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Measuring directed triadic closure with closure coefficients

Published online by Cambridge University Press:  01 June 2020

Hao Yin Open the ORCID record for Hao Yin [Opens in a new window] ,
Austin R. Benson  and
Johan Ugander
Hao Yin
Affiliation:
Institute for Computation and Mathematical Engineering, Stanford University, Stanford, CA, USA (e-mail: yinh@stanford.edu)
Austin R. Benson
Affiliation:
Department of Computer Science, Cornell University, Ithaca, NY, USA (e-mail: arb@cs.cornell.edu)
Johan Ugander*
Affiliation:
Department of Management Science and Engineering, Stanford University, Stanford, CA, USA
*
*Corresponding author. Email: jugander@stanford.edu
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Abstract

Recent work studying triadic closure in undirected graphs has drawn attention to the distinction between measures that focus on the “center” node of a wedge (i.e., length-2 path) versus measures that focus on the “initiator,” a distinction with considerable consequences. Existing measures in directed graphs, meanwhile, have all been center-focused. In this work, we propose a family of eight directed closure coefficients that measure the frequency of triadic closure in directed graphs from the perspective of the node initiating closure. The eight coefficients correspond to different labeled wedges, where the initiator and center nodes are labeled, and we observe dramatic empirical variation in these coefficients on real-world networks, even in cases when the induced directed triangles are isomorphic. To understand this phenomenon, we examine the theoretical behavior of our closure coefficients under a directed configuration model. Our analysis illustrates an underlying connection between the closure coefficients and moments of the joint in- and out-degree distributions of the network, offering an explanation of the observed asymmetries. We also use our directed closure coefficients as predictors in two machine learning tasks. We find interpretable models with AUC scores above 0.92 in class-balanced binary prediction, substantially outperforming models that use traditional center-focused measures.

Keywords

directed networkstriadic closureclosure coefficientsconfiguration model
Type
Research Article
Information
Network Science , Volume 8 , Issue 4 , December 2020 , pp. 551 - 573
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

Action Editor: Ulrik Brandes

References

Ahnert, S. E., & Fink, T. M. A. (2008). Clustering signatures classify directed networks. Physical Review E, 78(3), 036112.CrossRefGoogle ScholarPubMed
Backstrom, L., Huttenlocher, D., Kleinberg, J., & Lan, X. (2006). Group formation in large social networks: Membership, growth, and evolution. In Proceedings of the 12th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (pp. 44–54). ACM.CrossRefGoogle Scholar
Ball, B., & Newman, M. E. J. (2013). Friendship networks and social status. Network Science, 1(1), 16–30.CrossRefGoogle Scholar
Barrat, A., & Weigt, M. (2000). On the properties of small-world network models. The European Physical Journal B–Condensed Matter and Complex Systems, 13(3), 547–560.CrossRefGoogle Scholar
Bascompte, J., Melián, C. J., & Sala, E. (2005). Interaction strength combinations and the over fishing of a marine food web. Proceedings of the National Academy of Sciences, 102(15), 5443–5447.CrossRefGoogle Scholar
Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., & Hwang, D.-U. (2006). Complex networks: Structure and dynamics. Physics Reports, 424(4–5), 175–308.CrossRefGoogle Scholar
Brzozowski, M. J., & Romero, D. M. (2011). Who should I follow? Recommending people in directed social networks. In Fifth International AAAI Conference on Weblogs and Social Media.Google Scholar
Chen, N., & Olvera-Cravioto, M. (2013). Directed random graphs with given degree distributions. Stochastic Systems, 3(1), 147–186.CrossRefGoogle Scholar
Cheng, J., Romero, D. M., Meeder, B., & Kleinberg, J. (2011). Predicting reciprocity in social networks. In 2011 IEEE Third International Conference on Privacy, Security, Risk and Trust and 2011 IEEE Third International Conference on Social Computing (pp. 49–56). IEEE.CrossRefGoogle Scholar
Davis, J. A., & Leinhardt, S. (1972). The structure of positive relations in small groups. In Berger, J., Zelditch, M., & Anderson, B. (Eds.), Sociological theories in progress (vol. 2, pp. 218–251). Boston, MA: Houghton Mifflin.Google Scholar
Fagiolo, G. (2007). Clustering in complex directed networks. Physical Review E, 76(2), 026107.CrossRefGoogle ScholarPubMed
Fortunato, S. (2010). Community detection in graphs. Physics Reports, 486(3–5), 75–174.CrossRefGoogle Scholar
Fosdick, B. K., Larremore, D. B., Nishimura, J., & Ugander, J. (2018). Configuring random graph models with fixed degree sequences. SIAM Review, 60(2), 315–355.CrossRefGoogle Scholar
Friedman, J., Hastie, T., & Tibshirani, R. (2010). Regularization paths for generalized linear models via coordinate descent. Journal of Statistical Software, 33(1), 1.CrossRefGoogle ScholarPubMed
Garlaschelli, D., & Loffredo, M. I. (2004). Patterns of link reciprocity in directed networks. Physical Review Letters, 93(26), 268701.CrossRefGoogle ScholarPubMed
Gehrke, J., Ginsparg, P., & Kleinberg, J. (2003). Overview of the 2003 KDD Cup. ACM SIGKDD Explorations Newsletter, 5(2), 149–151.CrossRefGoogle Scholar
Gleich, D. F., & Seshadhri, C. (2012). Vertex neighborhoods, low conductance cuts, and good seeds for local community methods. In Proceedings of the 18th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (pp. 597–605). ACM.CrossRefGoogle Scholar
Greenhill, C. (2014). The switch Markov chain for sampling irregular graphs. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms (pp. 1564–1572). SIAM.Google Scholar
Henderson, K., Gallagher, B., Eliassi-Rad, T., Tong, H., Basu, S., Akoglu, L., Koutra, D., Faloutsos, C., & Li, L. (2012). RolX: Structural role extraction & mining in large graphs. In Proceedings of the 18th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (pp. 1231–1239). ACM.CrossRefGoogle Scholar
Homans, G. C. (1950).The human group. Harcourt: Brace & World.Google Scholar
Huang, H., Tang, J., Wu, S., & Liu, L. (2014). Mining triadic closure patterns in social networks. In Proceedings of the Twenty-Third International Conference on World Wide Web (pp. 499–504). ACM.CrossRefGoogle Scholar
Jackson, M. O., & Rogers, B. W. (2007). Meeting strangers and friends of friends: How random are social networks? American Economic Review, 97(3), 890–915.CrossRefGoogle Scholar
Kaiser, M. (2008). Mean clustering coefficients: The role of isolated nodes and leafs on clustering measures for small-world networks. New Journal of Physics, 10(8), 083042.CrossRefGoogle Scholar
LaFond, T., Neville, J., & Gallagher, B. (2014). Anomaly detection in networks with changing trends. In Outlier Detection and Description Under Data Diversity at the International Conference on Knowledge Discovery and Data Mining.Google Scholar
Lazega, E. (2001).The collegial phenomenon: The social mechanisms of cooperation among peers in a corporate law partnership. Oxford, UK: Oxford University Press.CrossRefGoogle Scholar
Leskovec, J., Kleinberg, J., & Faloutsos, C. (2005). Graphs over time: Densification laws, shrinking diameters and possible explanations. In Proceedings of the eleventh ACM SIGKDD International Conference on Knowledge Discovery in Data Mining (pp. 177–187). ACM.CrossRefGoogle Scholar
Leskovec, J., Lang, K. J., Dasgupta, A., & Mahoney, M. W. (2009). Community structure in large networks: Natural cluster sizes and the absence of large well-defined clusters. Internet Mathematics, 6(1), 29–123.CrossRefGoogle Scholar
Leskovec, J., Huttenlocher, D., & Kleinberg, J. (2010). Signed networks in social media. In Proceedings of the SIGCHI Conference on Human Factors in Computing Systems (pp. 1361–1370). ACM.CrossRefGoogle Scholar
Liao, W., Ding, J., Marinazzo, D., Xu, Q., Wang, Z., Yuan, C., Zhang, Z., Lu, G., & Chen, H. (2011). Small-world directed networks in the human brain: Multivariate Granger causality analysis of resting-state fMRI. Neuroimage, 54(4), 2683–2694.CrossRefGoogle ScholarPubMed
Lou, T., Tang, J., Hopcroft, J., Fang, Z., & Ding, X. (2013). Learning to predict reciprocity and triadic closure in social networks. ACM Transactions on Knowledge Discovery from Data (TKDD), 7(2), 5.CrossRefGoogle Scholar
Mangan, S., Zaslaver, A., & Alon, U. (2003). The coherent feedforward loop serves as a sign-sensitive delay element in transcription networks. Journal of Molecular Biology, 334(2), 197–204.CrossRefGoogle ScholarPubMed
Milo, R., Shen-Orr, S., Itzkovitz, S., Kashtan, N., Chklovskii, D., & Alon, U. (2002). Network motifs: Simple building blocks of complex networks. Science, 298(5594), 824–827.CrossRefGoogle ScholarPubMed
Minoiu, C., & Reyes, J. A. (2013). A network analysis of global banking: 1978–2010. Journal of Financial Stability, 9(2), 168–184.CrossRefGoogle Scholar
Molloy, M., & Reed, B. (1995). A critical point for random graphs with a given degree sequence. Random Structures & Algorithms, 6(2–3), 161–180.CrossRefGoogle Scholar
Newman, M. E. J. (2003). The structure and function of complex networks. SIAM Review, 45(2), 167–256.CrossRefGoogle Scholar
Newman, M. E. J., Strogatz, S. H., & Watts, D. J. (2001). Random graphs with arbitrary degree distributions and their applications. Physical Review E, 64(2), 026118.CrossRefGoogle ScholarPubMed
Newman, M. E. J., Forrest, S., & Balthrop, J. (2002). Email networks and the spread of computer viruses. Physical Review E, 66(3), 035101.CrossRefGoogle ScholarPubMed
Onnela, J.-P., Saramäki, J., Kertész, J., & Kaski, K. (2005). Intensity and coherence of motifs in weighted complex networks. Physical Review E, 71(6), 065103.CrossRefGoogle ScholarPubMed
Panzarasa, P., Opsahl, T., & Carley, K. M. (2009). Patterns and dynamics of users’ behavior and interaction: Network analysis of an online community. Journal of the American Society for Information Science and Technology, 60(5), 911–932.CrossRefGoogle Scholar
Rao, A. R., Jana, R., & Bandyopadhyay, S. (1996). A Markov chain Monte Carlo method for generating random (0, 1)-matrices with given marginals. Sankhyā: The Indian Journal of Statistics, Series A, 58, 225–242.Google Scholar
Rapoport, A. (1953). Spread of information through a population with socio-structural bias: I. Assumption of transitivity. The Bulletin of Mathematical Biophysics, 15(4), 523–533.CrossRefGoogle Scholar
Richardson, M., Agrawal, R., & Domingos, P. (2003). Trust management for the semantic web. In International Semantic Web Conference (pp. 351–368). Springer.CrossRefGoogle Scholar
Robles, P., Moreno, S., & Neville, J. (2016). Sampling of attributed networks from hierarchical generative models. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (pp. 1155–1164). ACM.CrossRefGoogle Scholar
Romero, D. M., & Kleinberg, J. (2010). The directed closure process in hybrid social-information networks, with an analysis of link formation on Twitter. In Fourth International AAAI Conference on Weblogs and Social Media.Google Scholar
Sarajlić, A., Malod-Dognin, N., Yaveroğlu, Ö. N., & Pržulj, N. (2016). Graphlet-based characterization of directed networks. Scientific Reports, 6, 35098.CrossRefGoogle ScholarPubMed
Seshadhri, C., Pinar, A., Durak, N., & Kolda, T. G. (2016). Directed closure measures for networks with reciprocity. Journal of Complex Networks, 5(1), 32–47.Google Scholar
Seshadhri, C., Kolda, T. G., & Pinar, A. (2012). Community structure and scale-free collections of Erdös-Rényi graphs. Physical Review E, 85(5), 056109.CrossRefGoogle ScholarPubMed
Simmel, G. (1908).Soziologie: Untersuchungen über die formen der vergesellschaftung. Leipzig, Germany: Duncker & Humblot.Google Scholar
Stegehuis, C. (2019). Closure coefficients in scale-free complex networks. arxiv preprint arxiv:1911.11410.Google Scholar
Ulanowicz, R. E., & DeAngelis, D. L. (2005). Network analysis of trophic dynamics in South Florida ecosystems. US Geological Survey Program on the South Florida Ecosystem, 114, 45.Google Scholar
Wasserman, S., & Faust, K. (1994). Social network analysis: Methods and applications. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
Watts, D. J., & Strogatz, S. H. (1998). Collective dynamics of ‘small-world’ networks. Nature, 393(6684), 440.CrossRefGoogle ScholarPubMed
Yin, H., Benson, A. R., Leskovec, J., & Gleich, D. F. (2017). Local higher-order graph clustering. In Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (pp. 555–564). ACM.CrossRefGoogle Scholar
Yin, H., Benson, A. R., & Leskovec, J. (2019). The local closure coefficient: A new perspective on network clustering. Proceedings of the Twelfth ACM International Conference on Web Search and Data Mining (pp. 303–311). ACM.CrossRefGoogle Scholar