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.