In the paper we show that the bibliographic data can be transformed into a collection of compatible networks. Using network multiplication different interesting derived networks can be obtained. In defining them an appropriate normalization should be considered. The proposed approach can be applied also to other collections of compatible networks. We also discuss the question when the multiplication of sparse networks preserves sparseness. The proposed approaches are illustrated with analyses of collection of networks on the topic "social network" obtained from the Web of Science.

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The node set of a two-mode network consists of two disjoint subsets and all its links are linking these two subsets. The links can be weighted. We developed a new method for identifying important subnetworks in two-mode networks. The method combines and extends the ideas from generalized cores in one-mode networks and from (p, q)cores for two-mode networks. In this paper we introduce the notion of generalized two-mode cores and discuss some of their properties. An efficient algorithm to determine generalized two-mode cores and an analysis of its complexity are also presented. For illustration some results obtained in analyses of real-life data are presented.Generalized Two-mode Cores, November 7, 2018 2/21 network analysis methods [3]. Direct methods for the analysis of two-mode networks are quite rare [4,5,6]. We can use bipartite statistics on degrees, generalized blockmodeling, (p, q)-cores, two-mode hubs and authorities, 4ring weights, bi-communities, two-mode clustering, bipartite cores, and some others. Many methods for identifying important subnetworks are available for one-mode networks (measures of centrality and importance, generalized cores, line islands, node islands, clustering, blockmodeling, etc.). We present a new direct method which can be used for the identification of important subnetworks in two-mode networks with respect to selected node properties.We combine the ideas from generalized cores in one-mode networks and from (p, q)-cores for two-mode networks into the notion of generalized twomode cores. We developed and implemented an algorithm for identifying generalized two-mode cores for selected node properties and given thresholds for both subsets of nodes. We also propose an algorithm to find the nested generalized two-mode cores for one fixed threshold value.In the next section we survey the works that contain the ideas we used for the development of our method. In Section 3 we present an algorithm for identifying the generalized two-mode cores. We list some node properties that are used as measures of importance. We also present some properties of generalized two-mode cores. We prove that for equivalent properties measured in ordinal scales the sets of generalized two-mode cores are the same. The algorithm, the proof of its correctness, and a simple analysis of its complexity are presented in Section 4. In Section 5 some results obtained in analyses of real-life data are presented.

We analyze the data about works (papers, books) from the time period 1990-2010 that are collected in Zentralblatt MATH database. The data were converted into four 2-mode networks (works × authors, works × journals, works × keywords and works × MSCs) and into a partition of works by publication year. The networks were analyzed using Pajek -a program for analysis and visualization of large networks. We explore the distributions of some properties of works and the collaborations among mathematicians. We also take a closer look at the characteristics of the field of graph theory as were realized with the publications.

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