(1979). Social Networks and Psychology. Connections, 2 - INSNA

- 1 0 8 -Clearly, in order to find such a minimal distance matrix min at least one of whose elements is lessthan the corresponding element in d, it will be necessary to have some choice of routes, i .e ., we musthave two or more informants . The method used to find min is given by Acton (1970) .We do not place too much reliance on min as representing any structure of interest to networkers .Instead, we define a category matrix cat ., which we claim represents the cognitive network from the viewpointof each informant . 13We obtain this as follows : in the course of obtaining min, several possible routes from i to j willhave been selected . In general there may be more than one such minimal route ; if a choice still exists wechoose the route needing fewest intermediaries . Then we define catij to be one more than the number ofintermediaries used to proceed from i to j . In other words, if i perceives direct interaction with j,defined here min-- = dij, then catij = 1 . If, however, mini] is such that i-7 k-aj then catij = 2, and soon . We can see the meaning of the category matrix diagrammatically (Figure 1) .FIGURE 1 . Diagrammatic description of category matrix : catii = 0 ; cat ij = cat ik = catie = 1 ;catim = . . . .= cat ir = 2 .The element catij is the number of the row in the network which i places j upon . Increasing the rownumber implies a more complicated interaction pattern, or that j is "not so easy to communicate withdirectly ."We claim that it is precisely the number of this row which represents the reality of the social network,rather than the route used to get there . This is especially true of the second row : in Fig . 1 wehave drawn 14 ja p ; i_-?k--Pp . Now both routes place p in the second row, and we appear to have overlappingpossibilities for the actual interaction route . This does not matter ; in fact, in physical structuredgroups there is often more than a 2 to 1 overlap ratio in the second row . In other words, on average, thetotal number of elements in the first rows of each of i's first row is usually 2 or more times the numberof distinct elements, implying a great deal of overlapping as described . It is tempting to hypothesizethat such overlap is essential to the healthy functioning of groups .We hypothesize that each informant knows, and understands the structure of his own first row (i .e .,he "knows who he knows"), and has a very good (70%) knowledge of his second row (a subset of his firstrow's first row) . This knowledge deteriorates as we descend the rows, as would be anticipated : clearlyan informant will have little or no knowledge of the interaction structure of a subset of the group withwhom he has little interaction . However, he will have sufficient knowledge to know how that little interactiontakes place, and who among his first or second rows is necessary to this interaction .We must stress the important fact that a position in a given row is a measure not of amount ofperceived interaction, but the degree of directness of such interaction .After the catij matrix is derived, we factor it, using the varimax rotation, by rows . Experimentationshowed that the varimax rotation gives the most intuitively useful results . Stingily, people with at leasta 0 .6 factor loading are chosen for inclusion in a "group ." CATIJ Row 1 links are then used to show theties between groups (and their members with all other groups and their members) . "Groups" are defined aspeople who have similar views on their relationship to the universe . This may result because of cliqueingor not, and in general ethnographic evidence must be relied upon to decide the meaning of factors and groupmembership .In two publications (Bernard and Killworth, 1973 ; Killworth and Bernard, 1974b) we have tested thevalidity of the CATIJ technique . We found that CATIJ filters out random error in dij (i .e ., that the CATIJmatrix is reasonably stable to perturbations in the original data) . (Of course, the errors in raw sociometricdata are probably not random . So far, we have no way to deal with this .) Furthermore, we havefound that factoring the CATIJ matrix produces excellent results, while factoring a sociogram producesvery poor results (i .e ., low connectivity even within subgroups) . Factoring CATIJ has turned up subgroupingswhich were not discerned by standard sociometric tests . In one memorable case, we had analyzed aprison living unit and found a totally unintuitive factor group of 3 inmates who had no apparent sociallinks . This bothered us until we learned, a week later, that the three had attempted an escape together .