Kurtosis: Is It an Appropriate Measure to Compare the Extent of Fat-tailedness of the Degree Distribution for Any Two Real- World Networks?

Kurtosis" has long been considered an appropriate measure to quantify the extent of fattailedness of the degree distribution of a complex real-world network. However, the Kurtosis values for more than one real-world network have not been studied in conjunction with other statistical measures that also capture the extent of variation in node degree. In this paper, we determine the Kurtosis values for a suite of 48 real-world networks along with measures such as SPR(K), Max(K)-Min(K), Max(K)-Avg(K), SD(K)/Avg(K), wherein SPR(K), Max(K), Min(K), Avg(K) and SD(K) represent the spectral radius ratio for node degree, maximum node degree, minimum node degree, average and standard deviation of node degree respectively. Contrary to the conceived notion in the literature, we observe that real-world networks whose degree distribution is Poisson in nature (characterized by lower values of SPR(K), Max(K)-Min(K), Max(K)-Avg(K), SD(K)/Avg(K)) could have Kurtosis values that are larger than that of realworld networks whose degree distribution is scale-free in nature (characterized by larger values of SPR(K), Max(K)-Min(K), Max(K)-Avg(K), SD(K)/Avg(K)). When evaluated for any two realworld networks among all the 48 real-world networks, the Kendall's concordance-based correlation coefficients between Kurtosis and each of SPR, Max(K)-Min(K), Max(K)-Avg(K) and SD(K)/Avg(K) are 0.40, 0.26, 0.34 and 0.50 respectively. Thus, we seriously question the appropriateness of using Kurtosis to compare the extent of fat-tailedness of the degree distribution of the vertices for any two real-world networks.