![Triadic closure](https://upload.wikimedia.org/wikipedia/commons/f/f7/Triadic_closure.svg)
Triadic closure is a concept in
social network
A social network is a social structure made up of a set of social actors (such as individuals or organizations), sets of dyadic ties, and other social interactions between actors. The social network perspective provides a set of methods for an ...
theory, first suggested by
German
German(s) may refer to:
* Germany (of or related to)
**Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ger ...
sociologist Georg Simmel
Georg Simmel (; ; 1 March 1858 – 26 September 1918) was a German sociologist, philosopher, and critic.
Simmel was influential in the field of sociology. Simmel was one of the first generation of German sociologists: his neo-Kantian approach l ...
in his 1908 book ''Soziologie''
'Sociology: Investigations on the Forms of Sociation'' 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.
[Easley, D, & Kleinberg, J. (2010). Networks, crowds, and markets: reasoning about a highly connected world. Cornell, NY: Cambridge Univ Pr.]
History
Triadic closure was made popular by
Mark Granovetter
Mark Sanford Granovetter (; born October 20, 1943) is an American sociologist and professor at Stanford University. He is best known for his work in social network theory and in economic sociology, particularly his theory on the spread of inform ...
in his 1973 article ''The Strength of Weak Ties''.
[Granovetter, M. (1973).]
The Strength of Weak Ties
", American Journal of Sociology, Vol. 78, Issue 6, May 1360-80. There he synthesized the theory of
cognitive balance In the psychology of motivation, balance theory is a theory of attitude change, proposed by Fritz Heider. It conceptualizes the cognitive consistency motive as a drive toward psychological balance. The consistency motive is the urge to maintain on ...
first introduced by
Fritz Heider
Fritz Heider (19 February 1896 – 2 January 1988) was an Austrian psychologist whose work was related to the Gestalt school. In 1958 he published ''The Psychology of Interpersonal Relations'', which expanded upon his creations of balance theory a ...
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
In graph theory, a clustering coefficient is a measure of the degree to which nodes in a graph tend to cluster together. Evidence suggests that in most real-world networks, and in particular social networks, nodes tend to create tightly knit groups ...
and transitivity for that graph.
Clustering coefficient
One measure for the presence of triadic closure is
clustering coefficient
In graph theory, a clustering coefficient is a measure of the degree to which nodes in a graph tend to cluster together. Evidence suggests that in most real-world networks, and in particular social networks, nodes tend to create tightly knit groups ...
, as follows:
Let
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
and
denote the number of vertices and edges in G, respectively, and let
be the
degree
Degree may refer to:
As a unit of measurement
* Degree (angle), a unit of angle measurement
** Degree of geographical latitude
** Degree of geographical longitude
* Degree symbol (°), a notation used in science, engineering, and mathematics
...
of vertex i.
We can define a triangle among the triple of vertices
,
, and
to be a set with the following three edges: .
We can also define the number of triangles that vertex
is involved in as
and, as each triangle is counted three times, we can express the number of triangles in G as
.
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
is
, assuming
. We can express
.
Now, for a vertex
with
, the
clustering coefficient
In graph theory, a clustering coefficient is a measure of the degree to which nodes in a graph tend to cluster together. Evidence suggests that in most real-world networks, and in particular social networks, nodes tend to create tightly knit groups ...
of vertex
is the fraction of triples for vertex
that are closed, and can be measured as
. Thus, the
clustering coefficient
In graph theory, a clustering coefficient is a measure of the degree to which nodes in a graph tend to cluster together. Evidence suggests that in most real-world networks, and in particular social networks, nodes tend to create tightly knit groups ...
of graph
is given by
, where
is the number of nodes with degree at least 2.
Transitivity
Another measure for the presence of triadic closure is transitivity, defined as
.
Causes and effects
In a trust network, triadic closure is likely to develop due to the transitive property. If a node A trusts node B, and node B trusts node C, node A will have the basis to trust node C. In a social network, strong triadic closure occurs because there is increased opportunity for nodes A and C with common neighbor B to meet and therefore create at least weak ties. Node B also has the incentive to bring A and C together to decrease the latent stress in two separate relationships.
[
Networks that stay true to this principle become highly interconnected and have very high clustering coefficients. However, networks that do not follow this principle turn out to be poorly connected and may suffer from instability once negative relations are included.
Triadic closure is a good model for how networks will evolve over time. While simple graph theory tends to analyze networks at one point in time, applying the triadic closure principle can predict the development of ties within a network and show the progression of connectivity.][
In ]social networks
A social network is a social structure made up of a set of social actors (such as individuals or organizations), sets of dyadic ties, and other social interactions between actors. The social network perspective provides a set of methods for an ...
, triadic closure facilitates cooperative behavior, but when new connections are made via referrals from existing connections the average global fraction of cooperators is less than when individuals choose new connections randomly from the population at large. Two possible effects for this are by the structural and informational construction. The structural construction arises from the propensity toward high clusterability. The informational construction comes from the assumption that an individual knows something about a friend's friend, as opposed to a random stranger.
Strong Triadic Closure Property and local bridges
Strong Triadic Closure Property is that if a node has strong ties to two neighbors, then these neighbors must have at least a weak tie between them.
References
{{DEFAULTSORT:Triadic Closure
Social systems
Sociological terminology
Social network analysis