Outward Influence and Cascade Size Estimation in Billion-scale Networks

Estimating cascade size and nodes' influence is a fundamental task in social, technological, and biological networks. Yet this task is extremely challenging due to the sheer size and the structural heterogeneity of networks. We investigate a new influence measure, termed outward influence (OI), defined as the (expected) number of nodes that a subset of nodes $S$ will activate, excluding the nodes in S. Thus, OI equals, the de facto standard measure, influence spread of S minus |S|. OI is not only more informative for nodes with small influence, but also, critical in designing new effective sampling and statistical estimation methods. Based on OI, we propose SIEA/SOIEA, novel methods to estimate influence spread/outward influence at scale and with rigorous theoretical guarantees. The proposed methods are built on two novel components 1) IICP an important sampling method for outward influence, and 2) RSA, a robust mean estimation method that minimize the number of samples through analyzing variance and range of random variables. Compared to the state-of-the art for influence estimation, SIEA is $\Omega(\log^4 n)$ times faster in theory and up to several orders of magnitude faster in practice. For the first time, influence of nodes in the networks of billions of edges can be estimated with high accuracy within a few minutes. Our comprehensive experiments on real-world networks also give evidence against the popular practice of using a fixed number, e.g. 10K or 20K, of samples to compute the "ground truth" for influence spread.Comment: 16 pages, SIGMETRICS 201

Importance Sketching of Influence Dynamics in Billion-scale Networks

The blooming availability of traces for social, biological, and communication networks opens up unprecedented opportunities in analyzing diffusion processes in networks. However, the sheer sizes of the nowadays networks raise serious challenges in computational efficiency and scalability. In this paper, we propose a new hyper-graph sketching framework for inflence dynamics in networks. The central of our sketching framework, called SKIS, is an efficient importance sampling algorithm that returns only non-singular reverse cascades in the network. Comparing to previously developed sketches like RIS and SKIM, our sketch significantly enhances estimation quality while substantially reducing processing time and memory-footprint. Further, we present general strategies of using SKIS to enhance existing algorithms for influence estimation and influence maximization which are motivated by practical applications like viral marketing. Using SKIS, we design high-quality influence oracle for seed sets with average estimation error up to 10x times smaller than those using RIS and 6x times smaller than SKIM. In addition, our influence maximization using SKIS substantially improves the quality of solutions for greedy algorithms. It achieves up to 10x times speed-up and 4x memory reduction for the fastest RIS-based DSSA algorithm, while maintaining the same theoretical guarantees.Comment: 12 pages, to appear in ICDM 2017 as a regular pape

Kirchhoff Index As a Measure of Edge Centrality in Weighted Networks: Nearly Linear Time Algorithms

Most previous work of centralities focuses on metrics of vertex importance and methods for identifying powerful vertices, while related work for edges is much lesser, especially for weighted networks, due to the computational challenge. In this paper, we propose to use the well-known Kirchhoff index as the measure of edge centrality in weighted networks, called $\theta$-Kirchhoff edge centrality. The Kirchhoff index of a network is defined as the sum of effective resistances over all vertex pairs. The centrality of an edge $e$ is reflected in the increase of Kirchhoff index of the network when the edge $e$ is partially deactivated, characterized by a parameter $\theta$. We define two equivalent measures for $\theta$-Kirchhoff edge centrality. Both are global metrics and have a better discriminating power than commonly used measures, based on local or partial structural information of networks, e.g. edge betweenness and spanning edge centrality. Despite the strong advantages of Kirchhoff index as a centrality measure and its wide applications, computing the exact value of Kirchhoff edge centrality for each edge in a graph is computationally demanding. To solve this problem, for each of the $\theta$-Kirchhoff edge centrality metrics, we present an efficient algorithm to compute its $\epsilon$-approximation for all the $m$ edges in nearly linear time in $m$. The proposed $\theta$-Kirchhoff edge centrality is the first global metric of edge importance that can be provably approximated in nearly-linear time. Moreover, according to the $\theta$-Kirchhoff edge centrality, we present a $\theta$-Kirchhoff vertex centrality measure, as well as a fast algorithm that can compute $\epsilon$-approximate Kirchhoff vertex centrality for all the $n$ vertices in nearly linear time in $m$

Bidirectional PageRank Estimation: From Average-Case to Worst-Case

We present a new algorithm for estimating the Personalized PageRank (PPR) between a source and target node on undirected graphs, with sublinear running-time guarantees over the worst-case choice of source and target nodes. Our work builds on a recent line of work on bidirectional estimators for PPR, which obtained sublinear running-time guarantees but in an average-case sense, for a uniformly random choice of target node. Crucially, we show how the reversibility of random walks on undirected networks can be exploited to convert average-case to worst-case guarantees. While past bidirectional methods combine forward random walks with reverse local pushes, our algorithm combines forward local pushes with reverse random walks. We also discuss how to modify our methods to estimate random-walk probabilities for any length distribution, thereby obtaining fast algorithms for estimating general graph diffusions, including the heat kernel, on undirected networks.Comment: Workshop on Algorithms and Models for the Web-Graph (WAW) 201

Non-linear regression models for Approximate Bayesian Computation

Approximate Bayesian inference on the basis of summary statistics is well-suited to complex problems for which the likelihood is either mathematically or computationally intractable. However the methods that use rejection suffer from the curse of dimensionality when the number of summary statistics is increased. Here we propose a machine-learning approach to the estimation of the posterior density by introducing two innovations. The new method fits a nonlinear conditional heteroscedastic regression of the parameter on the summary statistics, and then adaptively improves estimation using importance sampling. The new algorithm is compared to the state-of-the-art approximate Bayesian methods, and achieves considerable reduction of the computational burden in two examples of inference in statistical genetics and in a queueing model.Comment: 4 figures; version 3 minor changes; to appear in Statistics and Computin