Unified mathematical treatment of complex cascaded bipartite networks: the case of collections of journal papers

A mathematical treatment is proposed for analysis of entities and relations among entities in complex networks consisting of cascaded bipartite networks. This treatment is applied to the case of collections of journal papers, in which entities are papers, references, paper authors, reference authors, paper journals, reference journals, institutions, terms, and term definitions. An entity-relationship model is introduced that explicitly shows direct links between entity-types and possible useful indirect relations. From this a matrix formulation and generalized matrix arithmetic are introduced that allow easy expression of relations between entities and calculation of weights of indirect links and co-occurrence links. Occurrence matrices, equivalence matrices, membership matrices and co-occurrence matrices are described. A dynamic model of growth describes recursive relations in occurrence and co-occurrence matrices as papers are added to the paper collection. Graph theoretic matrices are introduced to allow information flow studies of networks of papers linked by their citations. Similarity calculations and similarity fusion are explained. Derivation of feature vectors for pattern recognition techniques is presented. The relation of the proposed mathematical treatment to seriation, clustering, multidimensional scaling, and visualization techniques is discussed. It is shown that most existing bibliometric analysis techniques for dealing with collections of journal papers are easily expressed in terms of the proposed mathematical treatment: co-citation analysis, bibliographic coupling analysis, author co-citation analysis, journal co-citation analysis, Braam-Moed-vanRaan (BMV) co-citation/co-word analysis, latent semantic analysis, hubs and authorities, and multidimensional scaling. This report discusses an extensive software toolkit that was developed for this research for analyzing and visualizing entities and links in a collection of journal papers. Additionally, an extensive case study is presented, analyzing and visualizing 60 years of anthrax research. When dealing with complex networks that consist of cascaded bipartite networks, the treatment presented here provides a general mathematical framework for all aspects of analysis of static network structure and network dynamic growth. As such, it provides a basic paradigm for thinking about and modeling such networks: computing direct and indirect links, expressing and analyzing statistical distributions of network characteristics, describing network growth, deriving feature vectors, clustering, and visualizing network structure and growth.

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