The structure of close communication, contacts and association in social networks is studied in the form of maximal subgraphs of diameter 2 (2-clubs), corresponding to three types of close communities: hamlets, social circles and coteries. The concept of borough of a graph is defined and introduced. Each borough is a chained union of 2-clubs of the network and any 2-club of the network belongs to one borough. Thus the set of boroughs of a network, together with the 2-clubs held by them, are shown to contain the structure of close communication in a network. Applications are given with examples from real world network data.
The structure of close communication, contacts and association in social networks is studied in the form of maximal subgraphs of diameter 2 (2-clubs), corresponding to three types of close communities: hamlets, social circles and coteries. The concept of borough of a graph is defined and introduced. Each borough is a chained union of 2-clubs of the network and any 2-club of the network belongs to one borough. Thus the set of boroughs of a network, together with the 2-clubs held by them, are shown to contain the structure of close communication in a network. Applications are given with examples from real world network data.
Corporate networks, as induced by interlocking directorates between corporations, provide structures of personal communication at the level of their boards. This paper studies such networks from a perspective of close communication in subnetworks, where each pair of nodes (boards of a corporation) are either neighbours, or have a common neighbour. These correspond to subgraphs of diameter at most 2, designated by us earlier as 2-clubs, with three types (coteries, social circles and hamlets) as degrees of close communication in social networks, within the concept of boroughs of a network. Boroughs are maximal areas and containers of close communication between nodes of a network. This framework is applied in this paper to an analysis of corporate board interlocks between the top 300 European corporations 2010, as studied by Heemkerk (2013), with data provided by him for that purpose. The paper gives results for several perspectives of close communication in the European corporate network of 2010, a year close to the global crash of 2008, as a further elaboration of those given in Heemskerk (2013). Heemskerk [2] gave an elaborate analysis of the network of interlocking directorates of the major European corporations in 2005 and 2010. In such corporate networks interlocks provide channels of personal access and communication between the boards of corporations, so that areas of close communication can be defined as sub-networks where each pair of nodes (corporations) are neighbours or have at least one common neighbour. These can be represented as (sub)graphs of diameter at most two. For such sub-networks Mokken [3][4][5] introduced the concept of 2-clubs of a network and its three types (coterie, social circle, hamlet), later extended by Laan et al [1] with the concept of boroughs, which contain the 2-clubs of a network. Recent advances in hardware and corresponding programming techniques provide means and opportunities to use them on large networks (e.g. [6; 7] We tried to apply these in the context of corporate networks, using software developed by one of us [8; 9] . From this perspective of close communication we extend Heemskerk's analysis of the major European corporations in 2010, using his data set. In the following sections we first introduce the conceptual and analytic framework. We then analyse Heemskerk's network of 286 major European companies, restricting ourselves to its major component of 259 firms. Conceptual framework 1Close communication in a network is defined here as access and communication between nodes directly between neighbours (1 st step) or through a common neighbour (2 nd step). Close communities in a network are areas where each pair of nodes are neighbours or have a common neighbour. These can be represented as graphs of diameter at most two. Conform [3; 4] we represent close communities in a network by its 2-clubs, which are maximal subgraphs of diameter at most two: they are not included in, or part of another subgraph of diameter at most two. Note that 2-clubs can and will overlap...
Corporate networks, as induced by interlocking directorates between corporations, provide structures of personal communication between their boards. This paper studies such networks using the framework of a previous paper by Laan et al. (Soc Netw Anal Min, 2016. doi:10. 1007/s13278-016-0326-0) where close communication is defined by sub-networks, so that each pair of nodes (boards of a corporation) are either neighbours or have at least one common neighbour. These correspond to sub-graphs of diameter at most 2, designated by us earlier as 2-clubs of three types (coteries, social circles and hamlets), and conform three levels of close communication in social networks. They are all contained within the disjoint boroughs of a network, supercommunities which envelope all close communication between nodes of a network. This framework is applied here to an analysis of corporate board interlocks between the top 300 European corporations 2010, using the data from an earlier study by one of us (Heemskerk in Econ Soc 42:74-101, 2013). While the results corroborate the main findings of the earlier studies, our approach also uncovers additional, thus far unrevealed patterns. A single dominant European borough with the Francophone network as its centre and that of Germany only regionally and internally connected. The UK business elite on the other hand is very present and prominent in this European structure of corporate close communication.
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