Abstract
Through an elegant geometrical interpretation, the multi-fractal analysis quantifies the spatial and temporal irregularities of the structural and dynamical formation of complex networks. Despite its effectiveness in unweighted networks, the multi-fractal geometry of weighted complex networks, the role of interaction intensity, the influence of the embedding metric spaces and the design of reliable estimation algorithms remain open challenges. To address these challenges, we present a set of reliable multi-fractal estimation algorithms for quantifying the structural complexity and heterogeneity of weighted complex networks. Our methodology uncovers that (i) the weights of complex networks and their underlying metric spaces play a key role in dictating the existence of multi-fractal scaling and (ii) the multi-fractal scaling can be localized in both space and scales. In addition, this multi-fractal characterization framework enables the construction of a scaling-based similarity metric and the identification of community structure of human brain connectome. The detected communities are accurately aligned with the biological brain connectivity patterns. This characterization framework has no constraint on the target network and can thus be leveraged as a basis for both structural and dynamic analysis of networks in a wide spectrum of applications.
Highlights
Ultra-small world networks are characterized by smaller shortest path distances that scale as dmin ~ loglogN
The major attraction of its application stems from its capability to characterize the spatial and temporal irregularities that euclidean geometry fails to capture in real world physical systems, by an elegant interpretation of power-law behaviors
Its demonstrated effectiveness in characterizing complex systems motivates us to extend its formalism to the analysis of complex networks
Summary
Ultra-small world networks are characterized by smaller shortest path distances that scale as dmin ~ loglogN. The uncovered self-similarity in complex networks connects to the important fractal and multi-fractal geometry domain where a family of objects are distinguished based on their self-repeating patterns and invariability under scale-length operations. Such objects are known as fractal objects. Multi-fractals could be seen as an extension to fractals with increased complexity They are invariant by translation a distortion factor q needs to be considered to distinguish the details of different regions of the objects as a consequence of inhomogenous mass distribution. Multi-fractal analysis (MFA, see Methods for details) is proposed to capture the localized and heterogenous self-similarity by learning a generalized fractal dimension D(q) under different distortion factors q
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