Strong Localization in Personalized PageRank Vectors

Nassar, Huda; Kloster, Kyle; Gleich, David F.

doi:10.1007/978-3-319-26784-5_15

Cited by 13 publications

(12 citation statements)

References 20 publications

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“…Before concluding, we discuss implications of our work regarding the topic of eigenvector localization in complex networks, which is an important topic in network science [42,43] for the study of centrality [44][45][46], spatial analysis [47], and core-periphery structure [48,49]. In particular, there is growing interest in extending these ideas to time-varying [50] and multilayer networks [51].…”

Section: Discussionmentioning

confidence: 97%

Super-Resolution Community Detection for Layer-Aggregated Multilayer Networks

2017

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Applied network science often involves preprocessing network data before applying a network-analysis method, and there is typically a theoretical disconnect between these steps. For example, it is common to aggregate time-varying network data into windows prior to analysis, and the trade-offs of this preprocessing are not well understood. Focusing on the problem of detecting small communities in multilayer networks, we study the effects of layer aggregation by developing random-matrix theory for modularity matrices associated with layer-aggregated networks with N nodes and L layers, which are drawn from an ensemble of Erdős–Rényi networks with communities planted in subsets of layers. We study phase transitions in which eigenvectors localize onto communities (allowing their detection) and which occur for a given community provided its size surpasses a detectability limit K*. When layers are aggregated via a summation, we obtain K∗∝O(NL/T), where T is the number of layers across which the community persists. Interestingly, if T is allowed to vary with L, then summation-based layer aggregation enhances small-community detection even if the community persists across a vanishing fraction of layers, provided that T/L decays more slowly than 𝒪(L−1/2). Moreover, we find that thresholding the summation can, in some cases, cause K* to decay exponentially, decreasing by orders of magnitude in a phenomenon we call super-resolution community detection. In other words, layer aggregation with thresholding is a nonlinear data filter enabling detection of communities that are otherwise too small to detect. Importantly, different thresholds generally enhance the detectability of communities having different properties, illustrating that community detection can be obscured if one analyzes network data using a single threshold.

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Section: Discussionmentioning

confidence: 97%

Super-Resolution Community Detection for Layer-Aggregated Multilayer Networks

2017

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“…When the graph is strongly connected π (i) is non-zero for all nodes. Nevertheless, we can obtain a good approximation by truncating small elements to zero since most of the probability mass in the personalized PageRank vectors π (i) is localized on a small number of nodes [5,23,36]. Thus, we can approximate π (i) with a sparse vector and in turn approximate Π ppr with a sparse matrix.…”

Section: Personalized Pagerank and Localizationmentioning

confidence: 99%

Scaling Graph Neural Networks with Approximate PageRank

Bojchevski

Klicpera

Perozzi

et al. 2020

Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery &Amp; Data Mining

290

136

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Graph neural networks (GNNs) have emerged as a powerful approach for solving many network mining tasks. However, learning on large graphs remains a challenge-many recently proposed scalable GNN approaches rely on an expensive message-passing procedure to propagate information through the graph. We present the PPRGo model which utilizes an efficient approximation of information diffusion in GNNs resulting in significant speed gains while maintaining state-of-the-art prediction performance. In addition to being faster, PPRGo is inherently scalable, and can be trivially parallelized for large datasets like those found in industry settings. We demonstrate that PPRGo outperforms baselines in both distributed and single-machine training environments on a number of commonly used academic graphs. To better analyze the scalability of large-scale graph learning methods, we introduce a novel benchmark graph with 12.4 million nodes, 173 million edges, and 2.8 million node features. We show that training PPRGo from scratch and predicting labels for all nodes in this graph takes under 2 minutes on a single machine, far outpacing other baselines on the same graph. We discuss the practical application of PPRGo to solve large-scale node classification problems at Google. 1 CCS CONCEPTS • Computing methodologies → Machine learning.

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“…Out of the plethora of algorithms for PageRank computations, the family of Gauss-Southwell methods [34,3,10,22,23,35,45] and other versions of coordinate descent (see e.g., [15]), clearly stand out due to their rapid convergence, often much faster than power iterations. Such methods have been originally proposed in 1940-s for solving linear systems by greedy elimination of the absolute value of the residual [41,42].…”

Section: Related Literaturementioning

confidence: 99%

“…In this case random walk typically stays close to the restart node, which makes this model useful for local graph clustering [3,43], similarity measures [4] and semi-supervised learning [6]. Several algorithms [3,43,35,47] take advantage of localization of PPR to compute its sparse approximation. Moreover, recently developed FAST-PPR [31] achieves even faster convergence by combining the Gauss-Southwell-type method from [2] with random walks.…”

Section: Related Literaturementioning

confidence: 99%

Red Light Green Light Method for Solving Large Markov Chains

Avrachenkov¹,

Brown²,

Litvak³

2020

Preprint

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Discrete-time discrete-state finite Markov chains are versatile mathematical models for a wide range of real-life stochastic processes. One of most common tasks in studies of Markov chains is computation of the stationary distribution. Without loss of generality, and drawing our motivation from applications to large networks, we interpret this problem as one of computing the stationary distribution of a random walk on a graph. We propose a new controlled, easily distributed algorithm for this task, briefly summarized as follows: at the beginning, each node receives a fixed amount of cash (positive or negative), and at each iteration, some nodes receive 'green light' to distribute their wealth or debt proportionally to the transition probabilities of the Markov chain; the stationary probability of a node is computed as a ratio of the cash distributed by this a node to the total cash distributed by all nodes together. Our method includes as special cases a wide range of known, very different, and previously disconnected methods including power iterations, Gauss-Southwell, and online distributed algorithms. We prove exponential convergence of our method, demonstrate its high efficiency, and derive scheduling strategies for the green-light, that achieve convergence rate faster than state-of-the-art algorithms.

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Strong Localization in Personalized PageRank Vectors

Cited by 13 publications

References 20 publications

Super-Resolution Community Detection for Layer-Aggregated Multilayer Networks

Super-Resolution Community Detection for Layer-Aggregated Multilayer Networks

Scaling Graph Neural Networks with Approximate PageRank

Red Light Green Light Method for Solving Large Markov Chains

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