Abstract
In the study of a given complex system, questions of interest can often be re-phrased in a useful manner as questions regarding some aspect of the structure or characteristics of a corresponding network graph. For example, various types of basic social dynamics can be represented by triplets of vertices with a particular pattern of ties among them (i.e., triads); questions involving the movement of information or commodities usually can be posed in terms of paths on the network graph and flows along those paths; certain notions of the ‘importance’ of individual system elements may be captured by measures of how ‘central’ the corresponding vertex is in the network; and the search for ‘communities’ and analogous types of unspecified ‘groups’ within a system frequently may be addressed as a graph partitioning problem.
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Notes
- 1.
Note that igraph and sna use different formats for storing network data objects. Here we handle the conversion by first producing an adjacency matrix for the karate club object karate. The two packages do, however, share several similar naming conventions. For example, to produce the analogous output of the sna functions degree, closeness, betweenness, and evcent, respectively, with the argument g, use the igraph functions degree, closeness, betweenness, and eigen_centrality with the argument karate (and, of course, appropriate choice of auxiliary arguments).
- 2.
Note that we first remove self-loops (of which the original AIDS blog network has three), since the notion of mutuality is well-defined only for dyads.
- 3.
Note that ‘clustering’ as used here, which is standard terminology in the broader data analysis community, differs from ‘clustering’ as used in Sect. 4.3.2, which arose in the social network community, in reference to the coefficient \(\mathsf{{cl}}_T\) used to summarize the relative density of triangles among connected triples.
- 4.
Note that for undirected graphs this matrix will be symmetric, and hence \(f_{k+} = f_{+k}\); for directed graphs, however, \(\mathbf {f}\) can be asymmetric.
- 5.
Note that these quantities are defined in complete analogy to the same quantities that underlie our definition of modularity.
References
P. Bonacich, “Factoring and weighting approaches to status scores and clique identification,” Journal of Mathematical Sociology, vol. 2, no. 1, pp. 113–120, 1972.
U. Brandes and T. Erlebach, Eds., Network Analysis: Methodological Foundations, ser. Lecture Notes in Computer Science. New York: Springer-Verlag, 2005, vol. 3418.
F. Chung, Spectral Graph Theory. American Mathematical Society, 1997.
A. Clauset, M. Newman, and C. Moore, “Finding community structure in very large networks,” Physical Review E, vol. 70, no. 6, p. 66111, 2004.
J. Davis and S. Leinhardt, “The structure of positive interpersonal relations in small groups,” in Sociological Theories in Progress, J. Berger, Ed. Boston: Houghton-Mifflin, 1972, vol. 2.
M. Fiedler, “Algebraic connectivity of graphs,” Czechoslovak Mathematical Journal, vol. 23, no. 98, pp. 298–305, 1973.
M. Fiedler, “Finding community structure in networks using the eigenvectors of matrices,” Physical Review E, vol. 74, no. 3, p. 36104, 2006.
M. Fiedler, “Mixing patterns in networks,” Physical Review E, vol. 67, no. 2, p. 26126, 2003.
A. Fornito, A. Zalesky, and E. Bullmore, Fundamentals of Brain Network Analysis. Academic Press, 2016.
L. Freeman, “A set of measures of centrality based on betweenness,” Sociometry, vol. 40, no. 1, pp. 35–41, 1977.
C. Godsil and G. Royle, Algebraic Graph Theory. New York: Springer, 2001.
P. Holland and S. Leinhardt, “A method for detecting structure in sociometric data,” American Journal of Sociology, vol. 70, pp. 492 – 513, 1970.
A. Jain, M. Murty, and P. Flynn, “Data clustering: a review,” ACM Computing Surveys, vol. 31, no. 3, pp. 264–323, 1999.
L. Katz, “A new status index derived from sociometric analysis,” Psychometrika, vol. 18, no. 1, pp. 39–43, 1953.
J. Kleinberg, “Authoritative sources in a hyperlinked environment,” Journal of the ACM, vol. 46, no. 5, pp. 604–632, 1999.
E. Kolaczyk, Statistical Analysis of Network Data: Methods and Models. Springer Verlag, 2009.
A. Lancichinetti and S. Fortunato, “Community detection algorithms: A comparative analysis,” Physical review E, vol. 80, no. 5, p. 056117, 2009.
B. Mohar, “The Laplacian spectrum of graphs,” in Graph Theory, Combinatorics, and Applications, Y. Alavi, G. Chartrand, O. Oellermann, and A. Schwenk, Eds. New York: Wiley, 1991, vol. 2, pp. 871–898.
M. Newman and M. Girvan, “Finding and evaluating community structure in networks,” Physical Review E, vol. 69, no. 2, p. 26113, 2004.
M. Newman, Networks: an Introduction. Oxford University Press, Inc., 2010.
G. Sabidussi, “The centrality index of a graph,” Psychometrika, vol. 31, pp. 581–683, 1966.
J. Scott, Social Network Analysis: A Handbook, Second Edition. Newbury Park, CA: Sage Publications, 2004.
S. Wasserman and K. Faust, Social Network Analysis: Methods and Applications. New York: Cambridge University Press, 1994.
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Kolaczyk, E.D., Csárdi, G. (2020). Descriptive Analysis of Network Graph Characteristics. In: Statistical Analysis of Network Data with R. Use R!. Springer, Cham. https://doi.org/10.1007/978-3-030-44129-6_4
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