Relationships among actors in a defined collective scheme are the primary source of information for social network analysis. Ties not only make the network structure but also provide the basis for the characterization of underlying processes occurring in the social system. Although social networks are typically characterized by a single type of relationship, social life is more complex and people are embedded with ‘different’ kinds of ties that are interlocked within the network relational structure.
These sorts of arrangements are known as
In this spirit, Breiger and Pattison ( 1986) proposed a type of equivalence among the network members that is built on local role algebras for the creation of the positional system. The goal with this paper is to extend this type of correspondence by suggesting an effective way to incorporate the attributes of the actors and their relationships into a single relational system representing the multiplex network structure. One important reason for such integration is that social conduct in networks does not always institute a link between individual subjects, and attribute-based information about the actors is often not ascribed to them, but depends on the individual’s own choices or circumstances.
Examples of actor attributes are the acquisition of a certain characteristic from the social environment such as innovation adoption; the acquisition of a certain attitude; the non-compulsory affiliation to a group; the individual’s personal wealth and political power, etc. Such attributes can play a significant role in the network relational structure and should be incorporated into the modeling process.
Representing social relations and actor attributes in an integrated system requires a formal definition of the social network concept. A
A multiplex network
Each element in
One of the theses of this paper is that non-ascribed attributes from the actors in the network can be an integrated part of the relational structure, which is typically represented by a semigroup of relations (Boorman and White, 1976; Pattison, 1993). In this sense, the incorporation of the changing attributes of the actors implies that subjects sharing a characteristic constitute a subset of self-reflexive ties associated to the social system represented by a matrix format that can be combined with the other elements in the relational structure.
In formal terms, actor attributes are represented by the elements of a
Accordingly, for a given attribute defined in
On the other hand,
As a result, the general representation of
The establishment of the indexed diagonal matrix implies that each type of attribute considered for the actors in the network is represented by its own array, and it constitutes an additional generator in the relational structure. When all network members share a given attribute, the result will be an identity matrix without any structural effect, whereas when none of the actors possesses the characteristic, the representation will be a null matrix with an annihilating effect where no composition is possible.
Clearly, we are mainly interested in the differentiation of the actors who share an attribute as opposed to those who do not share the trait because the resulting matrix that is neither a neutral nor an absorbing element in the algebraic structure has structuring consequences in the network relational system.
Although the concatenation of social ties used in the construction of the partial order structure is well established (Boorman and White, 1976; Pattison, 1993), there are caveats when producing algebraic systems with the diagonal matrices representing actor attributes. For instance, since social interactions are typically measured without loops and are represented by adjacency matrices with empty diagonals, these cannot be contained in an attribute relation with this form of representation. However, by grouping actors who are structurally equivalent, it is possible to obtain collective self-relations.
For example, take relations
0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | ||
1 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | ||
1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | ||
C | F | A |
A solution to this issue is to group the structurally equivalent actors in the network, which is the same as categorising actors with similar patterns of relationships. For instance, it is obvious that actors 1 and 2 are structurally equivalent (according to the definition made by Lorrain and White, 1971) because these actors are identically related with both social relations (
1 | 1 | 0 | 0 | 1 | 0 | ||
1 | 0 | 1 | 0 | 0 | 0 | ||
C | F | A |
Structural equivalence is the most stringent type of correspondence. Since its formal definition significant relaxations have been proposed for social networks; notably Automorphic equivalence (Everett, 1985; Winship and Mandel, 1983), Regular equivalence (Sailer, 1978; White and Reitz, 1983), and Generalized equivalence (Doreian et al., 1994, 2004). A common characteristic among these correspondence types is that they have a
While the grouping of the actors in social networks usually applies some relaxation to the equivalence criterion, in the case of multiplex networks it is desirable to preserve the
As an alternative to a global equivalence in the reduction of multiplex networks, we can apply a
To recognize local equivalences among the actors we rely on a three-dimensional array similar to
Relation-Box with an emphasized relation plane and its role relations
A horizontal slice in the Relation-Box is called a
A local role equivalence is also a way to characterize social roles in incomplete and in ego-centred networks while preserving the distinction of diverse types of relationships. Besides, Winship and Mandel ( 1983) point out that local role equivalence is a generalisation of Automorphic equivalence in the sense that both kinds of equivalence involve the same types of role relations. Automorphic equivalence would require not only the same types of role relations but also the same number of such relations, which implies equal role sets and local role algebras among correspondent actors (Pattison, 1993).
Although the Relation-Box theoretically permits consideration of compound relations of infinite lengths, the actors would not be aware of long chains of relations in their surrounding social environment. Thus, based on practical or substantial reasons, it is possible to perform the analysis with a ‘truncated’ version of this array with size
Breiger and Pattison ( 1986) developed one structural correspondence aimed at multiplex networks that is based on the individual perspectives of the actors. Although this equivalence type is referred to in the literature as ‘Ego algebra’ (Wasserman and Faust, 1994), we call it
The fact that CE generalizes local roles to the entire system implies that this type of correspondence works both at the local and at a ‘global’ level. That is, the establishment of roles and positions in the network are from the perspectives of individual actors, whereas the characterization of equivalence itself is made by considering the relational features that are common to all members in the network. However, this last feature works better with middle-sized networks, and hence CE can be regarded as a local to ‘middle-range’ type of correspondence (Pattison, personal communication).
The local portion of CE lies in the actors’ particular views of the system in terms of inclusions among the role relations of the actors’ immediate neighbours. Recall that the role relations are recorded in the columns of the individual relational planes, which means that there are in total
Isolated actors in a multiplex network are unable to ‘see’ any type of relationships among other actors through the defined links. This implies that role relations for isolates are empty no matter the type of tie or its length, and that any role relations in the relation plane are blank as well. However, connected actors have a different perspective where there is an inclusion among other actors. The collection of inclusions (or lack of them) for each actor or class is reflected in a square array size
In more formal terms, from the standpoint of a given actor
The last proposition implies that there is no inclusion between actors
On the other hand, the global part of CE occurs with the union of the different personal hierarchies into a
The partition of the network itself is then a product of a global type of equivalence that is performed on the cumulated person hierarchy. However, we should bear in mind that matrix
In the next section we illustrate the process of constructing the local and global hierarchies in detail with an example of the reduction of a multiplex network. As in Breiger and Pattison ( 1986), we study the Florentine families network classic dataset and, like the authors, we apply CE in the reduction of this social system.
The Florentine families network dataset (Breiger and Pattison, 1986; Kent, 1978; Padgett and Ansell, 1993) corresponds to a group of people from Florence who had a leading role in the creation of the modern banking system in early 15th-century Europe. There are two types of social ties in the network that correspond to Business and Marriage relations among the 16 prominent Florentine families, two of which stand out as being particularly powerful and rivals: the Data was retrieved from
Figure 2 depicts the network as a multigraph where different shapes in the edges represent the two kinds of relations. We note in the picture that eight bonds combine Business and Marriage ties in the system and that the network has one component and a single isolated actor represented by the
Multigraph of the Florentine families network
Solid edges are Marriage relations, dashed edges are Business ties, and node size reflects their financial Wealth. Plot made with a force-directed layout of the
A crucial part of the modelling of multiplex networks is the reduction of the social system because the corresponding relational structure represented by the semigroup is typically large and complex, even for small arrangements. For instance, Breiger and Pattison ( 1986, p. 221) report a semigroup size with an order of 81 for the Florentine families network, and this is only considering the two generator relations without attributes. Certainly, it is necessary to work with a more manageable structure in order to obtain better insights into its logic of interlock.
The reduction of the network implies constructing a relational structure based on a system of roles and positions, which leads to the
Next, we categorize the actors in the Florentine families network in terms of CE having actor attributes as generator relations. The first step is to look at the structure product of the actors’ views of their neighbours’ relations in terms of inclusions, and then we perform the modelling to produce the network positional system to an
Applying CE in the reduction of a multiplex network structure implies the construction of the Relation-Box, which provides the basis of the local part of this type of correspondence. Recall that the Relation-Box is defined by the number of actors in the network and the number of string relations that make up the actors’ immediate social ties and eventually the combination of these. Then all inclusions from the individual perspectives are combined into a single matrix that stands for the global part of CE.
To illustrate the process of constructing person hierarchies we restrict the analysis to the smallest case of the Relation-Box with no compounds so that for the Florentine families network the dimensions are 16 × 16 × 2. When we look at Figure 2, we see that apart from
For a two-chain relationship, the person hierarchy of
Each actor This is disregarding the isolated actor.
The partial order structure representing
Cumulated person hierarchy,
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
2 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
3 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
4 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
5 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
6 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
7 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
8 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
9 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
10 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
11 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 |
12 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 |
13 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 |
14 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 |
15 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 0 |
16 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
All computations are made with the
Therefore, the positional system can have either two classes of collective actors plus the isolated actor or four classes with pairwise individual positions in the system. Regardless of the option chosen, both reduced arrangements seem to be good representations of the network structure in terms of the patterned social relations, and they serve as the basis for the construction of the role structure of the Florentine families network. However, a number of attributes from the actors may play a significant part in the establishment of the network positional system, and hence we continue the rest of the analysis of this network by incorporating actor attributes in the establishment of the network role structure.
The core motivation for this paper is to incorporate in the modelling of the network relational structure significant actor characteristics, which do not have a structural character; that is, traits that are inherent to the actors and do not depend directly on their embedment in the network, such as individual centrality measures, dyadic attributes, etc. On the other hand, although actor attributes can be independent variables, they are not ascribed to the actors in the same way as age, gender or other demographic information, but are governed by the action of the actors themselves. The belief is that these kinds of actor attributes should be part of the modelling of the network positional system and also of the establishment of role structure when the attribute has a structural effect. Naturally, extreme cases, e.g. when all or no actors share the attribute, will not have an influence on the final structure since they are represented by the identity and the null matrix, respectively.
In the case of the banking families, the power and influence of these families in the 15th century constitute significant characteristics. Table 2 gives the Wealth and the number of Priorates of the Florentine families as reported in Wasserman and Faust ( 1994, p. 744), and these two attribute types, either together or individually, are candidates for the modelling of the network positional system and subsequent role structure. For this type of analysis each attribute is represented with an indexed matrix; hence reducing the network structure with the actor attributes resembles the process we just applied to the Marriage and Business relations with CE, except that now there are additional generators to the social ties representing the attributes.
Wealth and number of Priorates of the Florentine families
|
Number of ⪆ 34 |
|||
---|---|---|---|---|
|
10 | 0 | 53 | 1 |
|
36 | 0 | 65 | 1 |
|
55 | 1 | NA | 0 |
|
44 | 1 | 12 | 0 |
|
20 | 0 | 22 | 0 |
|
32 | 0 | NA | 0 |
|
8 | 0 | 21 | 0 |
|
42 | 1 | 0 | 0 |
|
103 | 1 | 53 | 1 |
|
48 | 1 | NA | 0 |
|
49 | 1 | 42 | 1 |
|
3 | 0 | 0 | 0 |
|
27 | 0 | 38 | 1 |
|
10 | 0 | 35 | 1 |
|
146 | 1 | 74 | 1 |
|
48 | 1 | NA | 0 |
We note in Table 2 that each category has two columns, one for the absolute values and another that marks the limits of these values according to a cut-off value. In one case we differentiate the very wealthy families from the ‘modestly’ rich actors in the network by adopting a cut-off value of 40,000 Lira, which approximates the average of their financial resources. Actually, the mean is 42.56, and the Lamberteschi family lies in this limit, but rounding the cut-off value to 40 makes more sense for the analysis.
We continue the analysis of the banking network by applying CE for grouping the actors in the construction of the positional system with actor attributes. The difference is that now the Relation-Box on which the person hierarchies are based includes the additional generators representing the attribute-based information. Since indexed matrices only have information on their diagonals, the different person hierarchies in the network include self-containments whenever the actor has the attribute. For example, while the person hierarchy of
Figure 3 shows
Hasse diagrams of
Top to bottom: with Wealth,
It is important to note, however, that although the different levels in the Hasse diagrams try to reflect the ranks in the partial order structures, there can be ambiguities in the placements depending on the diagram structure. For example, the
All partial orders shown are emerging structures with the smallest value of
These diagrams very clearly show that the attributes of the actors, such as their monetary wealth and political power, have an impact on the relational structure of this particular network. If we look at the diagrams in Figure 3, we note that there is a further differentiation in the network in all three cases when considering actor attributes in the modelling. Apart from the isolated actors, whose personal hierarchy corresponds to the null matrix, the cumulated person hierarchy for Wealth clearly involves three levels, whereas there are five levels in the diagrams for the number of Priorates, and for the two attributes together.
As a result, the positional system with Wealth differentiates three categories of actors plus the isolated node where the largest class in the previous classification is now divided into two categories. Thus the personal wealth has a structuring influence in the network, and this makes a lot of sense; the richest actor of the banking network is the
When we look at the number of Priorates there is even more differentiation in
Theory can guide us in the establishment of the categories in the positional system in the two last cases. We also need to determine which of the resulting role structures that are a product of the positional system provides the best insights into the relational interlock of the multiplex network structure. This task constitutes one of the last steps in the modelling of the system and we look at the reduced relational structures of the banking families’ network.
The main challenge in establishing the positional system of the network is to find the sets of collective relations that produce the most meaningful network role structure, i.e. a reduced system that provides an insight into the logic of interlock of the network relations. This is typically achieved with the role structure having the smallest possible dimensions. The logic of interlock is a kind of rationality that is shaped by different algebraic constraints expressed in the final relational structure where the different types of ties and the relevant actor attributes are interrelated in this case.
Although the class membership with the Wealth attribute with three defined classes of collective actors seems straightforward, there are ambiguities both as regards the number of Priorates and when the two features are combined. Such uncertainties arise because a number of actors in the network can be classed in different ways according to their respective locations in the partial order structures of That is why
Now we look closer at the inbetween actors in the two hierarchies where political power is involved. From Table 2 we obtain the assignment of these families with respect to the two attributes, and the next upper and lower vectors give the categories for Wealth and number of Priorates, respectively:
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|
|
|
|
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0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 |
Certainly, one possibility is that all these actors are grouped together into a single class irrespective of their economic or political power, and in this way we have a positional system with three categories of collective actors for both Priorates, and also for Wealth and Priorates. The arrangements of roles for Business and Marriage are then equal and all the positions are represented by actors who are both very wealthy and powerful in political terms (of course disregarding
A straightforward way to achieve a structuring effect of diagonal matrices is by separating the actors with ‘ones’ in the intermediate category from the actors with ‘zeros’ in the vector corresponding to this attribute type. Hence we end up with a positional system that has four categories of collective actors, and for the number of Priorates (the second row), for instance,
Conversely, if we model the network relational system with both attributes at the same time, we first differentiate Besides, assigning
A third possibility is to combine the Business and Marriage ties with Wealth in the analysis. In this case the class system of actors takes the levels given in the Hasse diagram of Figure 3 with Wealth. The positional system in this case implies that the Marriage ties do not follow a particular pattern in the role structure, whereas Business ties and Wealth role relations follow a core-periphery structure as shown in the matrices below:
1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | ||
1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 0 | ||
1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | ||
Business | Marriage | Wealth |
There are no ambiguities in the categorisation of actors in the banking network with the financial Wealth of the actors, which leads to a univocally substantial interpretation of the role structure for this positional system. However, the main advantage with these generators is that the role structure gets smaller than with the previous two settings, allowing a more transparent interpretation of the role interlock, even though we are aware that a different logic may arise in the role structure when considering the number of Priorates. The reader can refer to Ostoic ( 2018a) for an extended analysis of the role structure and role interlock of this particular network.
The structuring effect of attribute-based information in the reduction of multiplex networks constituted the most significant aspect covered in this paper, where one of the main challenges has been preserving the multiplicity of the different types of tie. In this sense, the notion of Compositional Equivalence defined by Breiger and Pattison ( 1986) allows us to reduce the network structure without dropping the relational differentiation, and we extend the positional analysis to non-ascribed characteristics of the actors in the network, which are included as generator relations in the form of diagonal matrices. There is a strong belief that attribute-based information enriches the substantial interpretation of the relational structure of the network, and this is so irrespective of whether the relational system is in a reduced or full format.
Even though the reduction of the network can bring some ambiguities, aggregated structures are more manageable for substantial interpretation of the relational logic in multiplex network structures, which are complex systems by definition. CE has proven to be a valuable option for mid-sized networks; however, theoretical guidance is required both for the selection of the attribute types and for the establishment of the positional system and subsequent role structure.
There are still some important issues that need to be accounted for. The first concern deals with directed multiplex networks, in which the application of CE typically requires counting with relational contrast reflected in the transpositions of the ties. A second aspect is the rationale behind relational structures, which is expressed by algebraic constraints governing the system, including sets of equations among strings, hierarchy in the relations, and interrelations between the different types of tie occurring in the network. These aspects are mentioned only briefly here and their treatment is out of the scope of this article. Finally, a statistical approach to the modelling is required for larger network structures, and statistical methods for multiplex networks can serve to complement the modelling process either in an early stage of the analysis or by providing relational and role structures having both fixed and random effects with attributes.