Triadic closure is a concept in social network theory, first suggested by German sociologist Georg Simmel in his 1908 book Soziologie (Sociology: Inquiries into the Construction of Social Forms). Triadic closure is the property among three nodes A, B, and C (representing people, for instance), that if the connections A-B and A-C exist, there is a tendency for the new connection B-C to be formed. Triadic closure can be used to understand and predict the growth of networks, although it is only one of many mechanisms by which new connections are formed in complex networks.
History Triadic closure was made popular by Mark Granovetter in his 1973 article The Strength of Weak Ties. There he synthesized the theory of cognitive balance first introduced by Fritz Heider in 1946 with a Simmelian understanding of social networks. In general terms, cognitive balance refers to the propensity of two individuals to want to feel the same way about an object. If the triad of three individuals is not closed, then the person connected to both of the individuals will want to close this triad in order to achieve closure in the relationship network.
Measurements
The two most common measures of triadic closure for a graph are (in no particular order) the clustering coefficient and transitivity for that graph.
Clustering coefficient One measure for the presence of triadic closure is clustering coefficient, as follows: Let G = ( V , E ) {\displaystyle G=(V,E)} be an undirected simple graph (i.e., a graph having no self-loops or multiple edges) with V the set of vertices and E the set of edges. Also, let N = | V | {\displaystyle N=|V|} and M = | E | {\displaystyle M=|E|} denote the number of vertices and edges in G, respectively, and let d i {\displaystyle d_{i}} be the degree of vertex i. We can define a triangle among the triple of vertices i {\displaystyle i} , j {\displaystyle j} , and k {\displaystyle k} to be a set with the following three edges: {(i,j), (j,k), (i,k)}. We can also define the number of triangles that vertex i {\displaystyle i} is involved in as δ ( i ) {\displaystyle \delta (i)} and, as each triangle is counted three times, we can express the number of triangles in G as δ ( G ) = 1 3 ∑ i ∈ V δ ( i ) {\displaystyle \delta (G)={\frac {1}{3}}\sum _{i\in V}\ \delta (i)} . Assuming that triadic closure holds, only two strong edges are required for a triple to form. Thus, the number of theoretical triples that should be present under the triadic closure hypothesis for a vertex i {\displaystyle i} is τ ( i ) = ( d i 2 ) {\displaystyle \tau (i)={\binom {d_{i}}{2}}} , assuming d i ≥ 2 {\displaystyle d_{i}\geq 2} . We can express τ ( G ) = 1 3 ∑ i ∈ V τ ( i ) {\displaystyle \tau (G)={\frac {1}{3}}\sum _{i\in V}\ \tau (i)} . Now, for a vertex i {\displaystyle i} with d i ≥ 2 {\displaystyle d_{i}\geq 2} , the clustering coefficient c ( i ) {\displaystyle c(i)} of vertex i {\displaystyle i} is the fraction of triples for vertex i {\displaystyle i} that are closed, and can be measured as δ ( i ) τ ( i ) {\displaystyle {\frac {\delta (i)}{\tau (i)}}} . Thus, the clustering coefficient C ( G ) {\displaystyle C(G)} of graph G {\displaystyle G} is given by C ( G ) = 1 N 2 ∑ i ∈ V , d i ≥ 2 c ( i ) {\displaystyle C(G)={\frac {1}{N_{2}}}\sum _{i\in V,d_{i}\geq 2}c(i)} , where N 2 {\displaystyle N_{2}} is the number of nodes with degree at least 2.
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