The relationship between the different triad types and the global network structures was extensively studied by the social network pioneers (Davis & Leinhardt, 1967; Holland & Leinhardt, 1970; Johnsen, 1985). Different triad types were used to test the existence of different global network structures in empirical networks (Holland & Leinhardt, 1970). This was done by comparing the number of different triad types in an observed network with the distribution of the number of triad types in random networks.
Nowadays, the term motif (Milo et al., 2002) is often used to study different aspects of global network structures. They are defined as “patterns of interconnections occurring in complex networks at numbers that are significantly higher than those in randomized networks” (Milo et al., 2002). The triad types can be considered a subset of all possible patterns. Different patterns with three or four nodes are typically used because considering patterns of size two would be insufficient, while considering patterns of a higher size might be computationally very intensive and less-informative in terms of global network structures. The triad types are seen as the smallest sociological unit from which the dynamic of a multi-person relationship can be observed (Davis & Leinhardt, 1967).
Although different triad types have often been used to describe the global network structures, they have yet to be systematically studied in the context of the most common blockmodel types. It is also known that the distribution of different triad types can be related to a given global network structure, although there is a lack of understanding of whether the distribution of different triad
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types can cause the selected global network structure to emerge. This issue is addressed in Chapter 3.
Different triad types cannot be seen as “mechanisms” or “local network mechanisms” as defined in Section 1.1 because the local network structures cannot be indirectly used to explain how the behaviour of the individuals affects the global network structure. However, triad types can be used as a form of help while contemplating the possible relationship between local network mechanisms and global network structures, as often occurs in the context of Exponential Random Graph Modelling.
The most common blockmodels were considered in this study, namely cohesive blockmodel, symmetric and asymmetric core-periphery blockmodel, transitivity blockmodel, transitive- cohesive blockmodel, hierarchical blockmodel and hierarchical-cohesive blockmodel. The three clusters were assumed in all cases but in the asymmetric and symmetric core-periphery blockmodel types only two clusters are possible by definition. For each studied blockmodel type, different triad types were classified in the sets of allowed and forbidden triad types. The classification was obtained by considering the networks containing the selected ideal blockmodel (without any inconsistency). The set of allowed triad types consists of those triad types that appear in such network structures while the set of forbidden triad types is made up of those triad types that do not appear in such network structures. It turned out that one can distinguish different blockmodel types by only looking at the sets of allowed and forbidden triad types.
By considering these sets of triad types, two different algorithms were used to generate the networks in order to increase the reliability of the results. The first algorithm was the proposed Relocation of links algorithm, whereas the second one was the MCMC algorithm implemented in the “ergm” package for the R computer language.
All of the studied blockmodels can be generated by considering different sets of triad types. In general, the fit of the generated global network structures to the ideal global network structures is not significantly worse when the set of forbidden triad types is used as opposed to the case when the set of allowed triad types is used, although a very important difference arises while generating networks by considering one set or another.
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When the networks are generated by the Relocation of links algorithm and the set of allowed triad types is used, the distribution of the triad types must be given. The information on the number of clusters and their sizes could be embedded in such a distribution of different triad types. On the other hand, considering the set of forbidden triad types only gives information on which types of triads are not allowed to appear in the network. This can still contain some information on the number of clusters (two vs. more than two clusters), but the amount of information the researcher needs to provide is much smaller.
Nevertheless, the generated networks with the target hierarchical blockmodel contained a greater amount of inconsistencies than the generated networks with other target blockmodel types, especially when the networks were generated using the MCMC algorithm. When using the RL algorithm by considering all triad types, the blockmodel was successfully generated, but the cluster of the nodes on the highest and the cluster of the nodes on the lowest hierarchical level were very small. Considering paths-of-length-three as an additional local network structure led to generated networks containing a very clear hierarchical blockmodel.
The main finding of this chapter is that the selected global network structures are able to emerge due to the selected local network structures even when the nodes’ attributes are not considered. This is a good indicator that more complex local network mechanisms that produce a given global network structures might exist. Such local network mechanisms are addressed in the later chapters.