Problems with Fitting to the Power-Law Distribution

This article describes a gradient complex network model whose weights are proportional to the difference between uniformly distributed ``fitness'' values assigned to the nodes. It is shown analytically and experimentally that the strength (i.e. the weighted node degree) density of such a network model can be well approximated by a power law with $\gamma \approx 0.35$. Possible implications for neuronal networks topology and dynamics are also discussed.

Power-law distributions are typical macroscopic features occurring in almost all complex systems observable in nature. As a result, researchers in quantitative analyses must often generate random synthetic variates obeying power-law distributions. The task is usually performed through standard methods that map uniform random variates into the desired probability space. Whereas all these algorithms are theoretically solid, in this paper we show that they are subject to severe ...

In their recent work "Scale-free networks are rare", Broido and Clauset address the problem of the analysis of degree distributions in networks to classify them as scale-free at different strengths of "scale-freeness." Over the last two decades, a multitude of papers in network science have reported that the degree distributions in many real-world networks follow power laws. Such networks were then referred to as scale-free. However, due to a lack of a precise definition, the...

In this paper, we propose an optimization-based mechanism to explain power law distributions, where the function that the optimization process is seeking to optimize is derived mathematically, then the behavior and interpretation of this function are analyzed. The derived function shows some similarity to the entropy function in representing order and randomness; however, it also represents the energy, where the optimization process is seeking to maximize the number of elemen...

This article describes a methodology for fitting experimental data to the discrete power-law distribution and provides the results of a detailed simulation exercise used to calculate accurate cutoff values used to assess the fit to a power-law distribution when using the maximum likelihood estimation for the exponent of the distribution. Using massively parallel programming computing, we were able to accelerate by a factor of 60 the computational time required for these calcu...

We propose a possible relation between complex networks and gravity. Our guide in our proposal is the power-law distribution of the node degree in network theory and the information approach to gravity. The established bridge may allow us to carry geometric mathematical structures, which are considered in gravitational theories, to probabilistic aspects studied in the framework of complex networks and vice versa.

The three Power-Laws proposed by Faloutsos et al(1999) are important discoveries among many recent works on finding hidden rules in the seemingly chaotic Internet topology. In this note, we want to point out that the first two laws discovered by Faloutsos et al(1999, hereafter, {\it Faloutsos' Power Laws}) are in fact equivalent. That is, as long as any one of them is true, the other can be derived from it, and {\it vice versa}. Although these two laws are equivalent, they pr...

Many biological networks have been labelled scale-free as their degree distribution can be approximately described by a powerlaw distribution. While the degree distribution does not summarize all aspects of a network it has often been suggested that its functional form contains important clues as to underlying evolutionary processes that have shaped the network. Generally determining the appropriate functional form for the degree distribution has been fitted in an ad-hoc fash...

We analyze about two hundred naturally occurring networks with distinct dynamical origins to formally test whether the commonly assumed hypothesis of an underlying scale-free structure is generally viable. This has recently been questioned on the basis of statistical testing of the validity of power law distributions of network degrees by contrasting real data. Specifically, we analyze by finite-size scaling analysis the datasets of real networks to check whether purported de...

Preferential attachment is an appealing edge generating mechanism for modeling social networks. It provides both an intuitive description of network growth and an explanation for the observed power laws in degree distributions. However, there are often limitations in fitting parametric network models to data due to the complex nature of real-world networks. In this paper, we consider a semi-parametric estimation approach by looking at only the nodes with large in- or out-degr...