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Multifractal analysis (MFA) is a useful tool to systematically describe thespatial heterogeneity of both theoretical and experimental fractal patterns.One of the widely used methods for fractal analysis is box-covering. It isknown to be NP-hard. More severely, in comparison with fractal analysisalgorithms, MFA algorithms have much higher computational complexity. Amongvarious MFA algorithms for complex networks, the sandbox MFA algorithm behaveswith the best computational efficiency. However, the existing sandbox algorithmis still computationally expensive. It becomes challenging to implement the MFAfor large-scale networks with tens of millions of nodes. It is also not clearwhether or not MFA results can be improved by a largely increased size of atheoretical network. To tackle these challenges, a computationally-efficientsandbox algorithm (CESA) is presented in this paper for MFA of large-scalenetworks. Our CESA employs the breadth-first search (BFS) technique to directlysearch the neighbor nodes of each layer of center nodes, and then to retrievethe required information. Our CESA's input is a sparse data structure derivedfrom the compressed sparse row (CSR) format designed for compressed storage ofthe adjacency matrix of large-scale network. A theoretical analysis revealsthat the CESA reduces the time complexity of the existing sandbox algorithmfrom cubic to quadratic, and also improves the space complexity from quadraticto linear. MFA experiments are performed for typical complex networks to verifyour CESA. Finally, our CESA is applied to a few typical real-world networks oflarge scale.

A computationally-efficient sandbox algorithm for multifractal analysis of large-scale complex networks with tens of millions of nodes