PolySocial Networks (PoSN)
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PolySocial Networks (PoSN) are the abstract networks representing the union of all instantiable networks for each person in an aggregate. The instantiation of a subset of these in a given context we call PolySocial Reality (PoSR). Each person in the network possesses a perspectual intersection of many potential resources for information and interaction, only some of which can be instantiated in a given instance, considering constraints of time, cognitive capacity, and mutual exclusion.
Multigraphs support one of the aspects of PoSN/PoSR: the multiplicity of different relations that may underly multiple intersecting social networks. Gjoka, et al (2011) propose a promising approach to this problem based on multigraphs: graphs whose vertices may be connected by more than one edge. Because of the multiplicity of edges, each type representing a different context for social relations, a projected multigraph will be more likely to have a higher degree of connectivity. (See Fig. 1)
Fig. 1. Multigraph integrating two networks.
Combining multigraphs with metagraphs (Basu and Blanning, 1992) appears a reasonable initial mathematical representation for an exploration of PoSR through the analysis of trails and aggregated trails. A metagraph is a graph where vertices are sets, and edges are connections between sets (see Fig. 2). A multigraph that is also a metagraph permits us, at least, to represent the data in a form that is interoperative and can be converted into different forms such as matricies, XML or relational data suitable for online analytic tools for which a range of algorithms for methods of analysis have been established.
Fig. 2. Multi-Metagraph.