  Multifractal analysis (MFA) is a useful tool to systematically describe the
spatial heterogeneity of both theoretical and experimental fractal patterns.
One of the widely used methods for fractal analysis is box-covering. It is
known to be NP-hard. More severely, in comparison with fractal analysis
algorithms, MFA algorithms have much higher computational complexity. Among
various MFA algorithms for complex networks, the sandbox MFA algorithm behaves
with the best computational efficiency. However, the existing sandbox algorithm
is still computationally expensive. It becomes challenging to implement the MFA
for large-scale networks with tens of millions of nodes. It is also not clear
whether or not MFA results can be improved by a largely increased size of a
theoretical network. To tackle these challenges, a computationally-efficient
sandbox algorithm (CESA) is presented in this paper for MFA of large-scale
networks. Our CESA employs the breadth-first search (BFS) technique to directly
search the neighbor nodes of each layer of center nodes, and then to retrieve
the required information. Our CESA's input is a sparse data structure derived
from the compressed sparse row (CSR) format designed for compressed storage of
the adjacency matrix of large-scale network. A theoretical analysis reveals
that the CESA reduces the time complexity of the existing sandbox algorithm
from cubic to quadratic, and also improves the space complexity from quadratic
to linear. MFA experiments are performed for typical complex networks to verify
our CESA. Finally, our CESA is applied to a few typical real-world networks of
large scale.
