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. 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=(L₁, L₂ ,L₃,....,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=(R₁, R₂, R₃.....,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 t₁ and t₂ are fuzzy ε(ε>=0) "bow tie similar" or "≅εBT" iff: t₁≅εBT t₂ iff |C_t1,_t2| >=ε and |C_t2,_t1| >=ε, where C_ti,_tj is the centrality score distribution of term t₁ for term t_j. A centrality score C_tj is defined iteratively as: (C_ti)_k =1 at k=0; (C_tj)_k+1=Max(Min((B° (C_tj)_k ° AT, BT ° (C_tj)_k ° A ))/|| B ° (C_ti)_k ° AT + BT ° (C_ti)_k ° A)||, where, ° is the symbol of composition of 2 relations, T is the transpose of the mapping, || is an absolute value, B is a fuzzy relation on L×{_t1,t₂}, where L=(L₁, L₂, L₃,.....,L_m), A is a fuzzy relation on {t₁,t₂} × R, where R=(R₁, R₂, R₃,.....,R_n). Remember that the centrality score definition stems from the following observation on bow tie graphs: (L₁, L₂, L₃, ....., L_m) is t₁ & (L₁, L₂, L₃, .....L_m) is t2 and t1 is (R₁, R₂, R₃,.....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 ε "bow tie similar" is a better similarity between terms than the equivalent non-fuzzy similarity measures.