to come from using approximation algorithms to deal with computational constraints or from partial measurements (last paragraph of introduction, lines 43--48). The usefulness of the Gaussian error seems to be that the solution to the optimization problem (Problem 2) is the MLE. The experimental results would be much more convincing if the noise was closer to the motivation. For example, the authors could compute approximations to the effective resistance with an approximation algorithm (the simplest case might be to run just use a few steps of an iterative solver for computing the effective resistances).  From the presentation of the introduction, I expected more theoretical results. However, the only formal theoretical statements I see in the main text are Theorem 1, which was proved in another paper, and Proposition 1, which is a gradient calculation. Overall, I think this paper would greatly benefit by tightening up all of the language around the problem formulation and including more experiments and description of the actual methodology (e.g., what is the SDP formulation?).  All of that being said, I enjoyed the main idea of the paper. The problem itself opens up some interesting avenues for both theoretical and empirical future work, and the numerical experiments do show that the proposed methods can be successful. I think the experiments (namely, choice of real-world datasets and types of noise) and presentation could be updated for an improved camera version. Thus, I lean towards accepting this paper.  Fianlly, perhaps I am missing a compelling reason that we should be using egonets as the main example, so please point this out to me if it is the case.   Some small issues:  1. In the explanation of privacy attacks (lines 36--42), it would be useful to know some scenarios under which the node similarities would be released, but the link information would not be released.  2. "we show that, when G is a tree, exact recovery is possible for Problem 1.2 when S is a superset of G's edges" Should this be subset of edges?  3. Please include a reference to [43] in the statement of Theorem 1. Right now, it seems like the theorem is a novel contribution of this paper, which, from my reading of the text, it is not.  4. "This approach quickly converges to near global minimums for many networks." I am not sure how I am supposed to draw this conclusion from the experiments.  5. "This modified problem is convex and can be solved via an SDP." Where is the SDP formulation?  6. Why is there no \sigma^2 = 0.1 case for FB Medium A and FB Large A.  7. Typos: Appendixx E (line 186), we learn via share (line 236)