Complex biological networks, encompassing metabolic pathways, gene regulatory systems, and protein-protein interaction networks, often exhibit scale-free structures characterized by heavy-tailed degree distributions. However, empirical studies reveal significant deviations from ideal power law behavior, underscoring the need for more flexible and accurate probabilistic models. In this work, we propose the Compounded Burr (CBurr) distribution, a novel four parameter family derived by compounding the Burr distribution with a discrete mixing process. This model is specifically designed to capture both the body and tail behavior of real-world network degree distributions with applications to biological networks. We rigorously derive its statistical properties, including moments, hazard and risk functions, and tail behavior, and develop an efficient maximum likelihood estimation framework. The CBurr model demonstrates broad applicability to networks with complex connectivity patterns, particularly in biological, social, and technological domains. Extensive experiments on large-scale biological network datasets show that CBurr consistently outperforms classical power-law, log-normal, and other heavy-tailed models across the full degree spectrum. By providing a statistically grounded and interpretable framework, the CBurr model enhances our ability to characterize the structural heterogeneity of biological networks.