Common application of linear algebra in sociology and psychology

Common application of linear algebra

At the website 1 ) from the list below the following problem is presented : it is necessary to propose a criterion in compliance with which one may select the strongest team basing on the results of the team competition . At the website 2 ) another problem is presented Suppose we have an organization (e .g . business , group of friends computer network , and we want to keep track of how "members (e .g departments , individuals , machines ) communicate with others . In both cases we can model the situation with a matrix

of 0 's and 1 's . In the i-th row and j-th column of such the matrix we put 1 ' if the i-th Team defeated j-th Team , or if the i-th Member of organization communicates with the j-th Member of organization without mediators . If the i-th Team defeated the j-th Team or the i-th Member of organization communicates with j-th Member of organization we claim that i-th element has one-step dominance over j-th dominance . If we add up all matrix elements from the k-th raw , thus in the case of athletic competition we will have the Team has will result or we will have the k-th member of organization with others

If there are two teams which have the same number of victories over other teams then we will face a question how to decide which team is the best ? With this aim we may calculate two-step dominances . For this purpose we have to calculate matrix A2 . It is very simple to understand (moreover the detailed explanation is presented at the website 1 , that ij-th element of this matrix will show how many two-step dominances the i-th team has over the j-th team . It is obvious that the sum of all elements in the i-th raw will show the number of two-step dominances of the i-th team over the all other teams . Thus , having two teams with the equal number of victories (one-step dominance ) we can suppose that team to the best which will have greater number of two-step dominance Alternatively , one could count both one-step and two-step dominances by computing A A2 . The team with greater one-step and two-step dominances should be considered to be the best . With respect to members of organization i-th and j-th , the matrix A2 records the fact that A has two different routes of 2-stage communication with B

Similarly the content of elements of matrixes A3 , A4 , etc becomes apparent

However , it is not so apparent how to use the methods of linear algebra in the case when among the possible results we may find not only victory or defeat but also draw . In this case the method considered for defining the best team requires to be a little modified

We also may apply the methods of linear algebra for studying graphs Graphs are the objects which have analogies in everyday life . The websites 3 ) and 4 ) provide with general idea of what a graph is Sociologists and psychologists use graphs to determine various kinds of relationships , such as influence , dominance , and communication , in groups . Graphs can be presented in the form of lines (called edges that connect points (called nodes . The application of graphs to the tasks of real life is very interesting

Graphs can be very difficult , so that they cannot be visually analyzed To study such graphs it is necessary to use the methods of linear algebra . Thus , a graph may represent communications network where vertices , or nodes represent subscribers of telephone network and edges represent phone line . Two points may or may not be connected with phone line . The study of such a telephone line is very complicated task while it may consists of thousands of subscribers . We may compose a matrix which will represent such a telephone line . In the i-th row and j-th column of this matrix we put 1 ' if there is a direct connection between i-th and j-th subscriber , and 0 ' otherwise

Now we may face the question is it possible for a subscriber i ' to reach in the given telephone network the subscriber j ? After the analysis of the information presented at the websites 1 ) and4 , we can easily understand the way by which we may receive the answer to this question

It 's necessary to see whether the element of the matrix aij is equal 0 or 1 . If it 's equal 1 , then the subscriber i 'can reach the subscriber j , if it 's equal 0 , then we cannot assert that the subscribers cannot reach each other over the telephone . So we must proceed to the second step

It 's necessary to obtain matrix A2 and find out whether the element in the i-th row and j-th column is equal 1 or 0 . If it 's equal 1 , then the subscribers can reach each other . In case the element is equal 0 , then again we cannot assert that the subscribers cannot reach each other over the telephone . We must go on to the step three

It is necessary to look for ij-th element of matrix A3 and so on till matrix An-1 , where n is the where k ?n . This may seem obvious , if we notice that the longest way by which the subscribers can be connected among themselves is that way when all other subscribers will be intermediate . This longest way will have the following appearance

i-th j-th Its length is equal n-1

Having constructed a graph for telephone network and the corresponding matrix we may ascertain is this network is integral or does it break into separate networks . We also may clarify whether there are defects in the network . However , if the connection between any subscribers can be executed directly not by the single way but by several ways as it is illustrated in the figure below then the given presentation of the graph needs certain modification It is obvious that these models are absolutely realistic and give us insights into dominance and similar phenomena . Using non-mathematical approaches it may be possible to understand such phenomena but it will take considerably more time and efforts than it will when applying mathematics . The application of linear algebra may considerably simplify studies of certain phenomena and allow , with relatively few efforts receiving the answers to set questions

1 ) HYPERLINK "http /media .pearsoncmg .com /aw /aw_lay_linearalg_3 /cs_apps /dominance .pdf br http /media .pearsoncmg .com /aw /aw_lay_linearalg_3 /cs_apps /dominance .pdf

2 ) HYPERLINK "http /users .wpi .edu vadim /LA_I /Projects /project1 .html http /users .wpi .edu vadim /LA_I /Projects /project1 .html

3 ) HYPERLINK "http /aix1 .uottawa .ca jkhoury /graph .htm http /aix1 .uottawa .ca jkhoury /graph .htm

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5 ) HYPERLINK "http /www .math .uiowa .edu jsimon /COURSES /M10Fall04 /CommunicationAndDom inanceMatrices .pdf http /www .math .uiowa .edu jsimon /COURSES /M10Fall04 /CommunicationAndDomi nanceMatrices .pdf ...