Search

• We consider comparisons between granularities induced by object similarity measures and classical Pawlak indiscernibility on decision tables. • We interpret object similarity measures as potential refinements of the granularity induced by the indiscernibility relation. • We introduce the notion of (A , ρ , ν)-object measure, where A is an attribute subset, and ρ and ν are collections of numerical maps depending of pairs of vales. • We apply the general results of the first part of our work to the study of several classical per-attribute similarities. • We analyze the relationship between granularity, the (A , ρ , ν)-object measures and the previous specific per-attribute similarities. In this paper we investigate the mathematical foundations of the notion of similarity between objects in relation to the granulations on a decision table D. First of all, we compare the endogenous granulation induced by Pawlak's indiscernibility with the exogenous granulation induced by a similarity measure ζ defined on pairs of objects and assuming values in the unit interval. To this aim, the starting point of our analysis is the introduction of the notion of refinement of the granulation induced by an attribute subset A through the object similarity measure ζ. More in detail, we say that ζ refines the granulation induced by A if ζ assumes value 1 on a pair of objects if and only if they are A -indiscernible. Next, starting from two given families ρ and ν of numerical maps defined on pairs of admissible values of D , we determine a broad class of potential similarity measures on the objects of D refining, sometimes under some specific additional hypotheses, the A -granulation on the object set of D. With regard to a such class of similarity measures, we establish several mathematical properties. Finally, we focus our attention to the analysis of specific pairs of numerical maps ρ and ν that have been classically studied in literature and, for each of them, we exhibit the main properties with respect to the aforementioned refinement of granulation. [ABSTRACT FROM AUTHOR]

Recently, we have shown that if the i th node of the Barabási-Albert (BA) network is characterized by the generalized degree q i (t) = k i (t) t i β / m , where k i (t) ~ tβ and m are its degree at current time t and at birth time t i , then the corresponding distribution function F (q, t) exhibits dynamic scaling. Applying the same idea to our recently proposed mediation-driven attachment (MDA) network, we find that it too exhibits dynamic scaling but, unlike the BA model where β = 1 / 2 , the exponent β of the MDA model assumes a spectrum of value 1/2 ≤ β ≤ 1. Moreover, we find that the scaling curves for small m are significantly different from those of the larger m and the same is true for the BA networks albeit in a lesser extent. We use the idea of the distribution of inverse harmonic mean (IHM) of the neighbours of each node and show that the number of data points that follow the power-law degree distribution increases as the skewness of the IHM distribution decreases. Finally, we show that both MDA and BA models become almost identical for large m. [ABSTRACT FROM AUTHOR]