comp_clean_W2

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Computer Science

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Feb 20, 2024

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Social and Information Networks University of Toronto CSC303 Winter/Spring 2023 Week 2: January 16-20 1/56
This week’s agenda The Strength of Weak Ties Triadic closure Defnition Clustering coefcient Driving forces Granovetter’s Thesis Strong & Weak Ties Bridges Strong Triadic Closure and it’s implications Social Capital Defnition Bonding vs. Bridging capital Relationship with graph structure Determining Strong Edges Sintos & Tsaparas algorithm Rozenshtein algorithm 2/56
Chapter 3: Strong and Weak Ties There are two themes that run throughout this chapter. 1 Strong vs. weak ties and “the strength of weak ties” is the specifc defning theme of the chapter. The chapter also starts a discussion of how networks evolve. 2 The larger theme is in some sense “the scientifc method” . Formalize concepts, construct models of behaviour and relationships, and test hypotheses. Models are not meant to be the same as reality but to abstract the important aspects of a system so that it can be studied and analysed. See the discussion of the strong triadic closure property in section 3.2 of text (pages 53 and 56 in my online copy). Informally strong ties : stronger links, corresponding to friends weak ties : weaker links, corresponding to acquaintances 3/56
Triadic closure (undirected graphs) B A C G F E D (a) Before B - C edge forms. B A C G F E D (b) After B - C edge forms. Figure: The formation of the edge between B and C illustrates the efects of triadic closure, since they have a common neighbour A . [E&K Figure 3.1] Triadic closure : mutual “friends” of say A are more likely (than “normally”) to become friends over time. How do we measure the extent to which triadic closure is occurring? How can we know why a new friendship tie is formed? (Friendship ties can range from “just knowing someone” to “a true friendship” .) 4/56
Measuring the extent of triadic closure The clustering coefcient of a node A is a way to measure (over time) the extent of triadic closure (perhaps without understanding why it is occurring). Let E be the set of an undirected edges of a network graph. (Forgive the abuse of notation where in the previous and next slide E is a node name.) For a node A , the clustering coefcient is the following ratio: vextendsingle vextendsingle braceleftbig ( B , C ) E : ( B , A ) E and ( C , A ) E bracerightbigvextendsingle vextendsingle vextendsingle vextendsingle braceleftbig { B , C } : ( B , A ) E and ( C , A ) E bracerightbigvextendsingle vextendsingle The numerator is the number of all edges ( B , C ) in the network such that B and C are adjacent to (i.e. mutual friends of) A . The denominator is the total number of all unordered pairs { B , C } such that B and C are adjacent to A . 5/56
Example of clustering coefcient B A C G F E D (a) Before new edges form. The clustering coefcient of node A in Fig. (a) is 1 / 6 (since there is only the single edge ( C , D ) among the six pairs of friends: { B , C } , { B , D } , { B , E } , { C , D } , { C , E } , and { D , E } ) 6/56
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