Self-similarity of complex networks

Abstract

Complex networks have been studied extensively owing to their relevance to many real systems such as the world-wide web, the Internet, energy landscapes and biological and social networks. A large number of real networks are referred to as 'scale-free' because they show a power-law distribution of the number of links per node. However, it is widely believed that complex networks are not invariant or self-similar under a length-scale transformation. This conclusion originates from the 'small-world' property of these networks, which implies that the number of nodes increases exponentially with the 'diameter' of the network, rather than the power-law relation expected for a self-similar structure. Here we analyse a variety of real complex networks and find that, on the contrary, they consist of self-repeating patterns on all length scales. This result is achieved by the application of a renormalization procedure that coarse-grains the system into boxes containing nodes within a given 'size'. We identify a power-law relation between the number of boxes needed to cover the network and the size of the box, defining a finite self-similar exponent. These fundamental properties help to explain the scale-free nature of complex networks and suggest a common self-organization dynamics.

Networks formed from interdependent networks

“…b, A loop-like NON; P 1 is shown as a function of p for k = 6 and two values of q. The results are obtained using equation (19). Note that increasing q yields a first-order transition.…”

“…Dynamical processes, such as flow and electrical transport in heterogeneous networks, were shown to be significantly more efficient when compared with ErdAEs-Rényi networks 64,65 . Furthermore, it was shown that networks can also possess self-similar properties, so that under proper coarse graining (or, renormalization) of the nodes the network properties remain invariant 19 .However, these complex systems were mainly modelled and analysed as single networks that do not interact with or depend on other networks. In interacting networks, the failure of nodes in one network generally leads to the failure of dependent nodes in other networks, which in turn may cause further damage to the first network, leading to cascading failures and catastrophic consequences.…”

“…Dynamical processes, such as flow and electrical transport in heterogeneous networks, were shown to be significantly more efficient when compared with ErdAEs-Rényi networks 64,65 . Furthermore, it was shown that networks can also possess self-similar properties, so that under proper coarse graining (or, renormalization) of the nodes the network properties remain invariant 19 .…”

Mining the Synergistic Core Allosteric Modules Variation and Sequencing Pharmacological Module Drivers in a Preclinical Model of Ischemia

“…With the “sandpile model” and the “avalanche effect” as typical representation,43 exogenous‐endogenous collective dynamics may determine the next moment of an open system. As an important feature of system far‐from‐equilibrium (self‐organized criticality), self‐similarity has been proven to be the intrinsic property of complex network through the renormalization group theory 44. Interestingly, this property has also been extended to characterizing the community of the complex network 45.…”

Preventable H5N1 avian influenza epidemics in the British poultry industry network exhibit characteristic scales

“…Dybiec et al (2004) claim that it is impossible to stop epidemics on scale-free networks unless a large proportion of the population is treated. Others offer hope in identifying the most highly connected sites (hubs) as the most vulnerable part of such systems (Albert et al 2000;Callaway et al 2000;May & Lloyd 2001;Song et al 2005;Jeger et al 2007).…”