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Transitivity correlation: A descriptive measure of network transitivity

Published online by Cambridge University Press:  23 September 2019

David Dekker Open the ORCID record for David Dekker [Opens in a new window] ,
David Krackhardt  and
Tom A. B. Snijders
David Dekker*
Affiliation:
Center for Business Network Analysis, University of Greenwich, London, UK
David Krackhardt
Affiliation:
Heinz College and the Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, USA (email: krack@cmu.edu)
Tom A. B. Snijders
Affiliation:
Nuffield College, University of Oxford, Oxford, UK Department of Sociology, University of Groningen, Groningen, The Netherlands (email: Tom.Snijders@nuffield.ox.ac.uk)
*
*Corresponding author. Email: david.dekker@gmail.com
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Abstract

This paper proposes that common measures for network transitivity, based on the enumeration of transitive triples, do not reflect the theoretical statements about transitivity they aim to describe. These statements are often formulated as comparative conditional probabilities, but these are not directly reflected by simple functions of enumerations. We think that a better approach is obtained by considering the probability of a tie between two randomly drawn nodes, conditional on selected features of the network. Two measures of transitivity based on correlation coefficients between the existence of a tie and the existence, or the number, of two-paths between the nodes are developed, and called “Transitivity Phi” and “Transitivity Correlation.” Some desirable properties for these measures are studied and compared to existing clustering coefficients, in both random (Erdös–Renyi) and in stylized networks (windmills). Furthermore, it is shown that in a directed graph, under the condition of zero Transitivity Correlation, the total number of transitive triples is determined by four underlying features: size, density, reciprocity, and the covariance between in- and outdegrees. Also, it is demonstrated that plotting conditional probability of ties, given the number of two-paths, provides valuable insights into empirical regularities and irregularities of transitivity patterns.

Keywords

network transitivitytransitive triplestransitivity covariancetransitivity correlationtransitivity Phiclustering coefficient
Type
Original Article
Information
Network Science , Volume 7 , Issue 3 , September 2019 , pp. 353 - 375
Copyright
© Cambridge University Press 2019 

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References

Butts, C. T. (2016). Package “sna”. Available at: https://cran.r-project.org/web/packages/sna/sna.pdfGoogle Scholar
Chatterjee, S., Diaconis, P., & Sly, A. (2011). Random graphs with a given degree sequence. The Annals of Applied Probability, 1400–1435.CrossRefGoogle Scholar
Coleman, J. S. (1990). Foundations of Social Theory. Cambridge, MA: Harvard University Press.Google Scholar
Coulomb, S., Bauer, M., Bernard, D., & Marsolier-Kergoat, M.-C. (2005). Gene essentiality and the topology of protein interaction networks. Proceedings of the Royal Society of London B: Biological Sciences, 272(1573), 1721–1725.CrossRefGoogle ScholarPubMed
Cross, R. L., & Parker, A. (2004). The Hidden Power of Social Networks: Understanding How Work Really Gets Done in Organizations. Boston, MA: Harvard Business Press.Google Scholar
Davis, A., Gardner, B. B., Gardner, M. R., & Warner, W. L. (1941). Deep South: A Sociological Anthropological Study of Caste and Class. Chicago: University of Chicago Press.Google Scholar
Davis, J. A. (1970). Clustering and hierarchy in interpersonal relations: Testing two graph theoretical models on 742 sociomatrices. American Sociological Review, 35, 843–852.CrossRefGoogle Scholar
Erdös, P., & Renyi, A. (1959). On random graphs. Publ Math Debrecen, 6(290–297), 290–297.Google Scholar
Falk, R., & Well, A. D. (1997). Many faces of the correlation coefficient. Journal of Statistics Education, 5(3), 1–18.CrossRefGoogle Scholar
Faust, K. (2007). Very local structure in social networks. Sociological methodology, 5(2), 148–256.Google Scholar
Faust, K. (2010). A puzzle concerning triads in social networks: Graph constraints and the triad census. Social Networks, 32(3), 221–233.CrossRefGoogle Scholar
Feld, S. L., & Elmore, R. (1982). Patterns of sociometric choices: Transitivity reconsidered. Social Psychology Quarterly, 45(2), 77–85.CrossRefGoogle Scholar
Frank, O. (1980). Sampling and inference in a population graph. International Statistical Review Revue Internationale de Statistique, 48(1), 33.Google Scholar
Frank, O. (1988). Random sampling and social networks: A survey of various approaches. Mathematiques Informatique et Sciences Humaines, 26, 19–33.Google Scholar
Freeman, L. C. (2017). Datasets. moreno.ss.uci.edu/data.html, Accessed February 1, 2017.Google Scholar
Goodreau, S. M., Kitts, J. A., & Morris, M. (2009). Birds of a feather, or friend of a friend? using exponential random graph models to investigate adolescent social networks. Demography, 46(1), 103–125.CrossRefGoogle ScholarPubMed
Granovetter, M. S. (1973). The Strength of Weak Ties. American Journal of Sociology, 78(6), 1360–1380.CrossRefGoogle Scholar
Han, J.-D. J., Dupuy, D., Bertin, N., Cusick, M. E., & Vidal, M. (2005). Effect of sampling on topology predictions of protein-protein interaction networks. Nature Biotechnology, 23(7), 839–844.CrossRefGoogle ScholarPubMed
Harary, F., & Kommel, H. J. (1979). Matrix measures for transitivity and balance. Journal of Mathematical Sociology, 6(2), 199–210.CrossRefGoogle Scholar
Hayes, B. (2006). Connecting the dots can the tools of graph theory and social-network studies unravel the next big plot? American Scientist, 94(5), 400–404.CrossRefGoogle Scholar
Heider, F. (1958). The Psychology of Interpersonal Relations. New York: John Wiley & Sons.CrossRefGoogle Scholar
Holland, P. W., & Leinhardt, S. (1970). A method for detecting structure in sociometric data. American Journal of Sociology, 76(3), 492–513.CrossRefGoogle Scholar
Holland, P. W., & Leinhardt, S. (1971). Transitivity in structural models of small groups. Small Group Research, 2(2), 107–124.Google Scholar
Holland, P. W., & Leinhardt, S. (1972). Holland and Leinhardt reply: Some evidence on the transitivity of positive interpersonal sentiment. American Journal of Sociology, 77(6), 1205–1209.CrossRefGoogle Scholar
Holland, P. W, & Leinhardt, S. (1976). Local structure in social networks. Sociological Methodology, 7, 1–45.CrossRefGoogle Scholar
Jackson, M. O. (2008). Social and Economic Networks. Vol. 3. Princeton: Princeton University Press.Google Scholar
Jeong, H., Mason, S. P., Barabási, A.-L., & Oltvai, Z. N. (2001). Lethality and centrality in protein networks. Nature, 411(6833), 41–42.CrossRefGoogle ScholarPubMed
Karlberg, M. (1999). Testing transitivity in digraphs. Sociological Methodology, 29(1), 225–251.CrossRefGoogle Scholar
Krackhardt, D. (1987). Cognitive social structures. Social Networks, 9, 109–134.CrossRefGoogle Scholar
Lazega, E. (2001). The Collegial Phenomenon: The Social Mechanisms of Cooperation Among Peers in a Corporate Law Partnership. Oxford University Press on Demand.CrossRefGoogle Scholar
Lusher, D., Koskinen, J., & Robins, G. (2012). Exponential Random Graph Models for Social Networks: Theory, Methods, and Applications. New York: Cambridge University Press.CrossRefGoogle Scholar
Lusseau, D. (2003). The emergent properties of a dolphin social network. Proceedings of the Royal Society of London B: Biological Sciences, 270(Suppl 2), S186–S188.CrossRefGoogle ScholarPubMed
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), 19.CrossRefGoogle ScholarPubMed
Newman, M. E. J. (2003). The structure and function of complex networks. Siam Review, 45(2), 167–256.CrossRefGoogle Scholar
Newman, M. E. J. (2006). Finding community structure in networks using the eigenvectors of matrices. Physical Review E, 74(3), 036104.CrossRefGoogle ScholarPubMed
Opsahl, T. (2017). Datasets. https://toreopsahl.com/datasets/, Accessed February 1, 2017.Google Scholar
Opsahl, T., & Panzarasa, P. (2009). Clustering in weighted networks. Social Networks, 31(2), 155–163.CrossRefGoogle Scholar
Padgett, J. F., & Ansell, C. K. (1993). Robust action and the rise of the medici, 1400-1434. American Journal of Sociology, 98(6), 1259–1319.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, 225–242.Google Scholar
Roberts, J. M. (2000). Simple methods for simulating sociomatrices with given marginal totals. Social Networks, 22(3), 273–283.CrossRefGoogle Scholar
Schelling, T. C. (1978). Micromotives and Macrobehavior. New York [etc.]: Norton.Google Scholar
Smith, D. A., & White, D. R. (1992). Structure and dynamics of the global economy: Network analysis of international trade 1965–1980. Social Forces, 857–893.CrossRefGoogle Scholar
Snijders, T. A. B. (1991). Enumeration and simulation methods for 0–1 matrices with given marginals. Psychometrika, 56(3), 397–417.CrossRefGoogle Scholar
Snijders, T. A. B., Pattison, P. E., Robins, G. L., & Handcock, M. S. (2006). New specifications for exponential random graph models. Sociological Methodology, 36, 99–153.CrossRefGoogle Scholar
Snijders, T. A. B. (2011). Statistical models for social networks. Annual Review of Sociology, 37, 131–153.CrossRefGoogle Scholar
Snijders, T. A. B. (2017). Manual for ZO version 2.3. Department of Sociology, University of Groningen, Groningen. http://www.stats.ox.ac.uk/~snijders/socnet.htm.Google Scholar
Stumpf, M. P. H., Wiuf, C., & May, R. M. (2005). Subnets of scale-free networks are not scale-free: Sampling properties of networks. Proceedings of the National Academy of Sciences of the United States of America, 102(12), 4221–4224.CrossRefGoogle Scholar
Tao, T. (2016). An improved MCMC algorithm for generating random graphs from constrained distributions. Network Science, 4, 117–139.CrossRefGoogle Scholar
Verhelst, N. D. (2008). An efficient MCMC algorithm to sample binary matrices with fixed marginals. Psychometrika, 73(4), 705–728.CrossRefGoogle Scholar
Wasserman, S., & Faust, K. (1994). Social Network Analysis: Methods and Applications. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Wasserman, S. S. (1975). Random directed graph distributions in the triad census in social networks. Journal of Mathematical Sociology, 5(1), 61–86.CrossRefGoogle Scholar
Watts, D. J., & Strogatz, S. H. (1998). Collective dynamics of ‘small-world’ networks. Nature, 393(6684), 440–442.CrossRefGoogle ScholarPubMed