Triadic closure
Encyclopedia
Triadic closure is a concept in social network
theory, first suggested by German
sociologist
Georg Simmel
in the early 1900s. Triadic closure is the property among 3 nodes A, B, and C, such that if a strong tie exists between A-B and A-C, there is a weak or strong tie between B-C. This property is too extreme to hold true across very large, complex networks, but it is a useful simplification of reality that can be used to understand and predict networks.
in his 1973 article The Strength of Weak Ties. There he synthesized the theory of cognitive balance first introduced by 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.
and transitivity for that graph.
, 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
of vertex i.
Then we can define a triangle among the triple of vertices
, 
, and 
to be a set with the following three edges: {(i,j), (j,k), (i,k)}. Then we can 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 and the number of triples of vertex i is 
, assuming 
. Thus we can express 
.
Now, for a vertex
with 
, the clustering coefficient
of vertex 
is the fraction of triples for vertex 
that are closed, and can be measured as 
. Thus, the clustering coefficient
of graph 
is given by 
, where 
is the number of nodes with degree at least 2.
.
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, 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.
Proof by contradiction:
Let node B be a local bridge between nodes A and C such that there is no weak tie between the nodes involved. Therefore B has a strong tie to both A and C. By the definition of Strong Triadic Closure, a weak tie would develop between nodes A and C. However, this contradicts the fact that B is a local gatekeeper. Thus at least one of the nodes involved in a local bridge needs to be a weak tie to prevent triadic closure from occurring.
Social network
A social network is a social structure made up of individuals called "nodes", which are tied by one or more specific types of interdependency, such as friendship, kinship, common interest, financial exchange, dislike, sexual relationships, or relationships of beliefs, knowledge or prestige.Social...
theory, first suggested by German
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
sociologist
Sociology
Sociology is the study of society. It is a social science—a term with which it is sometimes synonymous—which uses various methods of empirical investigation and critical analysis to develop a body of knowledge about human social activity...
Georg Simmel
Georg Simmel
Georg Simmel was a major German sociologist, philosopher, and critic.Simmel was one of the first generation of German sociologists: his neo-Kantian approach laid the foundations for sociological antipositivism, asking 'What is society?' in a direct allusion to Kant's question 'What is nature?',...
in the early 1900s. Triadic closure is the property among 3 nodes A, B, and C, such that if a strong tie exists between A-B and A-C, there is a weak or strong tie between B-C. This property is too extreme to hold true across very large, complex networks, but it is a useful simplification of reality that can be used to understand and predict networks.
History
Triadic closure was made popular by Mark GranovetterMark Granovetter
Professor Mark Granovetter is an American sociologist at Stanford University who has created theories in modern sociology since the 1970s. He is best known for his work in social network theory and in economic sociology, particularly his theory on the spread of information in social networks known...
in his 1973 article The Strength of Weak Ties. There he synthesized the theory of cognitive balance first introduced by 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 coefficientClustering coefficient
In graph theory, a clustering coefficient is a measure of 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 characterised by a relatively high density of ties...
and transitivity for that graph.
Clustering Coefficient
One measure for the presence of triadic closure is clustering coefficientClustering coefficient
In graph theory, a clustering coefficient is a measure of 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 characterised by a relatively high density of ties...
, 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 degreeDegree (graph theory)
In graph theory, the degree of a vertex of a graph is the number of edges incident to the vertex, with loops counted twice. The degree of a vertex v is denoted \deg. The maximum degree of a graph G, denoted by Δ, and the minimum degree of a graph, denoted by δ, are the maximum and minimum degree...
of vertex i.
Then we can define a triangle among the triple of vertices
, , and to be a set with the following three edges: {(i,j), (j,k), (i,k)}. Then we can 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 and the number of triples of vertex i is , assuming . Thus we can express .Now, for a vertex
with , the clustering coefficientClustering coefficient
In graph theory, a clustering coefficient is a measure of 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 characterised by a relatively high density of ties...
of vertex is the fraction of triples for vertex that are closed, and can be measured as . Thus, the clustering coefficientClustering coefficient
In graph theory, a clustering coefficient is a measure of 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 characterised by a relatively high density of ties...
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, 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. A local bridge occurs, on the other hand, when a node is acting as a gatekeeper between two neighboring nodes who are not otherwise connected. In a network that follows the Strong Triadic Closure Property, one of the ties between nodes involved in a local bridge needs to be a weak tie.Proof by contradiction:
Let node B be a local bridge between nodes A and C such that there is no weak tie between the nodes involved. Therefore B has a strong tie to both A and C. By the definition of Strong Triadic Closure, a weak tie would develop between nodes A and C. However, this contradicts the fact that B is a local gatekeeper. Thus at least one of the nodes involved in a local bridge needs to be a weak tie to prevent triadic closure from occurring.