Tree

I have been talking a lot about network structure in my previous blogs. In this blog I shall provide a few preliminaries about network structure and how they are measured. A virtual community network structure is viewed in terms of nodes and ties.

Nodes are individual actors within the network, and ties represent the flow of relationships between the actors. The relationships defined by linkages among units/nodes are a fundamental component of a virtual community (Wasserman & Faust, 1994). Social Network Analysis (SNA) techniques are used to visualize the patterns of interactions among participants on the virtual community. Metrics are often used to determine the roles of nodes in a network. Of these the most prominent are degree, closeness and betweenness. Degree is the sum of the links attached to a node. Closeness is the reciprocal of the sum of all the geodesic (shortest) distances from a given node to all others. A higher betweenness value for a node means that it is on higher number of shortest-paths between nodes, which is an indication of the node’s importance (Wasserman & Faust, 1994).

Virtual communities are characterized by both scale free and small world characteristics of a network (Klemm & Eguıluz, 2002). The scale free and small network characteristics are as follows:

Trees, Scale Free & Small World Networks

(a)     Tree Network

Networks that grow by attaching new nodes to existing nodes (by adding one edge only) form into trees. They have no cycles (Fig. a). If in this process the new nodes attach preferentially to existing nodes with a large number of edges, then the result is a scale-free network (Albert & Barabási, 2000). Scale-free networks are distinguished by three characteristics. First, they are highly clustered; if two vertices share a common neighbor, it is likely the two are themselves adjacent. Second, the average shortest path between two vertices is logarithmically small. And finally, the node degrees are distributed according to a power law.

(b)     Scale Free Network

 

In Scale Free networks the distribution of different network parameters act in an exponential fashion (Fig. b). The most interesting and most measured exponentially distributed parameter is the distribution of connections from each node outwards (Out Degree). This uneven distribution means that in these networks some of the members are connected to a lesser degree and some of the members are connected to a greater degree, which is how they hold a senior position in the network (Goh, et al., 2002). Networks of this type are relatively resilient, but are not at all immune to attack. In other words, a random removal of network members (a crash) will not hurt its stability, but a directed removal of keypoints will cause the network to quickly collapse. On Scale Free networks, the distribution of density or congestion is constant and is not dependent on the exponential coefficient of the distribution of the number of connections (Jeong, 2003).

(c)     Small World Network

 

A Small World network is a network in which most nodes are not neighbors of each other but most nodes can be reached by other nodes in the networks by hopping over a few nodes. These networks (Watts & Strogatz, 2003) form when long distance connections are added at random to regular networks (Fig. c). They are characterized by low paths lengths between nodes and high clustering coefficient. The Clustering Coefficient is the extent to which the nodes in the graph tend to create a unified group with many internal connections but few connections leading out of the group. The Clustering Coefficient can be seen as a measurement of the nodes’ isolation. The Characteristic Path Length (CPL) is a measurement of the average distance needed to pass from node to node. A network can be considered a Small World network when its CPL is similar to the CPL of a random network of the same length, but its CC is much larger (at least by a single order of magnitude) when compared to a similar random network. In other words, in Small World networks we will expect to find a large unified group; in networks such as these "hiding" is impossible (Herman, 2003).

I hope this clears up a few things and gives you an idea of what I mean by structure and how to measure the structure of a social network. Please feel free to ask me any clarifications and also share some of your methods that you use.