A surrogate for networks -- How scale-free is my scale-free network?

We bring rigor to the vibrant activity of detecting power laws in empirical degree distributions in real-world networks. We first provide a rigorous definition of power-law distributions, equivalent to the definition of regularly varying distributions that are widely used in statistics and other fields. This definition allows the distribution to deviate from a pure power law arbitrarily but without affecting the power-law tail exponent. We then identify three estimators of th...

This work introduces a method for fitting to the degree distributions of complex network datasets, such that the most appropriate distribution from a set of candidate distributions is chosen while maximizing the portion of the distribution to which the model is fit. Current methods for fitting to degree distributions in the literature are inconsistent and often assume a priori what distribution the data are drawn from. Much focus is given to fitting to the tail of the distrib...

The study of complex networks has pursued an understanding of macroscopic behavior by focusing on power-laws in microscopic observables. Here, we uncover two universal fundamental physical principles that are at the basis of complex networks generation. These principles together predict the generic emergence of deviations from ideal power laws, which were previously discussed away by reference to the thermodynamic limit. Our approach proposes a paradigm shift in the physics o...

Complex networks across various fields are often considered to be scale free -- a statistical property usually solely characterized by a power-law distribution of the nodes' degree $k$. However, this characterization is incomplete. In real-world networks, the distribution of the degree-degree distance $\eta$, a simple link-based metric of network connectivity similar to $k$, appears to exhibit a stronger power-law distribution than $k$. While offering an alternative character...

This short communication uses a simple experiment to show that fitting to a power law distribution by using graphical methods based on linear fit on the log-log scale is biased and inaccurate. It shows that using maximum likelihood estimation (MLE) is far more robust. Finally, it presents a new table for performing the Kolmogorov-Smirnof test for goodness-of-fit tailored to power-law distributions in which the power-law exponent is estimated using MLE. The techniques presente...

Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment...

Recently, one paper in Nature(Papadopoulos, 2012) raised an old debate on the origin of the scale-free property of complex networks, which focuses on whether the scale-free property origins from the optimization or not. Because the real-world complex networks often have multiple traits, any explanation on the scale-free property of complex networks should be capable of explaining the other traits as well. This paper proposed a framework which can model multi-trait scale-free ...

Networks are everywhere and their many types, including social networks, the Internet, food webs etc., have been studied for the last few decades. However, in real-world networks, it's hard to find examples that can be easily comparable, i.e. have the same density or even number of nodes and edges. We propose a flexible and extensible NetSim framework to understand how properties in different types of networks change with varying number of edges and vertices. Our approach ena...

We introduce a new approach to constructing networks with realistic features. Our method, in spite of its conceptual simplicity (it has only two parameters) is capable of generating a wide variety of network types with prescribed statistical properties, e.g., with degree- or clustering coefficient distributions of various, very different forms. In turn, these graphs can be used to test hypotheses, or, as models of actual data. The method is based on a mapping between suitably...

A projective network model is a model that enables predictions to be made based on a subsample of the network data, with the predictions remaining unchanged if a larger sample is taken into consideration. An exchangeable model is a model that does not depend on the order in which nodes are sampled. Despite a large variety of non-equilibrium (growing) and equilibrium (static) sparse complex network models that are widely used in network science, how to reconcile sparseness (co...