167 research outputs found

    Private Graphon Estimation for Sparse Graphs

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    We design algorithms for fitting a high-dimensional statistical model to a large, sparse network without revealing sensitive information of individual members. Given a sparse input graph GG, our algorithms output a node-differentially-private nonparametric block model approximation. By node-differentially-private, we mean that our output hides the insertion or removal of a vertex and all its adjacent edges. If GG is an instance of the network obtained from a generative nonparametric model defined in terms of a graphon WW, our model guarantees consistency, in the sense that as the number of vertices tends to infinity, the output of our algorithm converges to WW in an appropriate version of the L2L_2 norm. In particular, this means we can estimate the sizes of all multi-way cuts in GG. Our results hold as long as WW is bounded, the average degree of GG grows at least like the log of the number of vertices, and the number of blocks goes to infinity at an appropriate rate. We give explicit error bounds in terms of the parameters of the model; in several settings, our bounds improve on or match known nonprivate results.Comment: 36 page

    Revealing Network Structure, Confidentially: Improved Rates for Node-Private Graphon Estimation

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    Motivated by growing concerns over ensuring privacy on social networks, we develop new algorithms and impossibility results for fitting complex statistical models to network data subject to rigorous privacy guarantees. We consider the so-called node-differentially private algorithms, which compute information about a graph or network while provably revealing almost no information about the presence or absence of a particular node in the graph. We provide new algorithms for node-differentially private estimation for a popular and expressive family of network models: stochastic block models and their generalization, graphons. Our algorithms improve on prior work, reducing their error quadratically and matching, in many regimes, the optimal nonprivate algorithm. We also show that for the simplest random graph models (G(n,p)G(n,p) and G(n,m)G(n,m)), node-private algorithms can be qualitatively more accurate than for more complex models---converging at a rate of 1ϵ2n3\frac{1}{\epsilon^2 n^{3}} instead of 1ϵ2n2\frac{1}{\epsilon^2 n^2}. This result uses a new extension lemma for differentially private algorithms that we hope will be broadly useful

    First to Market is not Everything: an Analysis of Preferential Attachment with Fitness

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    In this paper, we provide a rigorous analysis of preferential attachment with fitness, a random graph model introduced by Bianconi and Barabasi. Depending on the shape of the fitness distribution, we observe three distinct phases: a first-mover-advantage phase, a fit-get-richer phase and an innovation-pays-off phase
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