Lecture 7: Quantitative Network Fundamentals II Selected Social Science MetricsSelected Social Science Metrics Degree Distributions and Power Laws February 25, 2010 Professor C. Magee, 2010 page 1tSocial Network Analysisy Many structural metrics have been invented and used by Social Scientists studying social networks over the past 70+ years. The Journal Social Networks is the research front These are well-covered in Wasserman and Faust –Social Network Analysis (1994) The following slides cover a few selected examples in one area from that book. The purpose is to give some feel for the application of such metrics which attempt to measure structural properties of direct interest for il k l isocial network analysis We should also note that transitivity (clustering) and almost all other metrics discussed in this lecture were familiar to and used by social network scientists before the recent upsurge in activity over the past 10 years. Professor C. Magee, 2010 Page 2Transitivity or Clustering coefficient, C Measures quantitatively the degree to which nodes which each have relationships with a common node are likely to have a direct relationship nodesoftriplesconnectedofnumber networkintrianglesofnumberxC 3 1 = are likely to have a direct relationship. nodesoftriplesconnectedofnumber 1 ∑CC ;2 ∑= i iCn C ioncenteredtriplesofnumber inodetoconnectedtrianglesofnumberCi = Professor C. Magee, 2010 Page 3Example calculation of transitivity coefficientstransitivity coefficients This network has one triangle and eight connected triples, and therefore has a clustering coefficient of 3 x 1/8 = 3/8 The individual vertices have local C clustering coefficient, , of 3 x 1/8 = 3/8 The individual vertices have local clustering coefficients, of 1, 1, 1/6, 0 and 0, for a mean value, = 13/30. 1C2CProfessor C. Magee, 2010 Page 4 Source: M. E. J. Newman, The Structure and Function of Complex Networks, SIAM Review, Vol. 45, No. 2, pp . 167–256, 2003 Society for Industrial and Applied Matt t t t tTransitivity or Clustering coefficient IIcoefficient, II (Almost) always > than expected from random networks thus offering some support for earlier assertions that real networks have some non-random “structure” Thus, assessing clustering is a quick check whether you have a random graph where C = <k>/n. Indeed the size dependence of i i i b f l l ltransitivity can be useful to calculate © 2009 Chris Magee, Engineering Systems Division, Massachusetts Institute of TechnologyueStructuralTypology (lecture 1)Structural Typology (lecture 1) Totally regular • Real things • The ones we i d • No internal structure • Perfect gases Grids/crys tals Pure are interested in • New methods or adaptations of • Crowds of people • Classical i i h Trees Layered trees adaptations of existing methods needed economics with invisible hand • Stochastic methods d trees Star graphs Dt i i ti used • Less regular -“Hub and spokes” “S ll ld ” Deterministic methods used -“Small Worlds” -Communities -Clusters Motifs Professor C. Magee, 2010 Page 6 -Motifst t o e eo e enTransitivity or Clustering coefficient II -continuedcoefficient, II continued (Almost) always > than expected from random networks thus offering some support for earlier assertions that real networks have some non-random “structure” Thus, assessing clustering is a quick check whether you have a random graph where C = <k>/n. Indeed the size dependence f t iti it b f l l lof transitivity can be useful to calculate Higher order clusters (groups of n related nodes) also of interest but no clean way (yet) to separate lower order and highe d tendencies Mo Whitne sho d inhigher order tendencies. Moreover, Whitney showed in lecture 4 that methods for calculating higher order clustering in large networks is unknown territory. In directed graphs 2 effects (the proportion of nodes that In directed graphs, n=2 effects (the proportion of nodes that point at each other) can be of interest and is labeled reciprocity. This is an important social network attribute. Professor C. Magee, 2010 Page 7or Centrality Numerous metrics exist in the Social Networks Literature for assessing the “centrality” of a social network. C t lit t i tt t t h t i th l l f Centrality metrics attempt to characterize the level of “centralization” of control or action on this network One application is to assess how important a given actor ( d ) i i th t k ( ki f d di t li k(node) is in the network (ranking of nodes according to link information) Another application is to assess overall how much of the tl f th t k i tll d b th “ i t t” control of the network is controlled by the “more important” actors (group or network centrality) The relative importance of single channels/links and groups of links has also been of inte estlinks has also been of interest. We will look at a several of the social science defined metrics and explore the definitions by looking at “ideal toy graphs”: Team (family full) graphs Circle (or line) graphs and Star graphs (family or full) graphs, Circle (or line) graphs and Star graphs.Degree Centrality Actor (can be individual, group or organization ∑ xij depending on what is being studied). The actor in the example we will use is a “F il ” M l i h 1)(' − = n nC j iD “Family”. Most central is the node with the most links. )]()([1 max −∑ nCnCn i iDD Group (all actors in network) = 1 for a star graph )]2[( )]()([1 − = ∑ = n C i iDD D = 0 for a circle graph or “team” = 1/(n-1) for line graph Professor C. Magee, 2010 Page 9 /( ) g pPadgett’s Florentine Families: 15th Century Marriage Relations15 Century Marriage Relations n5:Castellani n :Peruzzi n12:Pucci n4:Bischeri n15:Strozzi n11:Peruzzi n3:Barbadori n :Medici n10:Pazzi n13:Ridolfi n14:Salvatin16:Tornabuoni n1:Acciaiuoli n7:Guadagni n9:Medici 10 n8:Lamberteschi n2:Ablizzi Professor C. Magee, 2010 Page 10 n6:GenoriFlorentine Families Centrality Metrics I: DegreeMetrics I: Degree)(iDnC′)(iCnC′)(iBnC′)(iInC′AcciaiuoliAblizziBarbadoriBischeriCtlli0.0710.2140.1430.2140 214CastellaniGenoriGuadagniLamberteschiMedici0.2140.0710.2860.0710.429PazziPeruzziPucciRidolfiSalvati0.0710.214---0.2140 143SalvatiStrozziTornabuoniCentralization0.1430.2860.2140.257Professor C.
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