The $\beta$-Model—Maximum Likelihood, Cramér–Rao Bounds, and Hypothesis Testing

Wahlström, Johan; Skog, Isaac; Rosa, Patricio S. La; Händel, Peter; Nehorai, Arye

doi:10.1109/tsp.2017.2691667

Cited by 7 publications

(7 citation statements)

References 45 publications

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“…Then the research branched into a few main directions. Chen and Olvera-Cravioto (2013); Stein and Leng (2021); Yan (2017); Yan et al (2019Yan et al ( , 2016 extend the model for directed and bipartite networks; Karwa and Slavković (2016); Pan and Yan (2020); Yan (2021) study differential privacy in β-model; Gao (2020); Graham (2017); Leng (2020, 2021); Su et al (2018); Yan et al (2019) study various joint β-models that incorporate node or edge covariates; Wahlström et al (2017) calculates the Cramer-Rao bound for repeated observations. Most related to this paper's topic are the recent works Chen et al (2021); Leng (2020, 2021).…”

Section: Introduction 1the β-Model: Formulation Motivating Data Examp...mentioning

confidence: 99%

“…First, there exists little study on lower-bound results for β-model's estimation accuracy. Wahlström et al (2017) gives Cramer-Rao lower bound for β i for a fixed i with repeated edge-wise observations. But what about the entire estimator; what about loss functions other than MSE; and can we establish minimax-type lower-bounds for true parameters over some range?…”

Section: Introduction 1the β-Model: Formulation Motivating Data Examp...mentioning

confidence: 99%

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L-2 Regularized maximum likelihood for $β$-model in large and sparse networks

Wang¹,

Zhang²,

Yan³

et al. 2021

Preprint

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The β-model is a powerful tool for modeling network generation driven by node degree heterogeneity. Its simple yet expressive nature particularly well-suits large and sparse networks, where many network models become infeasible due to computational challenge and observation scarcity. However, existing estimation algorithms for β-model do not scale up; and theoretical understandings remain limited to dense networks. This paper brings several major improvements to the method and theory of β-model to address urgent needs of practical applications. Our contributions include: 1. method: we propose a new 2 penalized MLE scheme; we design a novel algorithm that can comfortably handle sparse networks of millions of nodes, much faster and more memory-parsimonious than any existing algorithm; 2. theory: we present new error bounds on beta-models under much weaker assumptions; we also establish new lower-bounds and new asymptotic normality results; distinct from existing literature, our results cover both small and large regularization scenarios and reveal their distinct asymptotic dependency structures; 3. application: we apply our method to large COVID-19 network data sets and discover meaningful results.

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Section: Introduction 1the β-Model: Formulation Motivating Data Examp...mentioning

confidence: 99%

Section: Introduction 1the β-Model: Formulation Motivating Data Examp...mentioning

confidence: 99%

L-2 Regularized maximum likelihood for $β$-model in large and sparse networks

Wang¹,

Zhang²,

Yan³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…These methods are designed for different purposes and reveal many fundamental network properties. After that, many related algorithms are proposed (for instances, modularity-based methods [12][13][14], dynamic algorithms [15], methods based on statistical inference [16,17], maximum likelihood [18,19] and network motifs [20][21][22]). In addition, some detection algorithms' clustering nodes according to their attribute similarities are also reported [23].…”

Section: Introductionmentioning

confidence: 99%

A Scheme to Design Community Detection Algorithms in Various Networks

Nayak

2019

Future Internet

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Network structures, consisting of nodes and edges, have applications in almost all subjects. A set of nodes is called a community if the nodes have strong interrelations. Industries (including cell phone carriers and online social media companies) need community structures to allocate network resources and provide proper and accurate services. However, most detection algorithms are derived independently, which is arduous and even unnecessary. Although recent research shows that a general detection method that serves all purposes does not exist, we believe that there is some general procedure of deriving detection algorithms. In this paper, we represent such a general scheme. We mainly focus on two types of networks: transmission networks and similarity networks. We reduce them to a unified graph model, based on which we propose a method to define and detect community structures. Finally, we also give a demonstration to show how our design scheme works.Future Internet 2019, 11, 41 2 of 17 derived from multiple backgrounds, which can have various ground truth results. In other words, for one graph, the detection results of an algorithm may be considered good under one background but bad under another one. This implies that there does not exist a general detection algorithm to solve all community detection problems [28].Although it has been proved that no general detection method exists, we believe the existence of a design procedure for deriving ad hoc algorithms. In this paper, we propose such a general design scheme. In particular, we represent a method for converting practical problems into graph models. We mainly focus on the networks constructed by material transmission and node attribute similarities. Based on the graph models, we discuss the common properties shared by various community definitions, followed by a method for detecting them. Finally, we provide a demonstration to show how our scheme works. The comparison results show that an algorithm derived from our scheme can produce reasonable detection results.We want to emphasize that our scheme does not provide a deterministic way to implement community detection. Instead, we discuss the common properties shared by the most of the algorithms. While our work can facilitate the development of new algorithms, readers need to derive algorithms according to their practical problems. Thus, our work does not conflict with but supplements the result reported in Ref. [28].The rest of the paper is organized as follows. Section 2 provides a brief review of the popular community detection algorithms. Section 3 discusses the general properties of the graph models that should be considered during the model construction. Based on this, in Section 4, we define the community structures and discuss their fundamental properties. We also provide a corresponding detection algorithm. Finally, we derive an algorithm through our scheme for demonstration in Section 5. We also discuss its performance by comparing it with other famous algorithms. Related WorkThe research of commu...

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“…Moreover, we show that the number of computations required by the proposed algorithm is sublinear in terms of the expected size of the local cluster of interest, and that it provides a good approximation of the heat kernel PageRank, with approximation errors bounded by a probabilistic guarantee. Numerical experiments verify that the local clustering algorithm using our approximate heat kernel PageRank achieves state-of-the-art performance.Networks are a standard representation of complex interactions among multiple objects, and network analysis has become a crucial method for understanding the features of a variety of complex systems [1][2][3][4][5][6][7][8] . Graph clustering is a fundamental technique for understanding mesoscopic structures in networks consisting of communities or clusters, that is, groups of nodes that are densely connected locally but sparsely connected to other groups in the network 9 .…”

mentioning

confidence: 99%

“…Networks are a standard representation of complex interactions among multiple objects, and network analysis has become a crucial method for understanding the features of a variety of complex systems [1][2][3][4][5][6][7][8] . Graph clustering is a fundamental technique for understanding mesoscopic structures in networks consisting of communities or clusters, that is, groups of nodes that are densely connected locally but sparsely connected to other groups in the network 9 .…”

mentioning

confidence: 99%

Local clustering via approximate heat kernel PageRank with subgraph sampling

Wahlström

Nehorai

2021

Sci Rep

Self Cite

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Graph clustering, a fundamental technique in network science for understanding structures in complex systems, presents inherent problems. Though studied extensively in the literature, graph clustering in large systems remains particularly challenging because massive graphs incur a prohibitively large computational load. The heat kernel PageRank provides a quantitative ranking of nodes, and a local cluster can be efficiently found by performing a sweep over the heat kernel PageRank vector. But computing an exact heat kernel PageRank vector may be expensive, and approximate algorithms are often used instead. Most approximate algorithms compute the heat kernel PageRank vector on the whole graph, and thus are dependent on global structures. In this paper, we present an algorithm for approximating the heat kernel PageRank on a local subgraph. Moreover, we show that the number of computations required by the proposed algorithm is sublinear in terms of the expected size of the local cluster of interest, and that it provides a good approximation of the heat kernel PageRank, with approximation errors bounded by a probabilistic guarantee. Numerical experiments verify that the local clustering algorithm using our approximate heat kernel PageRank achieves state-of-the-art performance.

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The $\beta$-Model—Maximum Likelihood, Cramér–Rao Bounds, and Hypothesis Testing

Cited by 7 publications

References 45 publications

L-2 Regularized maximum likelihood for $β$-model in large and sparse networks

L-2 Regularized maximum likelihood for $β$-model in large and sparse networks

A Scheme to Design Community Detection Algorithms in Various Networks

Local clustering via approximate heat kernel PageRank with subgraph sampling

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