Uncovering patterns in heavy-tailed networks : A journey beyond scale-free

Real-world heavy-tailed networks are claimed to be scale-free, meaning that the degree distributions follow the classical power-law. But it is evident from a closer observation that there exists a clearly identifiable non-linear pattern in the entire degree distribution in a log-log scale. Thus, the classical power-law distribution is often inadequate to fit the large-scale complex network data sets. The presence of this non-linearity can also be linked to the recent debate on scarcity versus the universality of scale-free networks. The search, therefore, continues to develop probabilistic models that can efficiently capture the crucial aspect of heavy-tailed and long-tailed behavior of the entire degree distribution of real-world complex networks. This paper proposes a new variant of the popular Lomax distribution, termed as modified Lomax (MLM) distribution, which can efficiently fit the entire degree distribution of real-world networks. The newly introduced MLM distribution arises from a hierarchical family of Lomax distributions and belongs to the basin of attraction of Frechet distribution. Some interesting statistical properties of MLM including characteristics of the maximum likelihood estimates have been studied. Finally, the proposed MLM model is applied over several real-world complex networks to showcase its excellent performance in uncovering the patterns of these heavy-tailed networks.