1. A New Approach to Structural Equivalence: Places and Networks of Places as Tools for Sociological Theory Narciso Pizarro Professor Facultad de Ciencias Políticas y Sociología Universidad Complutense de Madrid

2. Narciso Pizarro2 The initial problem 1.We may describe social relations at a given moment in time representing the as social networks. 2.Points in a network can be individuals or sets of individuals that can also be called events, groups or institutions. 3.We can also represent social relations as two way networks, where we have simultaneously dots representing individuals and others representing sets of individuals. 4.What we can not do, for the moment being, is to compare two or more social networks representing data proceeding from two or more instants separated in time by intervals of time of the order of decades.

3. Narciso Pizarro3 Social structure and time  If the expression “social structure” has a theoretical meaning, related to the data we collect and analyze, it should denote something that is invariant during long enough time intervals.  Data we collect concerning social relations- in the usual sense of this expression- concerns individuals and groups identified by their names. This is the prime matter of social networks construction. If collect data about meaningful social relations in two points of time separated by a couple of decades, we will find: 1.Individuals in the first and in the second data set have not the same names. 2. Institutions, groups and other sets of individuals have not the same identities.

4. Narciso Pizarro4 The concept of place Dealing with data about multiple memberships of thousand of individuals over decades or long periods of time, it is clear that the following definition is, for the moment being, a convenient research tool. If we have a set of individuals belonging each one to one or more socially (not only analytically) defined sets of individuals, here called institutions to stress with the word the socially built character of the set and noted: we will define a place P as a subset of E such that at least one of the individuals members of I belongs to every one and only to the institutions included in the subset P. That is to say

5. Narciso Pizarro5 Networks of places So defined, places are subsets of E, independent from individuals not only because we can find many individuals having the same place, but also because, thinking in time, an individual’s place can last longer than the individual himself, as others one can occupy a particular place after the first one disappears. Also, social structure can be defined as a network of places, using the following straightforward definition of a relation between two places: Definition 2: Two places P i and P j are in a relation R if P i  P j   Definition 3:The set P of all the places defined in E and the set R of their relations constitutes a network of places.

6. Narciso Pizarro6 Structural Equivalence and Networks of Places (1) i1i1 i3i3 i4i4 i5i5 i6i6 i7i7 e1e1 e2e2 i2i2 Let’s consider a very simple case, with 7 individuals and 2 institutions:

7. Narciso Pizarro7 Classes and places Equivalence classes {i 1, i 2, i 3 } {i 4, i 5 } {i 6, i 7 } Corresponding places {e 1 } {e 1, e 2 } {e 2 }

8. Narciso Pizarro8 Links in networks of places We consider an individual i r to be linked to an individual i s through a link e k if both individuals are members of e k : i1i1 i3i3 i2i2 i4i4 i5i5 Links of type e 1 between individuals i4i4 i5i5 i6i6 i7i7 Links of type e 2 between individuals

9. Narciso Pizarro9 Reduced networks and networks of places or the equivalent Reduced Network {i 1, i 2, i 3 } {i 4, i 5 } {i 6, i 7 } Network of Places {e 1 }P {e 1, e 2 } or Q {e 2 }R If structurally equivalent individuals are identified, a reduced network is obtained, where points are equivalence classes, that is to say places. In our example the reduced network has the following appearance, where links e 1 are represented by continuous arcs and those of e 2 type, by dotted arcs.

10. Narciso Pizarro10 In our example, the dual network P* is the following e1e1 e2e2 Q Q P R Dual network: institutions linked by places It is possible, also, to define a dual network: a network P* of institutions linked by places. In this dual network an institution e k is linked to an institution e l by a link P if these institutions belongs both to the place P:

11. Narciso Pizarro11 Chains of relations: morphisms Following the usage established by Lorrain and White (1971), we will call morphisms the relations e k between the points of a network of places, as well as any chain of such relations. These morphisms represent the direct or indirect relations between places. Among the morphisms, we can distinguish between the generator morphisms, e k, that are the elements of E, and the composed morphisms, chains of generator morphisms.

12. Narciso Pizarro12 Graphs of morphisms The graph of a generator morphism e k is the set of couples (X,Y), where X and Y are places belonging to e k ( containing e k ). For example, if A 1 = {P, Q } is the set of places belonging to e 1, The graph of e 1 is simply the cartesian product A 1  A 1, that is to say, the set of all the couples of elements of A 1 P Q R P Q R

13. Narciso Pizarro13 Composed morphisms and their graphs The composed morphism is defined simply because there exist individuals who belong simultaneously to both institutions, that is to say, because there exists a place (in this case, Q) containing both institutions. That is equivalent to say that the intersection A 1  A 2 is non empty. It is the same for any other composed morphism. The graph of the composed morphism is the set of all couples (X,Y), where X is a place belonging to e 1 (containing the element e 1 ) and Y a place belonging to e 2 (containing e 2 ). In other words, the graph of the composed morphism is the cartesian product A 1  A 2. P Q R P Q R Graph of

14. Narciso Pizarro14 Composition of morphisms While the composed morphism e k oe l oe m is defined (that is to say, if there is such a chain of relations between the points of the network of places P and, what is the same, that intersections A k  A l and A l  A m are non empty), then the graph of the composed morphism e k oe l oe m is simply equal to A k  A m. The same way, if the composed morphism e k oe l oe m oe n is defined, then its graph is simply A k  A n, and so on. Because as we have seen the morphism e k oe k o…oe k as well as all the defined morphisms of the type e k oe k have all the same graph that e k, we can consider all these morphisms as corresponding to the same social relation and, consequently, identify all those composed morphisms to e k : (1)e k o……oe k = e k oe k = e k (2)e k o……oe l = e k oe l The equations (1) and (2) express the law of the first and the last letters (LORRAIN: 1975).

15. Narciso Pizarro15 The law of composition in the case where there are only two generating morphisms, as in our example. To simplify the notation we will equate e l = a and e 2 = b and this will be the composition law: ab aob boa aaaob a bboabb aoba a boa bb

16. Narciso Pizarro16 The square band semigroup The set of morphisms (a, b, aob, boa ), together with their composition table constitutes what is called a semigroup. This semigroup S has been generated by the set of institutions (e l = a and e 2 = b) and each element of S is idempotent (e k oe k =e k ) and S is a rectangular band (CLIFFORD & PRESTON, 1961). In our case, it should be called a square band, a particular case of a rectangular band.

17. Narciso Pizarro17 Properties of the square band semigroup(1) One of them is that a and b have isomorphic places in the composition law: if we substitute everywhere in the composition table a by b and b by a, we got the same composition table, with a permutation of lines 1 and 2 and of lines 3 and 4, as well as columns 1 and 2 and columns 3 and 4. All this changes nothing to the behavior of morphisms. Then, nothing in the composition table allows us to distinguish between social relations a and b of our network of places. On the contrary, of course, graph of a in the network of places is different from graph of b, because of the different set of places related by a and the particular set of places that b relates, these two sets of places may contain different numbers of places and their intersection can be more or less important.

18. Narciso Pizarro18 Properties of the square band semigroup(2) It is possible to take as generating morphisms aob and boa and find, by means of the compositions laws, a and b as composed morphisms: (3) aoboboa=a and boaoaob= b This last fact is, from a sociological standpoint, of a great signification: we can consider the generating relations a and b, that is to say, the institutional memberships defining places, as derived from indirect relations, the composed morphisms. Intra institutional relations may be viewed as derived from inter institutional relations. We can, then, conceive places as defined by their mutual relations, which is the minimal condition of any structural thought in social sciences. Also, in what concerns social actors subjectivity, this last fact allow us to consider consciousness of membership as a fact derived from unconscious indirect social relations. Subjectivity is no longer a prior condition for social structure.

19. Narciso Pizarro19 A first application to real data We took our data base on the Spanish power elite in 1981. It contains, among other information, data on individual’s memberships in institutions. Using an experimental computer program written by Sebastien Delarre in SAS-IML, we found the following network of places, that we represent using Pajek:

20. Narciso Pizarro20 A network of places

21. Narciso Pizarro21 Out-Degree Partition

22. Narciso Pizarro22 In-Degree Partition

23. Narciso Pizarro23 A smaller network

24. Narciso Pizarro24 Out-Degree Partition

25. Narciso Pizarro25 In-Degree Partition

26. Narciso Pizarro26 Some of the open problems  The effect of groups definition on networks of places: social categories and social groups.  The choice of parameters in the programs: most frequent or most populated places.  How to use the algebra of square band semigroups to analyze networks of places.