A New Method for Generalizing Burr and Related Distributions
2022, Chakraborty, Tanujit, Das, Suchismita, Chattopadhyay, Swarup
A new method has been proposed to generalize Burr-XII distribution, also called Burr distribution, by adding an extra parameter to an existing Burr distribution for more flexibility. In this method, the exponent of the Burr distribution is modeled using a nonlinear function of the data and one additional parameter. The models of this newly introduced generalized Burr family can significantly increase the flexibility of the former Burr distribution with respect to the density and hazard rate shapes. Families expanded using the method proposed here is heavy-tailed and belongs to the maximum domain of attractions of the Frechet distribution. The method is further applied to yield three-parameter classical Pareto and generalized exponentiated distributions which shows the broader application of the proposed idea of generalization. A relevant model of the new generalized Burr family has been considered in detail, with particular emphasis on the hazard functions, stochastic orders, estimation procedures, and testing methods are derived. Finally, as empirical evidence, the new distribution is applied to the analysis of large-scale heavy-tailed network data and compared with other commonly used distributions available for fitting degree distributions of networks. Experimental results suggest that the proposed Burr distribution with nonlinear exponent better fits the large-scale heavy-tailed networks better than the popularly used Marhsall-Olkin generalization of Burr and exponentiated Burr distributions.
Searching for Heavy-Tailed Probability Distributions for Modeling Real-World Complex Networks
2022, Chakraborty, Tanujit, Chattopadhyay, Swarup, Das, Suchismita, Kumar, Uttam, Senthilnath, J.
Perhaps the most recent controversial topic in network science research is to determine whether real-world complex networks are scale-free or not. Recently, Broido and Clauset [A.D. Broido, A. Clauset, Nature Communication, 10, 1017 (2019)] asserted that the degree distributions of real-world networks are rarely power law under statistical tests. Such complex networks, including social, biological, information, temporal, and brain networks, are often heavy-tailed where the assumption on the scale-free nature of realworld heavy-tailed networks become insignificant as the complex system evolves over time. The failure of power law distribution in fitting the degree distribution data is mainly due to the presence of an identifiable non-linearity within the entire degree distribution in a log-log scale of a complex heavy-tailed network. In this study, we attempt to address this issue by proposing a new class of heavy-tailed probability distributions for modeling the entire degree distributions of complex networks. We introduce a new family of generalized Lomax models (GLM) to capture the non-linearity of these heavy-tailed networks. These newly introduced GLM-type distributions provide better fitting and greater flexibility to the entire node degree distribution of complex networks. Several statistical properties of the proposed model, such as extreme value and inferential statistical properties, are derived into this context. Interestingly, the GLM family belongs to the basin of attraction of Frechet distribution, a heavy-tailed extreme value distribution. Rigorous experimental analysis showcases the excellent performance of the proposed family of distributions while fitting the heavytailed real-world complex networks over fifty real-world datasets in comparison with benchmark probability models. Our results show that GLM-type distributions are not rare, able to model almost 90% of the tested networks accurately compared to benchmark probability models. INDEX TERMS Complex networks, heavy-tailed networks, degree distribution, Lomax distribution, extreme value properties.