Directed degree corrected mixed membership model and estimating community memberships in directed networks

Qing, Huan

doi:10.48550/arxiv.2109.07826

Cited by 2 publications

(4 citation statements)

References 38 publications

(73 reference statements)

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“…Note that since the row nodes can be different from the column nodes, we may have

(i.e., there are no common nodes between

and

), and

may not be equal to

(i.e., the row nodes are different from the column nodes), which is a more general case than

(i.e., all row nodes are same as column nodes), where ⌀ denotes the null set, and such a directed network

is also known as a bipartite graph (or bipartite network) in [ 18 , 19 ]. In this paper, we use the subscript r and c to distinguish the terms for the row nodes and column nodes, where works in [ 18 , 19 , 23 , 24 , 25 , 26 ] also consider the general bipartite setting, such that the row nodes may differ from the column nodes. Let

be the bi-adjacency matrix of directed network

, such that

if there is a directional edge from row node

to column node

, and

otherwise.…”

Section: The Overlapping and Non-overlapping Modelmentioning

confidence: 99%

“…Proof of Theorem 2 Proof. First, by the proof of Lemma 4.3 of [ 25 ], we have the below lemma. Lemma A2.…”

Section: Appendix B1 Proof Of Propositionmentioning

confidence: 99%

“…Proof of Corollary 2 Proof. For the row nodes, under the conditions of Corollary 2, we have

Under the conditions of Corollary 2,

and

for some

by Lemma 2 of [ 25 ]. Then, by Lemma A2, we have

which gives that

The reason for why we do not consider the case when

for row nodes is similar as that of Corollary 1, and we omit it here.…”

Section: Appendix B1 Proof Of Propositionmentioning

confidence: 99%

“…Under the conditions of Corollary 2,

and

for some

by Lemma 2 of [ 25 ]. Then, by Lemma A2, we have

which gives that

…”

Section: Appendix B1 Proof Of Propositionmentioning

confidence: 99%

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Studying Asymmetric Structure in Directed Networks by Overlapping and Non-Overlapping Models

Qing¹

2022

Entropy

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We consider the problem of modeling and estimating communities in directed networks. Models to this problem in the previous literature always assume that the sending clusters and the receiving clusters have non-overlapping property or overlapping property simultaneously. However, previous models cannot model the directed network in which nodes in sending clusters have overlapping property, while nodes in receiving clusters have non-overlapping property, especially for the case when the number of sending clusters is no larger than that of the receiving clusters. This kind of directed network exists in the real world for its randomness, and by the fact that we have little prior knowledge of the community structure for some real-world directed networks. To study the asymmetric structure for such directed networks, we propose a flexible and identifiable Overlapping and Non-overlapping model (ONM). We also provide one model as an extension of ONM to model the directed network, with a variation in node degree. Two spectral clustering algorithms are designed to fit the models. We establish a theoretical guarantee on the estimation consistency for the algorithms under the proposed models. A small scale computer-generated directed networks are designed and conducted to support our theoretical results. Four real-world directed networks are used to illustrate the algorithms, and the results reveal the existence of highly mixed nodes and the asymmetric structure for these networks.

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“…Note that since the row nodes can be different from the column nodes, we may have

(i.e., there are no common nodes between

and

), and

may not be equal to

(i.e., the row nodes are different from the column nodes), which is a more general case than

(i.e., all row nodes are same as column nodes), where ⌀ denotes the null set, and such a directed network

be the bi-adjacency matrix of directed network

, such that

if there is a directional edge from row node

to column node

, and

otherwise.…”

Section: The Overlapping and Non-overlapping Modelmentioning

confidence: 99%

“…Proof of Theorem 2 Proof. First, by the proof of Lemma 4.3 of [ 25 ], we have the below lemma. Lemma A2.…”

Section: Appendix B1 Proof Of Propositionmentioning

confidence: 99%

“…Proof of Corollary 2 Proof. For the row nodes, under the conditions of Corollary 2, we have

Under the conditions of Corollary 2,

and

for some

by Lemma 2 of [ 25 ]. Then, by Lemma A2, we have

which gives that

The reason for why we do not consider the case when

for row nodes is similar as that of Corollary 1, and we omit it here.…”

Section: Appendix B1 Proof Of Propositionmentioning

confidence: 99%

“…Under the conditions of Corollary 2,

and

for some

by Lemma 2 of [ 25 ]. Then, by Lemma A2, we have

which gives that

…”

Section: Appendix B1 Proof Of Propositionmentioning

confidence: 99%

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Studying Asymmetric Structure in Directed Networks by Overlapping and Non-Overlapping Models

Qing¹

2022

Entropy

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Overlapping and nonoverlapping models

Qing

2021

Preprint

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Consider a directed network with K r row communities and K c column communities. Previous works found that modeling directed networks in which all nodes have overlapping property requires K r = K c for identifiability. In this paper, we propose an overlapping and nonoverlapping model to study directed networks in which row nodes have overlapping property while column nodes do not. The proposed model is identifiable when K r ≤ K c . Meanwhile, we provide one identifiable model as extension of ONM to model directed networks with variation in node degree. Two spectral algorithms with theoretical guarantee on consistent estimations are designed to fit the models. A small scale of numerical studies are used to illustrate the algorithms.

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Directed degree corrected mixed membership model and estimating community memberships in directed networks

Cited by 2 publications

References 38 publications

Studying Asymmetric Structure in Directed Networks by Overlapping and Non-Overlapping Models

Studying Asymmetric Structure in Directed Networks by Overlapping and Non-Overlapping Models

Overlapping and nonoverlapping models

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