Abstract
Heavy-tailed networks are often characterized in the literature by their degree distribution's similarity to a power law. However, many heavy-tailed networks in real life do not have power-law degree distributions, and in many applications the scale-free nature of the network is irrelevant so long as the network possesses hubs. Here we present the Cooke-Nieboer index (CNI), a non-asymptotic measure of the heavy-tailedness of a network's degree distribution which does not presume a power-law form. The CNI is easy to calculate, and clearly distinguishes between networks with power-law, exponential, and symmetric degree distributions.
Highlights
Most heavy-tailed distributions of interest fall into a subcategory known as the subexponential distributions, defined as follows [20]: if X1, . . . , Xn are independent and identically distributed (i.i.d.) random variables chosen from a subexponential distribution, lim x→∞
The Cooke-Nieboer index (CNI) of a partial periodic lattice (PPL) can be written in closed form as a 4m − 1 degree polynomial, given by the expression mmmm lattice ( p) =
Do the networks vary significantly or not? If we look at the degree distributions [Fig. 12(a)] from two particular months (May 2006 and August 2006) with very different tail indices (α = 5.0 and 1.2, respectively), we see that the two histograms are quite similar, suggesting that the CNI may be a more accurate representation of their heavy-tailed nature
Summary
A phase change in network science research occurred at the end of the last century, with the discovery that the relationships in many real-life systems have properties which can not be captured by Erdos-Rényi random graphs [1]. Hub-dominated networks are usually referred to as “scale-free networks” in the literature, which implies that their degree distribution P(x) corresponds in some way to a power law: P(x) ∼ x−α−1, α > 0. [12]) they cannot be heavy-tailed, let alone scale-free Researchers work around this by imagining that real-world networks are subsamples of some underlying infinite distribution [6] or generated via some well-defined process which would create a power-law distribution in the infinite-size limit [11]. This assumes, that some such distribution or process exists. We will give several examples where the CNI may give a more accurate representation than the tail index of the underlying network
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