We provide a survey of the Kolkata index of social inequality, focusing in particular on income inequality. Based on the observation that inequality functions (such as the Lorenz function), giving the measures of income or wealth against that of the population, to be generally nonlinear, we show that the fixed point (like Kolkata index k) of such a nonlinear function (or related, like the complementary Lorenz function) offer better measure of inequality than the average quantities (like Gini index). Indeed the Kolkata index can be viewed as a generalized Hirsch index for a normalized inequality function and gives the fraction k of the total wealth possessed by the rich 1−k fraction of the population. We analyze the structures of the inequality indices for both continuous and discrete income distributions. We also compare the Kolkata index to some other measures like the Gini coefficient and the Pietra index. Lastly, we provide some empirical studies which illustrate the differences between the Kolkata index and the Gini coefficient.

We study the mathematical and economic structure of the Kolkata (k) index of income inequality. We show that the k-index always exists and is a unique fixed point of the complementary Lorenz function, where the Lorenz function itself gives the fraction of cumulative income possessed by the cumulative fraction of population (when arranged from poorer to richer). We show that the k-index generalizes Pareto's 80/20 rule. Although the k and Pietra indices both split the society into two groups, we show that k-index is a more intensive measure for the poor-rich split. We compare the normalized k-index with the Gini coefficient and the Pietra index and discuss when they coincide. We establish that for any income distribution the value of Gini coefficient is no less than that of the Pietra index and the value of the Pietra index is no less than that of the normalized k-index. While the Gini coefficient and the Pietra index are affected by transfers exclusively among the rich or among the poor, the k-index is only affected by transfers across the two groups.

Social inequalities are ubiquitous, and here we show that the values of the Gini ([Formula: see text]) and Kolkata ([Formula: see text]) indices, two generic inequality indices, approach each other (starting from [Formula: see text] and [Formula: see text] for equality) as the competitions grow in various social institutions like markets, universities and elections. It is further shown that these two indices become equal and stabilize at a value (at [Formula: see text]) under unrestricted competitions. We propose to view this coincidence of inequality indices as a generalized version of the (more than a) century old 80-20 law of Pareto. Furthermore, the coincidence of the inequality indices noted here is very similar to the ones seen before for self-organized critical (SOC) systems. The observations here, therefore, stand as a quantitative support toward viewing interacting socio-economic systems in the framework of SOC, an idea conjectured for years.

A popular measure for citation inequalities of individual scientists has been the Hirsch index (h). If for any scientist the number nc of citations is plotted against the serial number np of the papers having those many citations (when the papers are ordered from the highest cited to the lowest), then h corresponds to the nearest lower integer value of np below the fixed point of the non-linear citation function (or given by nc = h = np if both np and nc are a dense set of integers near the h value). The same index can be estimated (from h = s = ns) for the avalanche or cluster of size (s) distributions (ns) in the elastic fiber bundle or percolation models. Another such inequality index called the Kolkata index (k) says that (1 − k) fraction of papers attract k fraction of citations (k = 0.80 corresponds to the 80–20 law of Pareto). We find, for stress (σ), the lattice occupation probability (p) or the Kolkata Index (k) near the bundle failure threshold (σc) or percolation threshold (pc) or the critical value of the Kolkata Index kc a good fit to Widom–Stauffer like scaling h/[N/log⁡N] = f(N[σc−σ]α), h/[N/log⁡N]=f(N|pc−p|α) or h/[Nc/logNc]=f(Nc|kc−k|α), respectively, with the asymptotically defined scaling function f, for systems of size N (total number of fibers or lattice sites) or Nc (total number of citations), and α denoting the appropriate scaling exponent. We also show that if the number (Nm) of members of parliaments or national assemblies of different countries (with population N) is identified as their respective h − indexes, then the data fit the scaling relation Nm∼N/log⁡N, resolving a major recent controversy.

Social inequalities are ubiquitous and evolve towards a universal limit. Herein, we extensively review the values of inequality measures, namely the Gini (g) index and the Kolkata (k) index, two standard measures of inequality used in the analysis of various social sectors through data analysis. The Kolkata index, denoted as k, indicates the proportion of the `wealth’ owned by (1−k) fraction of the `people’. Our findings suggest that both the Gini index and the Kolkata index tend to converge to similar values (around g=k≈0.87, starting from the point of perfect equality, where g=0 and k=0.5) as competition increases in different social institutions, such as markets, movies, elections, universities, prize winning, battle fields, sports (Olympics), etc., under conditions of unrestricted competition (no social welfare or support mechanism). In this review, we present the concept of a generalized form of Pareto’s 80/20 law (k=0.80), where the coincidence of inequality indices is observed. The observation of this coincidence is consistent with the precursor values of the g and k indices for the self-organized critical (SOC) state in self-tuned physical systems such as sand piles. These results provide quantitative support for the view that interacting socioeconomic systems can be understood within the framework of SOC, which has been hypothesized for many years. These findings suggest that the SOC model can be extended to capture the dynamics of complex socioeconomic systems and help us better understand their behavior.