Calculate Matrix Multiplication in two different method

In today's blog, we will talk about something about Social Network Analysis (SNA). 

Social Network Analysis is the study of the pattern of interaction between actors. In SNA, Social network is generally defined as a set of social actors, or nodes, members that are connected by one or more typrs of relations.

Then let's see a simple social network. To represent the relationship among factors in social network simply and clearly, we use sociogram which is a diagram to represent a social network. For example, take the relation X of Alice, Bob, Carol, and David into consideration. All these four people have their own blog. They would like to comment on others' blogs. Alice comments on Carol and David's blogs. Bob comments on Carol and David's blogs. Carol comments on Alice's blog and David comments on Alice and Bob's blogs. Therefore, we can get a sociogram below as the graphical representation (sociogram) of this simple social network.

We can also represent the ralation X of the simple social network by the sociomatrix X. Where {Alice, Bob, Carol, David}corresponds to {n1, n2, n3, n4}.

As most SNA ignore these "self-choices"elements, this table uses "-" to represent "self-choices". Symbol "0" means one individual does not comment on the others ans symbol "1" means one individual comments on the others.

After mastering how to represent the relationship of a simple social network in sociogram and sociomatrix, we will start to analyse the matrix multiplication using two different method (X multiply X is used as an example in this analysis). The sociogram and the sociomatrix is shown below.

• The first method

We can use math method to do the matrix multiplication in first way. X multiply X, which means X² can be calculated according to the equation x_ij^[2]=∑x_ikx_ki (k from 1 to g). We can get X² represented by socimatrix.

• The second method

In the second method, we can do the matrix multiplication by sociogram. For instance, according to sociogram, n1 can reach n3 in one way with two steps, so we fill "1" in the first line of the third column. Similarly, n1 can arrive at n6 in two ways with two steps, so we fill "2" in the first line of the six column.

In this way, we can easily deal with the matrix multipliaction using relationship of the factors in the social network.

In general, X² is defined as the matrix product of X times itself, 2 times. This operation can be used to study walk and reachability in SNA. 

^{These above are the part of SNA I prefer to talk about...}