# American Institute of Mathematical Sciences

July  2014, 19(5): 1355-1372. doi: 10.3934/dcdsb.2014.19.1355

## Rethinking centrality: The role of dynamical processes in social network analysis

 1 Robert Bosch LLC, 4005 Miranda Ave, Palo Alto, CA 94304, United States 2 USC Information Sciences Institute, 4676 Admiralty Way, Marina del Rey, CA 90292, United States

Received  January 2012 Revised  June 2012 Published  April 2014

Many popular measures used in social network analysis, including centrality, are based on the random walk. The random walk is a model of a stochastic process where a node interacts with one other node at a time. However, the random walk may not be appropriate for modeling social phenomena, including epidemics and information diffusion, in which one node may interact with many others at the same time, for example, by broadcasting the virus or information to its neighbors. To produce meaningful results, social network analysis algorithms have to take into account the nature of interactions between the nodes. In this paper we classify dynamical processes as conservative and non-conservative and relate them to well-known measures of centrality used in network analysis: PageRank and Alpha-Centrality. We demonstrate, by ranking users in online social networks used for broadcasting information, that non-conservative Alpha-Centrality generally leads to a better agreement with an empirical ranking scheme than the conservative PageRank.
Citation: Rumi Ghosh, Kristina Lerman. Rethinking centrality: The role of dynamical processes in social network analysis. Discrete & Continuous Dynamical Systems - B, 2014, 19 (5) : 1355-1372. doi: 10.3934/dcdsb.2014.19.1355
##### References:
 [1] R. M. Anderson and R. May, Infectious Diseases of Humans: Dynamics and Control,, Oxford University Press, (1991). Google Scholar [2] N. Bailey, The Mathematical Theory of Infectious Diseases and its Applications,, Griffin, (1975). Google Scholar [3] E. Bakshy, J. M. Hofman, W. A. Mason and D. J. Watts, Everyone's an influencer: Quantifying influence on twitter,, in Proc. the fourth ACM Int. Conf. on Web search and data mining, (2011), 65. doi: 10.1145/1935826.1935845. Google Scholar [4] A. Barrat, M. Barthélemy and A. Vespignani, Dynamical Processes on Complex Networks,, 1st edition, (2008). doi: 10.1017/CBO9780511791383. Google Scholar [5] L. M. A. Bettencourt, A. Cintrón-Arias, D. I. Kaiser and C. Castillo-Chávez, The power of a good idea: Quantitative modeling of the spread of ideas from epidemiological models,, Physica A: Statistical Mechanics and its Applications, 364 (2006), 513. doi: 10.1016/j.physa.2005.08.083. Google Scholar [6] P. Boldi, M. Santini and S. Vigna, Pagerank as a function of the damping factor,, in Proc. the 14th Int. Conf. on World Wide Web, (2005), 557. doi: 10.1145/1060745.1060827. Google Scholar [7] P. Bonacich, Factoring and weighting approaches to status scores and clique identification,, Journal of Mathematical Sociology, 2 (1972), 113. doi: 10.1080/0022250X.1972.9989806. Google Scholar [8] P. Bonacich, Power and centrality: A family of measures,, The American Journal of Sociology, 92 (1987), 1170. doi: 10.1086/228631. Google Scholar [9] P. Bonacich and P. Lloyd, Eigenvector-like measures of centrality for asymmetric relations,, Social Networks, 23 (2001), 191. doi: 10.1016/S0378-8733(01)00038-7. Google Scholar [10] S. Borgatti, Centrality and network flow,, Social Networks, 27 (2005), 55. doi: 10.1016/j.socnet.2004.11.008. Google Scholar [11] S. Borgatti and M. Everett, A graph-theoretic perspective on centrality,, Social Networks, 28 (2006), 466. doi: 10.1016/j.socnet.2005.11.005. Google Scholar [12] J. J. Brown and P. H. Reingen, Social ties and Word-of-Mouth referral behavior,, The Journal of Consumer Research, 14 (1987), 350. doi: 10.1086/209118. Google Scholar [13] D. Centola and M. Macy, Complex contagions and the weakness of long ties,, American Journal of Sociology, 113 (2007), 702. doi: 10.1086/521848. Google Scholar [14] M. Cha, H. Haddadi, F. Benevenuto and K. P. Gummadi, Measuring User Influence in Twitter: The Million Follower Fallacy,, in Proc. 4th Int. Conf. on Weblogs and Social Media (ICWSM), (2010). Google Scholar [15] K. Dietz, The estimation of the basic reproduction number for infectious diseases,, Statistical methods in medical research, 2 (1993), 23. doi: 10.1177/096228029300200103. Google Scholar [16] E. Estrada, N. Hatano and M. Benzi, The physics of communicability in complex networks,, Physics Reports, 514 (2012), 89. doi: 10.1016/j.physrep.2012.01.006. Google Scholar [17] S. Fortunato and A. Flammini, Random walks on directed networks: The case of pageRank,, International Journal of Bifurcation and Chaos, 17 (2007), 2343. doi: 10.1142/S0218127407018439. Google Scholar [18] L. C. Freeman, A set of measures of centrality based on betweenness,, Sociometry, 40 (1977), 35. doi: 10.2307/3033543. Google Scholar [19] F. Gebali, Markov chains.,, Analysis of Computer and Communication Networks, (). Google Scholar [20] R. Ghosh and K. Lerman, Predicting Influential Users in Online Social Networks,, in Proc. KDD workshop on Social Network Analysis (SNAKDD), (2010). Google Scholar [21] R. Ghosh and K. Lerman, A Framework for Quantitative Analysis of Cascades on Networks,, in Proc. Web Search and Data Mining Conference (WSDM), (2011), 665. doi: 10.1145/1935826.1935917. Google Scholar [22] R. Ghosh and K. Lerman, Parameterized centrality metric for network analysis,, Physical Review E, 83 (2011). doi: 10.1103/PhysRevE.83.066118. Google Scholar [23] R. Ghosh, T. Surachawala and K. Lerman, Entropy-based classification of ‘retweeting' activity on twitter,, in Proc. KDD workshop on Social Network Analysis (SNA-KDD), (2011). Google Scholar [24] D. F. Gleich, P. G. Constantine, A. D. Flaxman and A. Gunawardana, Tracking the random surfer: Empirically measured teleportation parameters in PageRank,, in Proc. 19th international conference on World wide web, (2010), 381. doi: 10.1145/1772690.1772730. Google Scholar [25] S. Goel, D. J. Watts and D. G. Goldstein, The structure of online diffusion networks,, in Proc. 13th ACM Conference on Electronic Commerce (EC 2012), (2012), 623. doi: 10.1145/2229012.2229058. Google Scholar [26] J. Goldenberg, B. Libai and E. Muller, Talk of the network: A complex systems look at the underlying process of word-of-mouth,, Marketing Letters, (): 211. Google Scholar [27] H. W. Hethcote, The mathematics of infectious diseases,, SIAM Review, 42 (2000), 599. doi: 10.1137/S0036144500371907. Google Scholar [28] N. Hodas and K. Lerman, How limited visibility and divided attention constrain social contagion,, in submitted to Social Computing, (2012). Google Scholar [29] N. O. Hodas and K. Lerman, The simple rules of social contagion,, Scientific Reports, 4 (2014). doi: 10.1038/srep04343. Google Scholar [30] T. Hogg and K. Lerman, Stochastic models of user-contributory web sites,, in Proc. 3rd Int. Conf. on Weblogs and Social Media (ICWSM), (2009). Google Scholar [31] T. Hogg and K. Lerman, Social Dynamics of Digg,, EPJ Data Science, 5 (2012). Google Scholar [32] J. L. Iribarren and E. Moro, Impact of human activity patterns on the dynamics of information diffusion,, Physical Review Letters, 103 (2009). doi: 10.1103/PhysRevLett.103.038702. Google Scholar [33] G. Jeh and J. Widom, Scaling personalized web search,, in Proc. the 12th Int. Conf. on World Wide Web, (2003), 271. doi: 10.1145/775189.775191. Google Scholar [34] E. Katz and P. Lazarsfeld, Personal Influence: The Part Played by People in the Flow of Mass Communications,, Transaction Publishers, (2005). Google Scholar [35] L. Katz, A new status index derived from sociometric analysis,, Psychometrika, 18 (1953), 39. doi: 10.1007/BF02289026. Google Scholar [36] D. Kempe, J. Kleinberg and E. Tardos, Maximizing the spread of influence through a social network,, KDD '03 Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining, (2003), 137. doi: 10.1145/956755.956769. Google Scholar [37] C. Kiss and M. Bichler, Identiffication of influencers-measuring influence in customer networks,, Decision Support Systems, 46 (2008), 233. Google Scholar [38] S. Kotz and N. Balakrishnan, Advances in urn models during the past two decades,, in Advances in combinatorial methods and applications to probability and statistics, (1997), 203. Google Scholar [39] , R. Lambiotte, J. C. Delvenne and M. Barahona,, Laplacian dynamics and multiscale modular structure in networks., (). Google Scholar [40] R. Lambiotte, R. Sinatra, J. C. Delvenne, T. S. Evans, M. Barahona and V. Latora, Flow graphs: Interweaving dynamics and structure,, Physical Review E, 84 (2011). doi: 10.1103/PhysRevE.84.017102. Google Scholar [41] C. Lee, H. Kwak, H. Park and S. Moon, Finding Influentials from Temporal Order of Information Adoption in Twitter,, Proc. 19th World-Wide Web (WWW) Conference (Poster), (2010). Google Scholar [42] K. Lerman and R. Ghosh, Information Contagion: An Empirical Study of the Spread of News on Digg and Twitter Social Networks,, Proc. 4th Int. Conf. on Weblogs and Social Media (ICWSM), (2010). Google Scholar [43] L. Page, S. Brin, R. Motwani and T. Winograd, The PageRank Citation Ranking: Bringing Order to the Web,, Technical report, (1998). Google Scholar [44] R. Pastor-Satorras and A. Vespignani, Epidemic spreading in scale-free networks,, Physical Review Letters, 86 (2001), 3200. doi: 10.1103/PhysRevLett.86.3200. Google Scholar [45] B. A. Prakash, D. Chakrabartiy, M. Faloutsos, N. Valler and C. Faloutsos, Threshold conditions for arbitrary cascade models on arbitrary networks,, in Proc. the Int. Conf. on Data Mining, (2011), 537. doi: 10.1109/ICDM.2011.145. Google Scholar [46] E. M. Rogers, Diffusion of Innovations, 5th Edition,, Free Press, (2003). Google Scholar [47] D. M. Romero, W. Galuba, S. Asur and B. A. Huberman, Influence and passivity in social media,, in Proc. the 20th international Conference on World wide web, (2011), 113. doi: 10.1145/1963192.1963250. Google Scholar [48] H. Tong, C. Faloutsos and J. Pan, Fast random walk with restart and its applications,, in ICDM '06: Proc. the Sixth Int. Conf. on Data Mining, (2006), 613. doi: 10.1109/ICDM.2006.70. Google Scholar [49] M. Trusov, A. V. Bodapati and R. E. Bucklin, Determining influential users in internet social networks,, Journal of Marketing Research, 47 (2010), 643. doi: 10.1509/jmkr.47.4.643. Google Scholar [50] G. Ver Steeg, R. Ghosh and K. Lerman, What stops social epidemics?,, in Proc. 5th International AAAI Conference on Weblogs and Social Media (ICWSM), (2011). Google Scholar [51] Y. Wang, D. Chakrabarti, C. Wang and C. Faloutsos, Epidemic Spreading in Real Networks: An Eigenvalue Viewpoint,, Reliable Distributed Systems, 0 (2003). Google Scholar [52] D. J. Watts and P. S. Dodds, Influentials, networks, and public opinion formation,, Journal of Consumer Research, 34 (2007), 441. doi: 10.1086/518527. Google Scholar

show all references

##### References:
 [1] R. M. Anderson and R. May, Infectious Diseases of Humans: Dynamics and Control,, Oxford University Press, (1991). Google Scholar [2] N. Bailey, The Mathematical Theory of Infectious Diseases and its Applications,, Griffin, (1975). Google Scholar [3] E. Bakshy, J. M. Hofman, W. A. Mason and D. J. Watts, Everyone's an influencer: Quantifying influence on twitter,, in Proc. the fourth ACM Int. Conf. on Web search and data mining, (2011), 65. doi: 10.1145/1935826.1935845. Google Scholar [4] A. Barrat, M. Barthélemy and A. Vespignani, Dynamical Processes on Complex Networks,, 1st edition, (2008). doi: 10.1017/CBO9780511791383. Google Scholar [5] L. M. A. Bettencourt, A. Cintrón-Arias, D. I. Kaiser and C. Castillo-Chávez, The power of a good idea: Quantitative modeling of the spread of ideas from epidemiological models,, Physica A: Statistical Mechanics and its Applications, 364 (2006), 513. doi: 10.1016/j.physa.2005.08.083. Google Scholar [6] P. Boldi, M. Santini and S. Vigna, Pagerank as a function of the damping factor,, in Proc. the 14th Int. Conf. on World Wide Web, (2005), 557. doi: 10.1145/1060745.1060827. Google Scholar [7] P. Bonacich, Factoring and weighting approaches to status scores and clique identification,, Journal of Mathematical Sociology, 2 (1972), 113. doi: 10.1080/0022250X.1972.9989806. Google Scholar [8] P. Bonacich, Power and centrality: A family of measures,, The American Journal of Sociology, 92 (1987), 1170. doi: 10.1086/228631. Google Scholar [9] P. Bonacich and P. Lloyd, Eigenvector-like measures of centrality for asymmetric relations,, Social Networks, 23 (2001), 191. doi: 10.1016/S0378-8733(01)00038-7. Google Scholar [10] S. Borgatti, Centrality and network flow,, Social Networks, 27 (2005), 55. doi: 10.1016/j.socnet.2004.11.008. Google Scholar [11] S. Borgatti and M. Everett, A graph-theoretic perspective on centrality,, Social Networks, 28 (2006), 466. doi: 10.1016/j.socnet.2005.11.005. Google Scholar [12] J. J. Brown and P. H. Reingen, Social ties and Word-of-Mouth referral behavior,, The Journal of Consumer Research, 14 (1987), 350. doi: 10.1086/209118. Google Scholar [13] D. Centola and M. Macy, Complex contagions and the weakness of long ties,, American Journal of Sociology, 113 (2007), 702. doi: 10.1086/521848. Google Scholar [14] M. Cha, H. Haddadi, F. Benevenuto and K. P. Gummadi, Measuring User Influence in Twitter: The Million Follower Fallacy,, in Proc. 4th Int. Conf. on Weblogs and Social Media (ICWSM), (2010). Google Scholar [15] K. Dietz, The estimation of the basic reproduction number for infectious diseases,, Statistical methods in medical research, 2 (1993), 23. doi: 10.1177/096228029300200103. Google Scholar [16] E. Estrada, N. Hatano and M. Benzi, The physics of communicability in complex networks,, Physics Reports, 514 (2012), 89. doi: 10.1016/j.physrep.2012.01.006. Google Scholar [17] S. Fortunato and A. Flammini, Random walks on directed networks: The case of pageRank,, International Journal of Bifurcation and Chaos, 17 (2007), 2343. doi: 10.1142/S0218127407018439. Google Scholar [18] L. C. Freeman, A set of measures of centrality based on betweenness,, Sociometry, 40 (1977), 35. doi: 10.2307/3033543. Google Scholar [19] F. Gebali, Markov chains.,, Analysis of Computer and Communication Networks, (). Google Scholar [20] R. Ghosh and K. Lerman, Predicting Influential Users in Online Social Networks,, in Proc. KDD workshop on Social Network Analysis (SNAKDD), (2010). Google Scholar [21] R. Ghosh and K. Lerman, A Framework for Quantitative Analysis of Cascades on Networks,, in Proc. Web Search and Data Mining Conference (WSDM), (2011), 665. doi: 10.1145/1935826.1935917. Google Scholar [22] R. Ghosh and K. Lerman, Parameterized centrality metric for network analysis,, Physical Review E, 83 (2011). doi: 10.1103/PhysRevE.83.066118. Google Scholar [23] R. Ghosh, T. Surachawala and K. Lerman, Entropy-based classification of ‘retweeting' activity on twitter,, in Proc. KDD workshop on Social Network Analysis (SNA-KDD), (2011). Google Scholar [24] D. F. Gleich, P. G. Constantine, A. D. Flaxman and A. Gunawardana, Tracking the random surfer: Empirically measured teleportation parameters in PageRank,, in Proc. 19th international conference on World wide web, (2010), 381. doi: 10.1145/1772690.1772730. Google Scholar [25] S. Goel, D. J. Watts and D. G. Goldstein, The structure of online diffusion networks,, in Proc. 13th ACM Conference on Electronic Commerce (EC 2012), (2012), 623. doi: 10.1145/2229012.2229058. Google Scholar [26] J. Goldenberg, B. Libai and E. Muller, Talk of the network: A complex systems look at the underlying process of word-of-mouth,, Marketing Letters, (): 211. Google Scholar [27] H. W. Hethcote, The mathematics of infectious diseases,, SIAM Review, 42 (2000), 599. doi: 10.1137/S0036144500371907. Google Scholar [28] N. Hodas and K. Lerman, How limited visibility and divided attention constrain social contagion,, in submitted to Social Computing, (2012). Google Scholar [29] N. O. Hodas and K. Lerman, The simple rules of social contagion,, Scientific Reports, 4 (2014). doi: 10.1038/srep04343. Google Scholar [30] T. Hogg and K. Lerman, Stochastic models of user-contributory web sites,, in Proc. 3rd Int. Conf. on Weblogs and Social Media (ICWSM), (2009). Google Scholar [31] T. Hogg and K. Lerman, Social Dynamics of Digg,, EPJ Data Science, 5 (2012). Google Scholar [32] J. L. Iribarren and E. Moro, Impact of human activity patterns on the dynamics of information diffusion,, Physical Review Letters, 103 (2009). doi: 10.1103/PhysRevLett.103.038702. Google Scholar [33] G. Jeh and J. Widom, Scaling personalized web search,, in Proc. the 12th Int. Conf. on World Wide Web, (2003), 271. doi: 10.1145/775189.775191. Google Scholar [34] E. Katz and P. Lazarsfeld, Personal Influence: The Part Played by People in the Flow of Mass Communications,, Transaction Publishers, (2005). Google Scholar [35] L. Katz, A new status index derived from sociometric analysis,, Psychometrika, 18 (1953), 39. doi: 10.1007/BF02289026. Google Scholar [36] D. Kempe, J. Kleinberg and E. Tardos, Maximizing the spread of influence through a social network,, KDD '03 Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining, (2003), 137. doi: 10.1145/956755.956769. Google Scholar [37] C. Kiss and M. Bichler, Identiffication of influencers-measuring influence in customer networks,, Decision Support Systems, 46 (2008), 233. Google Scholar [38] S. Kotz and N. Balakrishnan, Advances in urn models during the past two decades,, in Advances in combinatorial methods and applications to probability and statistics, (1997), 203. Google Scholar [39] , R. Lambiotte, J. C. Delvenne and M. Barahona,, Laplacian dynamics and multiscale modular structure in networks., (). Google Scholar [40] R. Lambiotte, R. Sinatra, J. C. Delvenne, T. S. Evans, M. Barahona and V. Latora, Flow graphs: Interweaving dynamics and structure,, Physical Review E, 84 (2011). doi: 10.1103/PhysRevE.84.017102. Google Scholar [41] C. Lee, H. Kwak, H. Park and S. Moon, Finding Influentials from Temporal Order of Information Adoption in Twitter,, Proc. 19th World-Wide Web (WWW) Conference (Poster), (2010). Google Scholar [42] K. Lerman and R. Ghosh, Information Contagion: An Empirical Study of the Spread of News on Digg and Twitter Social Networks,, Proc. 4th Int. Conf. on Weblogs and Social Media (ICWSM), (2010). Google Scholar [43] L. Page, S. Brin, R. Motwani and T. Winograd, The PageRank Citation Ranking: Bringing Order to the Web,, Technical report, (1998). Google Scholar [44] R. Pastor-Satorras and A. Vespignani, Epidemic spreading in scale-free networks,, Physical Review Letters, 86 (2001), 3200. doi: 10.1103/PhysRevLett.86.3200. Google Scholar [45] B. A. Prakash, D. Chakrabartiy, M. Faloutsos, N. Valler and C. Faloutsos, Threshold conditions for arbitrary cascade models on arbitrary networks,, in Proc. the Int. Conf. on Data Mining, (2011), 537. doi: 10.1109/ICDM.2011.145. Google Scholar [46] E. M. Rogers, Diffusion of Innovations, 5th Edition,, Free Press, (2003). Google Scholar [47] D. M. Romero, W. Galuba, S. Asur and B. A. Huberman, Influence and passivity in social media,, in Proc. the 20th international Conference on World wide web, (2011), 113. doi: 10.1145/1963192.1963250. Google Scholar [48] H. Tong, C. Faloutsos and J. Pan, Fast random walk with restart and its applications,, in ICDM '06: Proc. the Sixth Int. Conf. on Data Mining, (2006), 613. doi: 10.1109/ICDM.2006.70. Google Scholar [49] M. Trusov, A. V. Bodapati and R. E. Bucklin, Determining influential users in internet social networks,, Journal of Marketing Research, 47 (2010), 643. doi: 10.1509/jmkr.47.4.643. Google Scholar [50] G. Ver Steeg, R. Ghosh and K. Lerman, What stops social epidemics?,, in Proc. 5th International AAAI Conference on Weblogs and Social Media (ICWSM), (2011). Google Scholar [51] Y. Wang, D. Chakrabarti, C. Wang and C. Faloutsos, Epidemic Spreading in Real Networks: An Eigenvalue Viewpoint,, Reliable Distributed Systems, 0 (2003). Google Scholar [52] D. J. Watts and P. S. Dodds, Influentials, networks, and public opinion formation,, Journal of Consumer Research, 34 (2007), 441. doi: 10.1086/518527. Google Scholar
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