Eigenvector Centrality: Illustrations Supporting the Utility of Extracting More Than One Eigenvector to Obtain Additional Insights into Networks and Interdependent Structures

Among the many centrality indices used to detect structures of actors’ positions in networks is the use of the first eigenvector of an adjacency matrix that captures the connections among the actors. This research considers the seeming pervasive current practice of using only the first eigenvector. It is shows that, as in other statistical applications of eigenvectors, subsequent vectors can also contain illuminating information. Several small examples, and Freeman’s EIES network, are used to illustrate that while the first eigenvector is certainly informative, the second (and subsequent) eigenvector(s) can also be equally tractable and informative.

[1]  Phillip Bonacich,et al.  Eigenvector-like measures of centrality for asymmetric relations , 2001, Soc. Networks.

[2]  Bryan F. J. Manly,et al.  Multivariate Statistical Methods : A Primer , 1986 .

[3]  Benjamin Wright,et al.  Direct Factor Analysis in Sociometry , 1961 .

[4]  Stanley Wasserman,et al.  Social Network Analysis: Methods and Applications , 1994 .

[5]  Noah E. Friedkin,et al.  Social influence and opinions , 1990 .

[6]  Sergey Brin,et al.  The Anatomy of a Large-Scale Hypertextual Web Search Engine , 1998, Comput. Networks.

[7]  Stephen P. Borgatti,et al.  Centrality and network flow , 2005, Soc. Networks.

[8]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[9]  Martin G. Everett,et al.  A Graph-theoretic perspective on centrality , 2006, Soc. Networks.

[10]  P. Bonacich.  Factoring and weighting approaches to status scores and clique identification , 1972 .

[11]  John Scott.  What is social network analysis , 2010 .

[12]  James Moody,et al.  Structural effects of network sampling coverage I: Nodes missing at random , 2013, Soc. Networks.

[13]  Gita Reese Sukthankar,et al.  Link Prediction in Heterogeneous Collaboration Networks , 2014, Social Network Analysis.

[14]  Peter D. Hoff,et al.  Latent Space Approaches to Social Network Analysis , 2002 .

[15]  Richard Rothenberg,et al.  Choosing a centrality measure: Epidemiologic correlates in the Colorado Springs study of social networks☆ , 1995 .

[16]  Noah E. Friedkin,et al.  Two steps to obfuscation , 2014, Soc. Networks.

[17]  Phillip Bonacich,et al.  Some unique properties of eigenvector centrality , 2007, Soc. Networks.

[18]  P. Bonacich.  Power and Centrality: A Family of Measures , 1987, American Journal of Sociology.

[19]  D. F. Morrison,et al.  Multivariate Statistical Methods , 1968 .

[20]  A. Comrey.  The Minimum Residual Method of Factor Analysis , 1962 .

[21]  Martin G. Everett,et al.  Models of core/periphery structures , 2000, Soc. Networks.

[22]  B. Tabachnick,et al.  Using Multivariate Statistics , 1983 .

[23]  Kathleen M. Carley,et al.  On the robustness of centrality measures under conditions of imperfect data , 2006, Soc. Networks.

[24]  Leo Katz,et al.  A new status index derived from sociometric analysis , 1953 .

[25]  Thomas W. Valente,et al.  The stability of centrality measures when networks are sampled , 2003, Soc. Networks.

[26]  Noah E. Friedkin,et al.  Theoretical Foundations for Centrality Measures , 1991, American Journal of Sociology.