Structural holes is a concept from social network research, originally developed by Ronald Stuart Burt. A structural hole is understood as a gap between two individuals who have complementary sources to information. The study of structural holes spans the fields of sociology, economics, and computer science. Burt introduced this concept in an attempt to explain the origin of differences in social capital. Burt's theory suggests that individuals hold certain positional advantages/disadvantages from how they are embedded in neighborhoods or other social structures.
The concept Most social structures tend to be characterized by dense clusters of strong connections, also known as network closure. The theory relies on a fundamental idea that the homogeneity of information, new ideas, and behavior is generally higher within any group of people as compared to that in between two groups of people. An individual who acts as a mediator between two or more closely connected groups of people could gain important comparative advantages. In particular, the position of a bridge between distinct groups allows him or her to transfer or gatekeep valuable information from one group to another. In addition, the individual can combine all the ideas he or she receives from different sources and come up with the most innovative idea among all. At the same time, a broker also occupies a precarious position, as ties with disparate groups can be fragile and time-consuming to maintain.
If we compare two nodes, node A is more likely to get novel information than node B, even though they have the same number of links. This is so because nodes connected to B are also highly connected between each other. Therefore, any information that any of them could get from B, it could easily get from other nodes as well. Furthermore, the information, which B gets from different connections, is likely to be overlapping, so connections involving node B are said to be redundant. On contrary, the position of node A makes it serve as a bridge or a 'broker' between three different clusters. Thus, node A is likely to receive some non-redundant information from its contacts. The term 'structural holes' is used for the separation between non-redundant contacts. As a result of the hole between two contacts, they provide network benefits to the third party (to node A).
Measures
Bridge counts Bridge count is a simple and intuitive measure of structural holes in a network. Bridge is defined as a relation between two individuals if there is no indirect connection between them through mutual contacts.
Effective size
Burt's formula Burt introduced the measure of a network's redundancy. He aims to estimate to what extent contact j is redundant with other contacts of node i. Redundancy is understood as an investment of time and energy in a relationship with another node q, with whom node j is strongly connected.
Redundancy = p i q m j q {\displaystyle {\text{Redundancy}}=p_{iq}m_{jq}}
Where piq is proportion of i's energy invested in relationship with q, And mjq is calculated as j's interaction with q divided by j's strongest relationship with anyone. The redundancy in network is calculated by summing up this product across all nodes q. One minus this expression expresses the non-redundant portion of relationship. Effective size of i's network is defined as a sum of the j's nonredundant contacts.
Effective size of i's network = ∑ j [ 1 − ∑ q p i q m j q ] , q ≠ i , j , {\displaystyle {\text{Effective size of i's network}}=\sum _{j}\left[1-\sum _{q}p_{iq}m_{jq}\right],\quad q\neq i,j,}
The more each node is disconnected from other primary contacts, the higher the effective size would be. This indicator varies from 1 (network only provides a single link) to the total number of links N (each contact is non-redundant).
Borgatti's reformulation of Burt's formula Borgatti developed a simplified formula to calculate effective size for unweighted networks.
Redundancy = 2 t n {\displaystyle {\text{Redundancy}}={\frac {2t}{n}}}
Where t is the number of the total ties in the egocentric network (excluding those ties to the ego) and n is the number of total nodes in the egocentric network (excluding the ego). This formula can be modified to calculate the effective size of the ego's network.
Effective size of ego's network = n − 2 t n {\displaystyle {\text{Effective size of ego's network}}=n-{\frac {2t}{n}}}
Constraint Network constraint of a network is a sum of each connection's constraints cij:
c i j = ( p i j + ∑ q p i q p q j ) 2 , i ≠ q ≠ j {\displaystyle c_{ij}=(p_{ij}+\sum _{q}p_{iq}p_{qj})^{2},\quad i\neq q\neq j}
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