**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

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 *I _{6}*.
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

Now, squaring *M*
gives

_{}

One can see that
individual *I _{8}* has 2-stage influence on half of the group,
although he has only one direct influence on

** **