Blockmodels

Today, considerable attention is being paid to the global network structures that can be described by blockmodels (Doreian et al., 2005) in both social network analysis and other scientific fields.

The development of blockmodeling was initially based on social theories in the last century (Homans, 1950; Lewin, 1936; Nadel, 1957). Later, blockmodeling was used in various scientific fields (Alderson & Beckfield, 2004; Barnett & Danowski, 1992; Glückler & Panitz, 2016;

Kronegger, Ferligoj, & Doreian, 2011; L. Prota & Doreian, 2016; Laura Prota, D’Esposito, De Stefano, Giordano, & Vitale, 2013; Žiberna & Lazega, 2016).

The term “blockmodel” reflects the fact that if a network is represented by a matrix, which is then split according to a partition (a set of non-overlapping clusters), blocks (submatrices) are formed in the matrix. The term “block” refers to a submatrix showing the links between nodes from the same or different clusters. Two selected nodes are structurally equivalent (also see subsection 2.4.1) if they have links to the same set of other nodes (Batagelj, Ferligoj, & Doreian, 1992; Lorrain

& White, 1971). This is not the only definition of equivalence. Structural equivalence (Lorrain &

White, 1971) and its generalization – regular equivalence (White & Reitz, 1983) – are the most common equivalencies. When structural equivalence is used, only null and complete blocks are possible. In ideal complete blocks, all possible links are present while there are no links in ideal null blocks.

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An assumption often made with blockmodeling that similar (or equivalent) nodes are of the same type and therefore share the same rules on social behaviour (Lorrain & White, 1971; Michaelson

& Contractor, 1992). The decision on whether to use one type of equivalence or another in the blockmodeling context depends on the topic under study and on whether the blockmodeling is being used as a data-reduction technique or serves by way of operationalization of a social role (Borgatti & Everett, 1992). In the latter case, structural equivalency is often criticized for being too restrictive (Borgatti & Everett, 1992) yet it is also worth mentioning that actors can have the same social role without being equivalently linked to the same others (Mizruchi, 1993).

Several blockmodels are well-known (Doreian et al., 2005, p. 236) and studied. They are visualized in Figure 1.3 and described in more detail in the following paragraphs.

Figure 1.3: Different blockmodel types represented by a graph and image matrix

Cohesive blockmodel

According to Doreian, Batagelj & Ferligoj (2005), a cohesive blockmodel is defined by several clusters of nodes which are internally highly linked, but where no links exist between the nodes from different clusters.

This very basic global network structure was studied e.g. in the context of the structural organization of the brain (Shen, Hutchison, Bezgin, Everling, & McIntosh, 2015). This is

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(approximately, in some cases) the global structure also found by community-detection methods (see Lancichinetti & Fortunato (2009)).

Symmetric and asymmetric core-periphery blockmodel

Even though the core-periphery structure is one of the most typical and most often analysed, it is defined differently by various authors (Borgatti & Everett, 2000; Nordlund, 2018). Borgatti &

Everett (2000) summarize the three intuitive views of the core-periphery structure. The first assumes the existence of one cluster to which the nodes belong to a greater or lesser extent. Similar to the first definition is the definition whereby links between the core nodes exist and where are no links between the peripheral nodes. The links between the core and peripheral nodes can either exist or not. The third definition is spatially defined: the core nodes which are positioned in the centre of the Euclidean space are close to all nodes in the network while those on the outskirts are close only to the centre. These definitions all consider one core and one periphery, whereas definitions referring to several cores also can be found (Cugmas, Ferligoj, & Kronegger, 2016;

Kronegger, Mali, Ferligoj, & Doreian, 2012). Rombach et al. (2017) considered the situation where several core-periphery structures (as defined by Doreian, Batagelj & Ferligoj (2005)) appear in a single network.

In this study, the definition of core-periphery blockmodel comes from Doreian, Batagelj & Ferligoj (2005). In their definition, a core-periphery blockmodel consists of one internally well-linked cluster of core nodes and one cluster of peripheral nodes, which are not linked to each other. There are links between the core and the periphery. The core-periphery blockmodel is called symmetric in this study⁷ when the links between the peripheral and core nodes are mutual (Figure 1.3b), and asymmetric when only the peripheral nodes are linked to the core ones (Figure 1.3c). Another version of the asymmetric core-periphery blockmodel is where only the core nodes are linked to the peripheral ones (Figure 1.3h)⁸. The definition used in this study is today one of the most common definitions of the core-periphery model (Nordlund, 2018).

7 Doreian, Batagelj & Ferligoj (2005) call the asymmetric versions of the core-periphery blockmodel type a

“centralized blockmodel”.

8 In this dissertation, the first version of the asymmetric core-periphery blockmodel is referred to by the name

“asymmetric core-periphery blockmodel”.

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The core-periphery structure lies in the middle of several extreme properties, e.g. clique vs. star configurations, network assortativity vs. network disassortativity, hierarchy vs. non-hierarchy etc.

(Csermely, London, Wu, & Uzzi, 2013).

A clear core-periphery blockmodel was found among high school students, where a relation was defined by one student asking another student to borrow their study notes (Batagelj, Mrvar, Ferligoj, & Doreian, 2004). This blockmodel was also found when studying individual creative performances in the Hollywood film industry (Cattani & Ferriani, 2008), in the analysis of metabolic networks (Da Silva, Ma, & Zeng, 2008), and in many studies of scientific co-authorships (Chinchilla-Rodríguez, Ferligoj, Miguel, Kronegger, & de Moya-Anegón, 2012; Cugmas et al., 2016; Hu & Racherla, 2008).

Hierarchical blockmodel and hierarchical-cohesive blockmodel

When a hierarchical blockmodel consists of three clusters, then a cluster of internally non-linked nodes exists which are all linked to the second cluster of internally non-linked nodes and the nodes from the second cluster are all linked to the third cluster of nodes which are also not internally linked to each other.

A hierarchical-cohesive blockmodel is characterized by complete blocks on the diagonal of an image matrix, which means that the nodes belonging to a certain cluster are internally highly linked. Also, in the case of a hierarchical blockmodel, the clusters of nodes can be hierarchically ordered.

A hierarchical structure is often associated with companies' organizational structure. Oberg &

Walgenbach (2008) analysed employee communications in a given company. Even though the company's policies encourage the principles of non-hierarchical functioning (including ways of communicating among the employees), they confirmed that their day-to-day communication on the intranet indicates the existence of a hierarchy within the organization.

Transitivity blockmodel and transitive-cohesive blockmodel

A transitivity blockmodel is similar to a hierarchical blockmodel, except that in the case of a transitivity blockmodels links also exist from the clusters on the lowest level to all clusters on the highest levels (or vice versa). In the literature, both hierarchical and transitive global network structures are often called hierarchical.

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A transitive-cohesive blockmodel is similar to a transitivity blockmodel, but with the former one there are links between nodes from the same cluster.

The definitions of the transitivity, transitive-cohesive, hierarchical and hierarchical-cohesive blockmodel types, used in this dissertation, consider the links from the nodes on a lower hierarchical level to those on a higher hierarchical level. Well-known are also definitions, where the links goes from the nodes from a higher hierarchical position to the nodes from the lower hierarchical level.