A social network is a connected structure which consists of nodes and links. Nodes represent actors and links reflect a relationship between actors that exist in the network. In a social structure actors are humans and semantics of the relationship depend on the specific domain and scope to be covered in the model. Thus, a graph is the most attractive representation for a social network.
Graphs may be classified in various ways based on the types of links and the types of nodes. Based
23
on links, two main categories of graphs are directed and undirected graphs as shown in Figure 2.2.
In directed graphs each link has a source and destination, and each node is characterised by two types of degree: indegree of a node, say n, is computed based on the total number of links having node n as destination and outdegree of node n is determined based on the total number of links having node n as source. On the other hand, source and destination are not specified in an undirected graph. Accordingly, a node n in an undirected graph has a degree which is computed as the total number of links connected to it.
Figure 2.3: Examples of one-mode and two-mode networks
By considering nodes, one way to classify graphs is based on the number of sets of nodes which satisfy the property: whether no two nodes in the same set are connected, instead nodes in a given set are only connected to nodes in other sets. The most common graph type contains a homogeneous set of nodes which may be arbitrarily connected. Another common graph type is a bipartite graph which includes two sets of nodes and connections are allowed only between two nodes if they do not belong to the same set. The number of sets of nodes a graph may contain is the factor used to decide on the mode of the corresponding network; one-mode and two-mode networks are the most commonly used as shown in Figure 2.3.
Using a graph to represent a social network enriches the model because such representation facilitates using graph theory measures in the analysis and in processing the network structure.
24
For instance, a network may be represented as an adjacency matrix which is one of the most common methods to represent a graph. Once a network is represented as a matrix, all matrix algebra may be applied to study and analyse the content in link to the semantics underlying the corresponding network model.
Figure 2.4: Example of a two-mode network and its folding into a one mode network A two-mode network may be transformed into two one-mode networks by applying a process called folding [3], which depends on two operations, namely matrix transpose and multiplication.
This process is illustrated in Figure 2.4. Explicitly speaking, given the adjacency matrix of a network that connects two sets of entities, say users and web pages, the folding process multiplies the adjacency matrix of (users, pages) with its transpose represented as (pages, users) to produce a new adjacency matrix represented as (users, users). The latter adjacency matrix reflects the strength of the links between users based on the number of pages they have in common. On the other hand, multiplying the transpose (pages, users) by the original matrix (users, pages) will produce a new adjacency matrix of a one-mode network which reflects the strength of connectivity between each pair of pages based on the frequency of users who accessed the two pages.
25
To analyse a social network, a wide variety of measures may be used, e.g., degree, closeness, betweenness, density, diameter, cliques, etc. Degree reflects the number of links connected to a node and may be classified into indegree and outdegree for directed networks, as detailed above.
Further, normalised degree is computed by dividing the actual degree by (N−1), where N ≥ 2 is the number of nodes or actors in the network. For instance, the degree of node p in the one-mode network shown in Figure 2.3 is 4 and the normalised degree is 45 .
Closeness centrality of a node n represents the minimum total number of links that must be travelled in order to visit each node in the network. In a complete graph of N nodes, the closeness of each node is (N −1) or the normalised closeness is 1. On the other hand, the closeness of node p in the one-mode network shown in Figure 2.3 is 6 (because two links are needed to visit node q which is the only node without direct link to p).
Betweenness centrality of a node or link depends on the number of shortest paths passing through the node or link. A shortest path between two nodes u and v is the minimum number of consecutive links that must be traversed between v and u such that no intermediate link or node is visited more than once. For instance, the shortest path between the two nodes m and s has the length 2 because at least two links must be traversed between the two nodes. The two node p and q in Figure 2.3 have the same importance because every pair of the four other nodes have two alternative shortest paths, one passes through p and the other passes through q. Finally, the diameter of this graph is 2 because the longest shortest path in the graph consists of two links.
In a directed graph, it is possible to distinguish nodes into hubs and authorities. A hub is a node with high outdegree and an authority is a node with high indegree. These two measures are important when it comes to analyse the importance of specific nodes especially in graphs derived from the pages of web sites.
26