Application to sociology

Introduction Sociologists interested in various kinds of communications in a group of individuals often use graphs to represent and analyze relations inside the group. For terminology and some results about graph theory that we will use here, check the application of linear algebra to graph theory. The idea is to associate a vertex to each individual in the group, and if individual A influences or dominates individual B, we draw a directed edge from A to B. Note that the obtained graph can have at most one directed edge between two distinct vertices.

Example Consider a group of eight individuals I₁,…, I₈. The following digraph represents the dominance relationship among the individuals of the group:

The adjacency matrix of this graph is:

The row with most 1’s in the above matrix corresponds to the most influential individual in the group; in our case it is I₆. In the above graph, walks of length 1 (one edge) correspond to direct influence in the group, whereas walks of greater length correspond to indirect influence. For instance I₃ directly influences I₅ and I₅ directly influences I₄, therefore I₃ indirectly influences I₄.

Now, squaring M gives

One can see that individual I₈ has 2-stage influence on half of the group, although he has only one direct influence on I₆.