Novel matrix forms of rough set flow graphs with applications to data integration

Pawlak's flow graphs have attracted both practical and theoretical researchers because of their ability to visualize information flow. In this paper, we invent a new schema to represent throughflow of a flow graph and three coefficients of both normalized and combined normalized flow graphs in matrix form. Alternatively, starting from a flow graph with its throughflow matrix, we reform Pawlak's formulas to calculate these three coefficients in flow graphs by using matrix properties. While traditional algorithms for computing these three coefficients of the connection are exponential in l, an algorithm using our matrix representation is polynomial in l, where l is the number of layers of a flow graph. The matrix form can simplify computation, improve time complexity, alleviate problems due to missing coefficients and hence help to widen the applications of flow graphs. Practically, data sets often reside at different sources (heterogeneous data sources). Their individual analysis at each source is inadequate and requires special treatment. Hence, we introduce a composition method for flow graphs and corresponding formulas for calculating their coefficients which can omit some data sharing. We provide a real-world experiment on the Promotion of Academic Olympiads and Development of Science Education Foundation (POSN) data set which illustrates a desirable outcome and the advantages of the proposed matrix forms and the composition method.