Exactly scale-free scale-free networks

Many complex natural and physical systems exhibit patterns of interconnection that conform, approximately, to a network structure referred to as scale-free. Preferential attachment is one of many algorithms that have been introduced to model the growth and structure of scale-free networks. With so many different models of scale-free networks it is unclear what properties of scale-free networks are typical, and what properties are percularities of a particular growth or construction process. We propose that a maximum entropy process provides the best representation of what are typical properties of scale-free networks, and provides a standard against which real and algorithmically generated networks can be compared. We find that preferential attachment growth models do not yield typical realizations of scale-free networks because there is a latent structure, which we term “hub-centric”. We provide a method to generate or remove this latent hub-centric bias — thereby demonstrating exactly which features of preferential attachment networks are atypical of the broader class of scale-free networks. We are also able to statistically demonstrate whether real networks are typical realizations of scale-free networks; using a new surrogate generation method for complex networks, exactly analogous the the widely used surrogate tests of nonlinear time series analysis. ∗ [email protected] † [email protected] 1 ar X iv :1 30 9. 09 61 v3 [ ph ys ic s. so cph ] 1 5 N ov 2 01 3