Abstract Machine learning applications on large-scale network-structured data commonly encode network information in the form of node embeddings. Network embedding algorithms map the nodes into a low-dimensional space such that the nodes that are “similar” with respect to network topology are also close to each other in the embedding space. Real-world networks often have multiple versions or can be “multiplex” with multiple types of edges with different semantics. For such networks, computation of Consensus Embeddings based on the node embeddings of individual versions can be useful for various reasons, including privacy, efficiency, and effectiveness of analyses. Here, we systematically investigate the performance of three dimensionality reduction methods in computing consensus embeddings on networks with multiple versions: singular value decomposition, variational auto-encoders, and canonical correlation analysis (CCA). Our results show that (i) CCA outperforms other dimensionality reduction methods in computing concensus embeddings, (ii) in the context of link prediction, consensus embeddings can be used to make predictions with accuracy close to that provided by embeddings of integrated networks, and (iii) consensus embeddings can be used to improve the efficiency of combinatorial link prediction queries on multiple networks by multiple orders of magnitude.
{"title":"Consensus embedding for multiple networks: Computation and applications","authors":"Mengzhen Li, Mustafa Coşkun, Mehmet Koyutürk","doi":"10.1017/nws.2022.17","DOIUrl":"https://doi.org/10.1017/nws.2022.17","url":null,"abstract":"Abstract Machine learning applications on large-scale network-structured data commonly encode network information in the form of node embeddings. Network embedding algorithms map the nodes into a low-dimensional space such that the nodes that are “similar” with respect to network topology are also close to each other in the embedding space. Real-world networks often have multiple versions or can be “multiplex” with multiple types of edges with different semantics. For such networks, computation of Consensus Embeddings based on the node embeddings of individual versions can be useful for various reasons, including privacy, efficiency, and effectiveness of analyses. Here, we systematically investigate the performance of three dimensionality reduction methods in computing consensus embeddings on networks with multiple versions: singular value decomposition, variational auto-encoders, and canonical correlation analysis (CCA). Our results show that (i) CCA outperforms other dimensionality reduction methods in computing concensus embeddings, (ii) in the context of link prediction, consensus embeddings can be used to make predictions with accuracy close to that provided by embeddings of integrated networks, and (iii) consensus embeddings can be used to improve the efficiency of combinatorial link prediction queries on multiple networks by multiple orders of magnitude.","PeriodicalId":51827,"journal":{"name":"Network Science","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2022-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42375078","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract For interventions that affect how individuals interact, social network data may aid in understanding the mechanisms through which an intervention is effective. Social networks may even be an intermediate outcome observed prior to end of the study. In fact, social networks may also mediate the effects of the intervention on the outcome of interest, and Sweet (2019) introduced a statistical model for social networks as mediators in network-level interventions. We build on their approach and introduce a new model in which the network is a mediator using a latent space approach. We investigate our model through a simulation study and a real-world analysis of teacher advice-seeking networks.
{"title":"A hierarchical latent space network model for mediation","authors":"T. Sweet, S. Adhikari","doi":"10.1017/nws.2022.12","DOIUrl":"https://doi.org/10.1017/nws.2022.12","url":null,"abstract":"Abstract For interventions that affect how individuals interact, social network data may aid in understanding the mechanisms through which an intervention is effective. Social networks may even be an intermediate outcome observed prior to end of the study. In fact, social networks may also mediate the effects of the intervention on the outcome of interest, and Sweet (2019) introduced a statistical model for social networks as mediators in network-level interventions. We build on their approach and introduce a new model in which the network is a mediator using a latent space approach. We investigate our model through a simulation study and a real-world analysis of teacher advice-seeking networks.","PeriodicalId":51827,"journal":{"name":"Network Science","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2022-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49241172","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Several papers have highlighted the potential of network science to appeal to a younger audience of high school children and provided lesson material on network science for high school children. However, network science also provides a great topic for outreach activities for primary school children. Therefore, this article gives a short summary of an outreach activity on network science for primary school children aged 8–12 years. The material provided in this article contains presentation material for a lesson of approximately 1 hour, including experiments, exercises, and quizzes, which can be used by other scientists interested in popularizing network science. We then discuss the lessons learned from this material.
{"title":"Bringing network science to primary school","authors":"C. Stegehuis","doi":"10.1017/nws.2022.15","DOIUrl":"https://doi.org/10.1017/nws.2022.15","url":null,"abstract":"Abstract Several papers have highlighted the potential of network science to appeal to a younger audience of high school children and provided lesson material on network science for high school children. However, network science also provides a great topic for outreach activities for primary school children. Therefore, this article gives a short summary of an outreach activity on network science for primary school children aged 8–12 years. The material provided in this article contains presentation material for a lesson of approximately 1 hour, including experiments, exercises, and quizzes, which can be used by other scientists interested in popularizing network science. We then discuss the lessons learned from this material.","PeriodicalId":51827,"journal":{"name":"Network Science","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2022-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42657281","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Shared memberships, social statuses, beliefs, and places can facilitate the formation of social ties. Two-mode projections provide a method for transforming two-mode data on individuals’ memberships in such groups into a one-mode network of their possible social ties. In this paper, I explore the opposite process: how social ties can facilitate the formation of groups, and how a two-mode network can be generated from a one-mode network. Drawing on theories of team formation, club joining, and organization recruitment, I propose three models that describe how such groups might emerge from the relationships in a social network. I show that these models can be used to generate two-mode networks that have characteristics commonly observed in empirical two-mode social networks and that they encode features of the one-mode networks from which they were generated. I conclude by discussing these models’ limitations and future directions for theory and methods concerning group formation.
{"title":"The duality of networks and groups: Models to generate two-mode networks from one-mode networks","authors":"Z. Neal","doi":"10.1017/nws.2023.3","DOIUrl":"https://doi.org/10.1017/nws.2023.3","url":null,"abstract":"\u0000 Shared memberships, social statuses, beliefs, and places can facilitate the formation of social ties. Two-mode projections provide a method for transforming two-mode data on individuals’ memberships in such groups into a one-mode network of their possible social ties. In this paper, I explore the opposite process: how social ties can facilitate the formation of groups, and how a two-mode network can be generated from a one-mode network. Drawing on theories of team formation, club joining, and organization recruitment, I propose three models that describe how such groups might emerge from the relationships in a social network. I show that these models can be used to generate two-mode networks that have characteristics commonly observed in empirical two-mode social networks and that they encode features of the one-mode networks from which they were generated. I conclude by discussing these models’ limitations and future directions for theory and methods concerning group formation.","PeriodicalId":51827,"journal":{"name":"Network Science","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2022-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57044220","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Per Block, James Hollway, Christoph Stadtfeld, J. Koskinen, T. Snijders
Abstract We review the empirical comparison of Stochastic Actor-oriented Models (SAOMs) and Temporal Exponential Random Graph Models (TERGMs) by Leifeld & Cranmer in this journal [Network Science 7(1):20–51, 2019]. When specifying their TERGM, they use exogenous nodal attributes calculated from the outcome networks’ observed degrees instead of endogenous ERGM equivalents of structural effects as used in the SAOM. This turns the modeled endogeneity into circularity and obtained results are tautological. In consequence, their out-of-sample predictions using TERGMs are based on out-of-sample information and thereby predict the future using observations from the future. Thus, their analysis rests on erroneous model specifications that invalidate the article’s conclusions. Finally, beyond these specific points, we argue that their evaluation metric—tie-level predictive accuracy—is unsuited for the task of comparing model performance.
{"title":"Circular specifications and “predicting” with information from the future: Errors in the empirical SAOM–TERGM comparison of Leifeld & Cranmer","authors":"Per Block, James Hollway, Christoph Stadtfeld, J. Koskinen, T. Snijders","doi":"10.1017/nws.2022.6","DOIUrl":"https://doi.org/10.1017/nws.2022.6","url":null,"abstract":"Abstract We review the empirical comparison of Stochastic Actor-oriented Models (SAOMs) and Temporal Exponential Random Graph Models (TERGMs) by Leifeld & Cranmer in this journal [Network Science 7(1):20–51, 2019]. When specifying their TERGM, they use exogenous nodal attributes calculated from the outcome networks’ observed degrees instead of endogenous ERGM equivalents of structural effects as used in the SAOM. This turns the modeled endogeneity into circularity and obtained results are tautological. In consequence, their out-of-sample predictions using TERGMs are based on out-of-sample information and thereby predict the future using observations from the future. Thus, their analysis rests on erroneous model specifications that invalidate the article’s conclusions. Finally, beyond these specific points, we argue that their evaluation metric—tie-level predictive accuracy—is unsuited for the task of comparing model performance.","PeriodicalId":51827,"journal":{"name":"Network Science","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85850193","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We welcome our new editors and provide background on an unusual duo of articles in this issue.
摘要我们欢迎我们的新编辑，并提供本期两篇不同寻常的文章的背景。
{"title":"Editors’ Note","authors":"Stanley Wasserman, Ulrik Brandes","doi":"10.1017/nws.2022.8","DOIUrl":"https://doi.org/10.1017/nws.2022.8","url":null,"abstract":"Abstract We welcome our new editors and provide background on an unusual duo of articles in this issue.","PeriodicalId":51827,"journal":{"name":"Network Science","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43648656","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
a set of theoretical differences between the models and a proposed for model comparison based on out-of-sample prediction. the theoretical comparison or simulation framework. be using the processes, the of the to the and the impossibility of model comparison using dyadic prediction is by evidence, the discussion: Does the contain theory, and how can its inherent be
{"title":"The stochastic actor-oriented model is a theory as much as it is a method and must be subject to theory tests","authors":"Philip Leifeld, S. Cranmer","doi":"10.1017/nws.2022.7","DOIUrl":"https://doi.org/10.1017/nws.2022.7","url":null,"abstract":"a set of theoretical differences between the models and a proposed for model comparison based on out-of-sample prediction. the theoretical comparison or simulation framework. be using the processes, the of the to the and the impossibility of model comparison using dyadic prediction is by evidence, the discussion: Does the contain theory, and how can its inherent be","PeriodicalId":51827,"journal":{"name":"Network Science","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43466587","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Block, P., Hollway, J., Stadtfeld, C., Koskinen, J., & Snijders, T. (2022). Circular specifications and “predicting” with information from the future: Errors in the empirical SAOM–TERGM comparison of Leifeld & Cranmer. Network Science, 10(1). https://doi.org/10.1017/nws.2022.6 Leifeld, P., & Cranmer, S. (2019a). A theoretical and empirical comparison of the temporal exponential random graph model and the stochastic actor-oriented model. Network Science, 7(1), 20–51. https://doi.org/10.1017/nws.2018.26 Leifeld, P., & Cranmer, S. (2019b). Replication Data for: A Theoretical and Empirical Comparison of the Temporal Exponential Random Graph Model and the Stochastic Actor-Oriented Model, https://doi.org/10.7910/DVN/NEM2XU, Harvard Dataverse, V1. Leifeld, P., & Cranmer, S. (2022). The stochastic actor-oriented model is a theory as much as it is a method and must be subject to theory tests. Network Science, 10(1). https://doi.org/10.1017/nws.2022.7 Wasserman, S., & Brandes, U. (2022) Editors’ Note. Network Science, 10(1). https://doi.org/10.1017/nws.2022.8
{"title":"A theoretical and empirical comparison of the temporal exponential random graph model and the stochastic actor-oriented model – Corrigendum","authors":"Philip Leifeld, S. Cranmer","doi":"10.1017/nws.2022.11","DOIUrl":"https://doi.org/10.1017/nws.2022.11","url":null,"abstract":"Block, P., Hollway, J., Stadtfeld, C., Koskinen, J., & Snijders, T. (2022). Circular specifications and “predicting” with information from the future: Errors in the empirical SAOM–TERGM comparison of Leifeld & Cranmer. Network Science, 10(1). https://doi.org/10.1017/nws.2022.6 Leifeld, P., & Cranmer, S. (2019a). A theoretical and empirical comparison of the temporal exponential random graph model and the stochastic actor-oriented model. Network Science, 7(1), 20–51. https://doi.org/10.1017/nws.2018.26 Leifeld, P., & Cranmer, S. (2019b). Replication Data for: A Theoretical and Empirical Comparison of the Temporal Exponential Random Graph Model and the Stochastic Actor-Oriented Model, https://doi.org/10.7910/DVN/NEM2XU, Harvard Dataverse, V1. Leifeld, P., & Cranmer, S. (2022). The stochastic actor-oriented model is a theory as much as it is a method and must be subject to theory tests. Network Science, 10(1). https://doi.org/10.1017/nws.2022.7 Wasserman, S., & Brandes, U. (2022) Editors’ Note. Network Science, 10(1). https://doi.org/10.1017/nws.2022.8","PeriodicalId":51827,"journal":{"name":"Network Science","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44610212","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the detection and the reconstruction of a large very dense subgraph in a social graph with n nodes and m edges given as a stream of edges, when the graph follows a power law degree distribution, in the regime when $m=O(n. log n)$ . A subgraph S is very dense if it has $Omega(|S|^2)$ edges. We uniformly sample the edges with a Reservoir of size $k=O(sqrt{n}.log n)$ . Our detection algorithm checks whether the Reservoir has a giant component. We show that if the graph contains a very dense subgraph of size $Omega(sqrt{n})$ , then the detection algorithm is almost surely correct. On the other hand, a random graph that follows a power law degree distribution almost surely has no large very dense subgraph, and the detection algorithm is almost surely correct. We define a new model of random graphs which follow a power law degree distribution and have large very dense subgraphs. We then show that on this class of random graphs we can reconstruct a good approximation of the very dense subgraph with high probability. We generalize these results to dynamic graphs defined by sliding windows in a stream of edges.
{"title":"Large very dense subgraphs in a stream of edges","authors":"Claire Mathieu, Michel de Rougemont","doi":"10.1017/nws.2021.17","DOIUrl":"https://doi.org/10.1017/nws.2021.17","url":null,"abstract":"<p>We study the detection and the reconstruction of a large very dense subgraph in a social graph with <span>n</span> nodes and <span>m</span> edges given as a stream of edges, when the graph follows a power law degree distribution, in the regime when <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220125124825863-0283:S2050124221000175:S2050124221000175_inline1.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$m=O(n. log n)$\u0000</span></span>\u0000</span>\u0000</span>. A subgraph <span>S</span> is very dense if it has <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220125124825863-0283:S2050124221000175:S2050124221000175_inline2.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$Omega(|S|^2)$\u0000</span></span>\u0000</span>\u0000</span> edges. We uniformly sample the edges with a Reservoir of size <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220125124825863-0283:S2050124221000175:S2050124221000175_inline3.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$k=O(sqrt{n}.log n)$\u0000</span></span>\u0000</span>\u0000</span>. Our detection algorithm checks whether the Reservoir has a giant component. We show that if the graph contains a very dense subgraph of size <span>\u0000<span>\u0000<img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220125124825863-0283:S2050124221000175:S2050124221000175_inline4.png\"/>\u0000<span data-mathjax-type=\"texmath\"><span>\u0000$Omega(sqrt{n})$\u0000</span></span>\u0000</span>\u0000</span>, then the detection algorithm is almost surely correct. On the other hand, a random graph that follows a power law degree distribution almost surely has no large very dense subgraph, and the detection algorithm is almost surely correct. We define a new model of random graphs which follow a power law degree distribution and have large very dense subgraphs. We then show that on this class of random graphs we can reconstruct a good approximation of the very dense subgraph with high probability. We generalize these results to dynamic graphs defined by sliding windows in a stream of edges.</p>","PeriodicalId":51827,"journal":{"name":"Network Science","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2022-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138524499","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
$m=O(n. log n)$ . A subgraph S is very dense if it has 
$Omega(|S|^2)$ edges. We uniformly sample the edges with a Reservoir of size 
$k=O(sqrt{n}.log n)$ . Our detection algorithm checks whether the Reservoir has a giant component. We show that if the graph contains a very dense subgraph of size 
$Omega(sqrt{n})$ , then the detection algorithm is almost surely correct. On the other hand, a random graph that follows a power law degree distribution almost surely has no large very dense subgraph, and the detection algorithm is almost surely correct. We define a new model of random graphs which follow a power law degree distribution and have large very dense subgraphs. We then show that on this class of random graphs we can reconstruct a good approximation of the very dense subgraph with high probability. We generalize these results to dynamic graphs defined by sliding windows in a stream of edges.
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