A fuzzy similarity measure based on the centrality scores of fuzzy terms

The idea of bow tie graphs had generated interest in web graphs, web mining algorithms, and the study of hubs and authorities in web graphs[3]. A bow tie graph is a graph such that at the center of the bow tie is the knot, which is "the strongly connected core", the left bow consists of pages that eventually allow users to reach the core, but that cannot themselves be reached from the core, i.e., L, where L=(L-1,L-2,L-3,....,L-m), and the right bow consists of "termination" pages that can be accessed via links from the core but that do not link back into the core, i.e., R where R=(R-1,R-2,R-3,....,R-n), and where n is different from m. We use this idea to construct a new fuzzy similarity measure between 2 fuzzy concepts. We say that 2 fuzzy terms t(1) and t(2) are fuzzy epsilon(epsilon greater than or equal to 0) "Bow Tie similar" or "congruent toepsilonBT" iff. t(1) congruent toepsilonBT t(2) iff \C-t1,C-t2\ greater than or equal to epsilon and \C-t2,C-t1\ greater than or equal to epsilon where C-ti,C-tj is the centrality score distribution of term t(i) for term t(j). A centrality score C-ti is defined iteratively as: (C-ti)(k) = 1 at k = 0 (C-t1)(k+1) = Max(Min((Bdegrees (C-ti)(k) degrees AT, BT degrees (C-ti)(k) degrees A)) / parallel to Bdegrees (C-ti)(k) degrees AT + BT degrees(C-ti)(k) degrees A)parallel to where: degrees is the symbol of composition of 2 relations,T is the transpose of the mapping, parallel to is an absolute value, B is a fuzzy relation on Lx{(t1),t(2)}, where L = (L-1, L-2 L-3,...., L-m) (See above) ,A is a fuzzy relation on {t(1),t(2)}x R, where R = (R-1,R-2,R-3,...., R-n) (See above). Remember that the centrality score definition stems from the following observation on Bow Tie Graphs: (L-1, L-2, L-3,...., L-m) is t1 & (L-1, L-2, L-3,...., L-m) is t2 and t1 is (R-1, R-2, R-3,...... R-n) & t2 is (R-1,R-2,R-3,....., R-n) Our fuzzy similarity measure is a new fuzzy similarity measure and was never used before. We study different kinds of fuzzy membership functions for the relation A and B. We apply our fuzzy similarity measure to different terms that we picked from the dictionary at http: //www.eleves.ens.fr/home/senellar/. We compare those results to existing non-fuzzy similarity measures. Our results show that similarity between terms based on fuzzy epsilon "Bow Tie similar" is a better similarity between terms than the equivalent non-fuzzy similarity measures.