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A dyadic reciprocity index for repeated interaction networks*

Published online by Cambridge University Press:  15 April 2013

CHENG WANG ,
OMAR LIZARDO ,
DAVID HACHEN ,
ANTHONY STRATHMAN ,
ZOLTÁN TOROCZKAI  and
NITESH V. CHAWLA
CHENG WANG
Affiliation:
Department of Sociology, University of Notre Dame, 810 Flanner Hall, Notre Dame, IN 46556, USA Interdisciplinary Center for Network Science and Applications (ICeNSA), 384E Nieuwland Science Hall, Notre Dame, IN 46556, USA (e-mail: cwang3@nd.edu, olizardo@nd.edu, dshachen@nd.edu)
OMAR LIZARDO
Affiliation:
Department of Sociology, University of Notre Dame, 810 Flanner Hall, Notre Dame, IN 46556, USA Interdisciplinary Center for Network Science and Applications (ICeNSA), 384E Nieuwland Science Hall, Notre Dame, IN 46556, USA (e-mail: cwang3@nd.edu, olizardo@nd.edu, dshachen@nd.edu)
DAVID HACHEN
Affiliation:
Department of Sociology, University of Notre Dame, 810 Flanner Hall, Notre Dame, IN 46556, USA Interdisciplinary Center for Network Science and Applications (ICeNSA), 384E Nieuwland Science Hall, Notre Dame, IN 46556, USA (e-mail: cwang3@nd.edu, olizardo@nd.edu, dshachen@nd.edu)
ANTHONY STRATHMAN
Affiliation:
Department of Physics, University of Notre Dame, 225 Nieuwland Science Hall, Notre Dame, IN 46556, USA Interdisciplinary Center for Network Science and Applications (ICeNSA), 384E Nieuwland Science Hall, Notre Dame, IN 46556, USA (e-mail: astrathm@nd.edu, toro@nd.edu)
ZOLTÁN TOROCZKAI
Affiliation:
Department of Physics, University of Notre Dame, 225 Nieuwland Science Hall, Notre Dame, IN 46556, USA Interdisciplinary Center for Network Science and Applications (ICeNSA), 384E Nieuwland Science Hall, Notre Dame, IN 46556, USA (e-mail: astrathm@nd.edu, toro@nd.edu)
NITESH V. CHAWLA
Affiliation:
Department of Computer Science and Engineering, College of Engineering, University of Notre Dame, 384 Fitzpatrick Hall, Notre Dame, IN 46556, USA Interdisciplinary Center for Network Science and Applications (ICeNSA), 384E Nieuwland Science Hall, Notre Dame IN 46556, USA (e-mail: nchawla@nd.edu)
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Abstract

A wide variety of networked systems in human societies are composed of repeated communications between actors. A dyadic relationship made up of repeated interactions may be reciprocal (both actors have the same probability of directing a communication attempt to the other) or non-reciprocal (one actor has a higher probability of initiating a communication attempt than the other). In this paper we propose a theoretically motivated index of reciprocity appropriate for networks formed from repeated interactions based on these probabilities. We go on to examine the distribution of reciprocity in a large-scale social network built from trace-logs of over a billion cell-phone communication events across millions of actors in a large industrialized country. We find that while most relationships tend toward reciprocity, a substantial minority of relationships exhibit large levels of non-reciprocity. This is puzzling because behavioral theories in social science predict that persons will selectively terminate non-reciprocal relationships, keeping only those that approach reciprocity. We point to two structural features of human communication behavior and relationship formation—the division of contacts into strong and weak ties and degree-based assortativity—that either help or hinder the ability of persons to obtain communicative balance in their relationships. We examine the extent to which deviations from reciprocity in the observed network are partially traceable to the operation of these countervailing tendencies.

Keywords

reciprocityweighted networksinteractionweak tiesassortativity
Type
Research Article
Information
Network Science , Volume 1 , Issue 1 , April 2013 , pp. 31 - 48
Copyright
Copyright © Cambridge University Press 2013

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Footnotes

*

Research was sponsored by the Army Research Laboratory and was accomplished in part under Cooperative Agreement Number W911NF-09-2-0053, by the Defense Threat Reduction Agency (DTRA) grant HDTRA 1-09-1-0039 (Anthony Strathman and Zoltán Toroczkai), and by the National Science Foundation (NSF) grant BCS-0826958. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the US Government. The US Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation hereon. Special acknowledgments go to Albert László Barabási for providing the source data.

References

Almaas, E., Kovács, B., Vicsek, T., Oltvai, Z. N., & Barabási, A. L. (2004). Letters to nature-global organization of metabolic fluxes in the bacterium ewhedchia coli. Nature, 427 (6977), 839–842.Google Scholar
Aral, S., & Van Alstyne, M. (2011). The diversity-bandwidth tradeoff. American Journal of Sociology, 117 (1), 90–171.CrossRefGoogle Scholar
Ball, B., & Newman, M. E. J. (2013). Friendship networks and social status. Network Science, 1 (1), (this issue).CrossRefGoogle Scholar
Barabasi, A. L., & Albert, R. (1999). Emergence of scaling in random networks. Science, 286 (5439), 509.Google Scholar
Barrat, A., Barthelemy, M., Pastor-Satorras, R., & Vespignani, A. (2004). The architecture of complex weighted networks. Proceedings of the National Academy of Sciences, 101 (11), 3747–3752.CrossRefGoogle ScholarPubMed
Barthelemy, M., Gondran, B., & Guichard, E. (2003). Spatial structure of the internet traffic. Physica A: Statistical Mechanics and Its Applications, 319, 633–642.Google Scholar
Barthélemy, M., Barrat, A., Pastor-Satorras, R., & Vespignani, A. (2005). Characterization and modeling of weighted networks. Physica A: Statistical Mechanics and Its Applications, 346 (1–2), 34–43.Google 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.Google Scholar
Borgatti, S. P. (2005). Centrality and network flow. Social Networks, 27 (1), 55–71.CrossRefGoogle Scholar
Borgatti, S. P., Mehra, A., Brass, D. J., & Labianca, G. (2009). Network analysis in the social sciences. Science, 323 (5916), 892.CrossRefGoogle ScholarPubMed
Carley, K. M., & Krackhardt, D. (1996). Cognitive inconsistencies and non-symmetric friendship. Social Networks, 18 (1), 1–27.CrossRefGoogle Scholar
Catanzaro, M., Caldarelli, G., & Pietronero, L. (2004). Assortative model for social networks. Physical Review E, 70 (3), 037101.Google ScholarPubMed
Colizza, V., Flammini, A., Serrano, M. A., & Vespignani, A. (2006). Detecting rich-club ordering in complex networks. Nature Physics, 2 (2), 110–115.CrossRefGoogle Scholar
Csermely, P. (2004). Strong links are important, but weak links stabilize them. Trends in Biochemical Sciences, 29 (7), 331–334.Google Scholar
Csermely, P. (2006). Weak links: stabilizers of complex systems from proteins to social networks. Heidelberg, Germany: Springer-Verlag.Google Scholar
Davis, James A. (1963). Structural balance, mechanical solidarity, and interpersonal relations. American Journal of Sociology, 68, 444–462.Google Scholar
Davis, J. A. (1979). The Davis/Holland/Leinhardt studies: An overview. In Holland, P. W. & Leinhardt, S. (Eds.), Perspectives on social network research (pp. 51–62). New York: Academic Press.CrossRefGoogle Scholar
Doreian, P. (2002). Event sequences as generators of social network evolution. Social Networks, 24 (2), 93–119.Google Scholar
Eagle, N., Pentland, A. S., & Lazer, D. (2008). Mobile phone data for inferring social network structure. In Liu, H., Salerno, J. J. & Young, M. J. (Eds.), Social Computing, Behavioral Modeling and Prediction (pp. 79–88). New York: Springer.Google Scholar
Eagle, N., Pentland, A. S., & Lazer, D. (2009). Inferring friendship network structure by using mobile phone data. Proceedings of the National Academy of Sciences, 106 (36), 15274.Google Scholar
Feld, Scott L. (1981). The focused organization of social ties. American Journal of Sociology, 86, 1015–1035.Google Scholar
Freeman, L. C., Borgatti, S. P., & White, D. R. (1991). Centrality in valued graphs: A measure of betweenness based on network flow. Social Networks, 13 (2), 141–154.Google Scholar
Garlaschelli, D., & Loffredo, M. I. (2004). Patterns of link reciprocity in directed networks. Physical Review Letters, 93 (26), 268701.Google Scholar
Gould, Roger V. (2002). The origins of status hierarchies: A formal theory and empirical test. American Journal of Sociology, 107 (5), 1143–1178.CrossRefGoogle Scholar
Gouldner, Alvin W. (1960). The norm of reciprocity: A preliminary statement. American Sociological Review, 25 (2), 161–178.Google Scholar
Granovetter, Mark S. (1973). The strength of weak ties. American Journal of Sociology, 78 (6), 1360–1380.CrossRefGoogle Scholar
Hallinan, Maureen T. (1978). The process of friendship formation. Social Networks, 1, 193–210.Google Scholar
Hallinan, M. T., & Hutchins, E. E. (1980). Structural effects on dyadic change. Social Forces, 59 (1), 225–245.Google Scholar
Hammer, M. (1985). Implications of behavioral and cognitive reciprocity in social network data. Social Networks, 7, 189–201.Google Scholar
Heider, F. (1958). The psychology of interpersonal relations. New York: Wiley.Google Scholar
Herfindahl, Orris C. (1950). Concentration in the steel industry. PhD thesis, Columbia University, New York, NY.Google Scholar
Karsai, M., Kaski, K., & Kertész, J. (2012). Correlated dynamics in egocentric communication networks. Plos One, 7 (7), e40612.Google Scholar
Kossinets, G., & Watts, D. J. (2006). Empirical analysis of an evolving social network. Science, 311 (5757), 88–90.Google Scholar
Kovanen, L., Saramaki, J., & Kaski, K. (2010). Reciprocity of mobile phone calls. arxiv preprint, arxiv:1002.0763.Google Scholar
Krackhardt, D. (1987). Cognitive social structures. Social Networks, 9 (2), 109–134.CrossRefGoogle Scholar
Mandel, M. (2000). Measuring tendency towards mutuality in a social network. Social Networks, 22 (4), 285–298.CrossRefGoogle Scholar
Marsden, Peter V. (1987). Core discussion networks of Americans. American Sociological Review, 52 (1), 122–131.Google Scholar
Marsden, P. V., & Campbell, K. E. (1984). Measuring tie strength. Social Forces, 63, 482–501.Google Scholar
Martin, J. L. (2009). Social structures. Princeton, NJ: Princeton University Press.Google Scholar
Maslov, S., & Sneppen, K. (2002). Specificity and stability in topology of protein networks. Science, 296 (5569), 910–913.Google Scholar
Newcomb, T. M. (1961). The acquaintance process. New York: Holt, Rinehart and Winston.Google Scholar
Newcomb, Theodore M. (1968). Interpersonal balance. In Abelson, R. P., Aronson, E., McGiure, W. J., Newcomb, T. M., Rosenberg, M. J. & Tannenbaum, O. H. (Eds.), Theories of cognitive consistency (pp. 28–51). New York: Holt, Rinchart & Winston.Google Scholar
NewcombTheodore, M. Theodore, M. (1979). Reciprocity of interpersonal attraction: A non-confirmation of a plausible hypothesis. Social Psychology Quarterly, 42, 299–306.Google Scholar
Newman, M. E. J. (2002). Assortative mixing in networks. Physical Review Letters, 89 (20), 208701.CrossRefGoogle ScholarPubMed
Newman, Mark E. J. (2003). Mixing patterns in networks. Physical Review E, 67 (2), 26126.Google Scholar
Newman, M. E. J., & Park, J. (2003). Why social networks are different from other types of networks. Physical Review E, 68 (3), 36122.Google Scholar
Park, J., & Newman, M. E. J. (2003). Origin of degree correlations in the internet and other networks. Physical Review E, 68 (2), 26112.Google Scholar
Peay, E. R. (1980). Connectedness in a general model for valued networks. Social Networks, 2 (4), 385–410.Google Scholar
Schaefer, David R. (2012). Homophily through nonreciprocity: Results of an experiment. Social Forces, 90 (4), 1271–1295.CrossRefGoogle Scholar
Serrano, M. A., Boguñá, M., & Vespignani, A. (2009). Extracting the multiscale backbone of complex weighted networks. Proceedings of the National Academy of Sciences, 106 (16), 6483–6488.CrossRefGoogle ScholarPubMed
Skvoretz, J., & Agneessens, F. (2007). Reciprocity, multiplexity, and exchange: Measures. Quality and Quantity, 41 (3), 341–357.Google Scholar
Waller, W. (1937). The rating and dating complex. American Sociological Review, 2 (5), 727–734.CrossRefGoogle Scholar
Wasserman, S., & Faust, K. (1994). Social network analysis: methods and applications. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
Wellman, B., & Wortley, S. (1990). Different strokes from different folks: Community ties and social support. American Journal of Sociology, 96, 558–588.CrossRefGoogle Scholar
Wu, Y., Zhou, C., Xiao, J., Kurths, J., & Schellnhuber, H. J. (2010). Evidence for a bimodal distribution in human communication. Proceedings of the National Academy of Sciences, 107 (44), 18803–18808.Google Scholar
Xu, X. K., Wang, J. B., Wu, Y., & Small, M. (2012). Pairwise interaction pattern in the weighted communication network. arxiv preprint, arxiv:1203.1105.Google Scholar
Yang, S., & Knoke, D. (2001). Optimal connections: strength and distance in valued graphs. Social Networks, 23 (4), 285–295.Google Scholar
Yook, S. H., Jeong, H., Barabasi, A. L., & Tu, Y. (2001). Weighted evolving networks. Physical Review Letters, 86 (25), 5835–5838.Google Scholar
Zamora-López, G., Zlatić, V., Zhou, C., Štefančić, H., & Kurths, J. (2008). Reciprocity of networks with degree correlations and arbitrary degree sequences. Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, 77 (1), 016106.CrossRefGoogle ScholarPubMed