Network Cohesion

Measures for Network Cohesion

The simplest way to measure network cohesion is to examine how many ties that a network contains. In this sense, network cohesion can be simply expressed as the sum of all observed edges from a network. This index has a disadvantage in that the sum of edges is dependent on the size of a network. A poorly connected network with more actors may have the same total number of edges as a small cohesive network. For this reason, in order to compare across networks of different size, a standardized index is needed. The standardized index can be obtained by dividing the sum of edges by the maximum possible edges of a network. In social network analysis, this standardized index is also known as network density, denoted as D. Let N be the size of a network, and E denote the number of observed edges. Then, the network density for an undirected network can be expressed as

$$D=\frac{2E}{N \left(N-1\right)},$$

and for a directed network:

$$D=\frac{E}{N \left(N-1\right)}.$$

Network cohesion can also be measured by the average degree of the network (d). In social network analysis, degree can be simply interpreted as how active an actor is in a network and can be measured by the number of ties the actor has to other actors. Thus, the average degree is simply the average number of ties each actor has. For an undirected network,

$$\overline{d}=\frac{2E}{N }=D\times \left(N-1\right),$$

and for a directed network,

$$\overline{d}=\frac{E}{N }=D\times \left(N-1\right).$$

Actually, both network density and average degree are measures that are directly built upon the dyadic cohesion. Measures for network cohesion can also be built upon the subgroup-level cohesion, namely, structural cohesion. These measures not only consider the number of ties but also the structure and clustering among ties. Sometimes the subgroup is also termed as a component, which refers to the substructure of networks connected internally but disconnected between each other. One of the measures under this umbrella is component ratio (CR). Let C denote the number of components and N denote the number of actors in a network, then:

$$CR= \frac{C-1}{N-1}.$$

Another component-based measure for network cohesion is fragmentation (F). Fragmentation is defined as the proportion of pairs of actors that are not located in the same component. Let rij be any pair of actors i and j in a network. If i and j are observed in the same component, rij = 1; otherwise, rij = 0. Let N be the number of actors, and the fragmentation can then be calculated

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