We demonstrate that the self-similarity of some scale-free networks with respect to a simple degreethresholding renormalization scheme finds a natural interpretation in the assumption that network nodes exist in hidden metric spaces. Clustering, i.e., cycles of length three, plays a crucial role in this framework as a topological reflection of the triangle inequality in the hidden geometry. We prove that a class of hidden variable models with underlying metric spaces are able to accurately reproduce the self-similarity properties that we measured in the real networks. Our findings indicate that hidden geometries underlying these real networks are a plausible explanation for their observed topologies and, in particular, for their self-similarity with respect to the degree-based renormalization. PACS numbers: 89.75.Hc, 05.45.Df Self-similarity and scale invariance are traditionally known as characteristics of certain geometric objects, such as fractals [1], or of field theories describing system dynamics near critical points of phase transitions [2] . In these cases, objects or physical systems are intrinsically embedded in metric spaces and distance scales in these spaces are natural scaling factors. In complex networks [3], scale invariance is traditionally restricted to the scale-free property of the distributions of node degrees, which in a vast majority of complex networks follow power laws of the form P (k) ∼ k −γ , γ ∈ [2, 3]. In search for more complete self-similar descriptions, several recent works [4] introduced box-covering renormalization procedures, applied them to a few real networks, and found that certain networks, e.g., the Web and some biological networks, have finite fractal dimensions and degree distributions that remain invariant. Despite this promising progress, self-similarity and scale invariance of complex networks are still not well defined in a proper geometric sense. The reason is that many complex networks are not explicitly embedded in any physical space. As such, they lack any metric structure, except the one that their graph abstractions induce by the collection of lengths of shortest paths between nodes. However, this observable topological metric is a poor source of length-based scaling factors. It does not have large lengths as it exhibits the small-world [5] or even ultrasmall-world [6] property, meaning that the characteristic path lengths grow not polynomially but (sub)logarithmically with the network size. The apparent absence of any other metric structures supports the common belief that complex networks cannot be invariant under geometric length scale transformations.
Serrano, Krioukov and Boguna REF prove that a class of hidden variable models with underlying metric spaces are able to accurately reproduce the self-similarity properties in real-world networks.
Self-similarity of complex networks and hidden metric spaces
Physical Review Letters 100, 078701 (2008)
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