A Random Dot Product Model for Weighted Networks

DeFord, Daryl R.; Rockmore, Daniel N.

doi:10.48550/arxiv.1611.02530

“…Extensions to digraphs are straightforward [18], but we carefully study those ambiguities inherent to the model (not discussed in previous work) which may challenge downstream CPD methods. Unlike previous RDPG proposals for weigthed graphs [19], [20], our new non-parametric model in Section IV-B does not require a priori specification of the weights' distribution to perform provably consistent inference and estimation. We believe this contribution is significant in its own right, and beyond CPD it can e.g., impact node classification and visualization of network data.…”

Section: B Paper Outline and Contributionsmentioning

confidence: 99%

“…A couple of works have proposed similar adaptations of the vanilla RDPG model to the weighted case; see [19], [20]. The basic ideas therein are outlined next.…”

Section: B Weighted Rdpgmentioning

confidence: 99%

Online Change Point Detection for Weighted and Directed Random Dot Product Graphs

Marenco

¹

,

Bermolen

²

,

Fiori

³

et al. 2022

IEEE Trans. on Signal and Inf. Process. over Networks

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Given a sequence of random (directed and weighted) graphs, we address the problem of online monitoring and detection of changes in the underlying data distribution. Our idea is to endow sequential change-point detection (CPD) techniques with a graph representation learning substrate based on the versatile Random Dot Product Graph (RDPG) model. We consider efficient, online updates of a judicious monitoring function, which quantifies the discrepancy between the streaming graph observations and the nominal RDPG. This reference distribution is inferred via spectral embeddings of the first few graphs in the sequence. We characterize the distribution of this running statistic to select thresholds that guarantee error-rate control, and under simplifying approximations we offer insights on the algorithm's detection resolution and delay. The end result is a lightweight online CPD algorithm, that is also explainable by virtue of the well-appreciated interpretability of RDPG embeddings. This is in stark contrast with most existing graph CPD approaches, which either rely on extensive computation, or they store and process the entire observed time series. An apparent limitation of the RDPG model is its suitability for undirected and unweighted graphs only, a gap we aim to close here to broaden the scope of the CPD framework. Unlike previous proposals, our non-parametric RDPG model for weighted graphs does not require a priori specification of the weights' distribution to perform inference and estimation. This network modeling contribution is of independent interest beyond CPD. We offer an open-source implementation of the novel online CPD algorithm for weighted and direct graphs, whose effectiveness and efficiency are demonstrated via (reproducible) synthetic and real network data experiments.

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“…We choose this session as all n = 100 senators were active throughout the entire session. Similar latent space models [30,31] and network analyses of community detection from voting data have been studied in [19,25,26,27,32,40]. The edge probability P ij is defined to be the fraction of votes in which i and j voted the same way.…”

Section: Realistic Network Experimentsmentioning

confidence: 99%

Maximum a Posteriori Inference of Random Dot Product Graphs via Conic Programming

Wu¹,

Palmer²,

DeFord³

2021

Preprint

Self Cite

0

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We present a convex cone program to infer the latent probability matrix of a random dot product graph (RDPG). The optimization problem maximizes the Bernoulli maximum likelihood function with an added nuclear norm regularization term. The dual problem has a particularly nice form, related to the well-known semidefinite program relaxation of the maxcut problem. Using the primal-dual optimality conditions, we bound the entries and rank of the primal and dual solutions. Furthermore, we bound the optimal objective value and prove asymptotic consistency of the probability estimates of a slightly modified model under mild technical assumptions. Our experiments on synthetic RDPGs not only recover natural clusters, but also reveal the underlying low-dimensional geometry of the original data. We also demonstrate that the method recovers latent structure in the Karate Club Graph and synthetic U.S. Senate vote graphs and is scalable to graphs with up to a few hundred nodes.

show abstract

Online Change Point Detection for Weighted and Directed Random Dot Product Graphs

Marenco,

Bermolen,

Fiori

et al. 2022

Preprint

0

View full text Add to dashboard Cite

Given a sequence of random (directed and weighted) graphs, we address the problem of online monitoring and detection of changes in the underlying data distribution. Our idea is to endow sequential change-point detection (CPD) techniques with a graph representation learning substrate based on the versatile Random Dot Product Graph (RDPG) model. We consider efficient, online updates of a judicious monitoring function, which quantifies the discrepancy between the streaming graph observations and the nominal RDPG. This reference distribution is inferred via spectral embeddings of the first few graphs in the sequence. We characterize the distribution of this running statistic to select thresholds that guarantee error-rate control, and under simplifying approximations we offer insights on the algorithm's detection resolution and delay. The end result is a lightweight online CPD algorithm, that is also explainable by virtue of the well-appreciated interpretability of RDPG embeddings. This is in stark contrast with most existing graph CPD approaches, which either rely on extensive computation, or they store and process the entire observed time series. An apparent limitation of the RDPG model is its suitability for undirected and unweighted graphs only, a gap we aim to close here to broaden the scope of the CPD framework. Unlike previous proposals, our non-parametric RDPG model for weighted graphs does not require a priori specification of the weights' distribution to perform inference and estimation. This network modeling contribution is of independent interest beyond CPD. We offer an open-source implementation of the novel online CPD algorithm for weighted and direct graphs, whose effectiveness and efficiency are demonstrated via (reproducible) synthetic and real network data experiments.

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A Random Dot Product Model for Weighted Networks

Cited by 3 publications

References 35 publications

Online Change Point Detection for Weighted and Directed Random Dot Product Graphs

Online Change Point Detection for Weighted and Directed Random Dot Product Graphs

Maximum a Posteriori Inference of Random Dot Product Graphs via Conic Programming

Online Change Point Detection for Weighted and Directed Random Dot Product Graphs

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