A series of papers by Hickey (1982, 1983, 1984) presents a stochastic ordering based on randomness. This paper extends the results by introducing a novel methodology to derive models that preserve stochastic ordering based on randomness. We achieve this by presenting a new family of pseudometric spaces based on a majorization property. This class of pseudometrics provides a new methodology for deriving the randomness measure of a random variable. Using this, the paper introduces the Gini randomness measure and states its essential properties. We demonstrate that the proposed measure has certain advantages over entropy measures. The measure satisfies the value validity property, provides an adequate extension to continuous random variables, and is often more appropriate (based on sensitivity) than entropy in various scenarios.

Measuring inequalities in a multidimensional framework is a challenging problem, which is common to most field of science and engineering. Nevertheless, despite the enormous amount of researches illustrating the fields of application of inequality indices, and of the Gini index in particular, very few consider the case of a multidimensional variable. In this paper, we consider in some details a new inequality index, based on the Fourier transform, that can be fruitfully applied to measure the degree of inhomogeneity of multivariate probability distributions. This index exhibits a number of interesting properties that make it very promising in quantifying the degree of inequality in datasets of complex and multifaceted social phenomena.

Recently, an alternative measure of uncertainty called extropy is proposed by Lad et al. [12]. The extropy is a dual of entropy which has been considered by researchers. In this article, we introduce an alternative measure of uncertainty of random variable which we call it cumulative residual extropy. This measure is based on the cumulative distribution function F. Some properties of the proposed measure, such as its estimation and applications, are studied. Finally, some numerical examples for illustrating the theory are included.

In Balaji and Mahmoud [1], the authors introduced a distance-based Gini index for rooted trees. In this paper, we introduce a degree-based Gini index (or just simply degree Gini index) for graphs. The latter index is a topological measure on a graph capturing the proximity to regular graphs. When applied across the random members of a class of graphs, we can identify an average measure of regularity for the class. Whence, we can compare the classes of graphs from the vantage point of closeness to regularity.

We develop a simplified computational formula for the degree Gini index and study its extreme values. We show that the degree Gini index falls in the interval [0, 1). The main focus in our study is the degree Gini index for the class of binary trees. Via a left-packing transformation, we show that, for an arbitrary sequence of binary trees, the Gini index has inferior and superior limits in the interval [0, 1/4]. We also show, via the degree Gini index, that uniform rooted binary trees are more regular than binary search trees grown from random permutations.

In this paper, we investigate the degree profile and Gini index of random caterpillar trees (RCTs). We consider RCTs which evolve in two different manners: uniform and nonuniform. The degrees of the vertices on the central path (i.e., the degree profile) of a uniform RCT follows a multinomial distribution. For nonuniform RCTs, we focus on those growing in the fashion of preferential attachment. We develop methods based on stochastic recurrences to compute the exact expectations and the dispersion matrix of the degree variables. A generalized Pólya urn model is exploited to determine the exact joint distribution of these degree variables. We apply the methods from combinatorics to prove that the asymptotic distribution is Dirichlet. In addition, we propose a new type of Gini index to quantitatively distinguish the evolutionary characteristics of the two classes of RCTs. We present the results via several numerical experiments.

We propose two distance-based topological indices (level index and Gini index) as measures of disparity within a single tree and within tree classes. The level index and the Gini index of a single tree are measures of balance within the tree. On the other hand, the Gini index for a class of random trees can be used as a comparative measure of balance between tree classes. We establish a general expression for the level index of a tree. We compute the Gini index for two random classes of caterpillar trees and see that a random multinomial model of trees with finite height has a countable number of limits in [0, ⅓], whereas a model with independent level numbers fills the spectrum (0, ⅓].