Sampling-based box-covering algorithm for renormalization of networks

Wei, Zong-Wen; Wang, Bing–Hong; Wu, Xingtong; He, Yu; Liao, Hao; Zhou, Mingyang

doi:10.1063/1.5093174

Cited by 9 publications

(8 citation statements)

References 19 publications

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“…Although kernel designs [48,49] are included in the RRG to guarantee the modelling capacity of different unit correlations, the modelled correlation patterns in kernel spaces can be efficiently approximated using random projections [53][54][55][56], hashing functions [51,52], and approximate nearest neighbor search [57,58]. Therefore, the RRG can realize significant reductions in time and memory cost compared with existing works [22,[35][36][37][38][39][40][41][42][43][44] (Fig. 5).…”

Section: Discussionmentioning

confidence: 99%

“…This demonstration serves as a starting point to explore more possibilities in future studies. As we have discussed, optimization-free [22,[35][36][37][38][39][40][41][42][43][44] and optimization-dependent RGs [23][24][25][26][27][28][29][30][31][32][33][34] have their own advantages and disadvantages in solving physics questions. While we have shown a scheme towards ideal optimization-free RG designs in the present work, there is no reason to ignore the possibility of combining the advantages of both kinds of RGs.…”

Section: Discussionmentioning

confidence: 99%

“…RGs). Different from the classic approaches proposed only for structure [35][36][37][38][39][40][41][42] or dynamics renormalization [22,43,44], the RRG can be applied to study criticality and symmetry in both system structures and dynamics. Its kernel space designs [48,49] enhance the expression ability of different kinds of correlations among system units to meet diverse research demands.…”

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confidence: 99%

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Fast renormalizing the structures and dynamics of ultra-large systems via random renormalization group

Xu,

Tian,

Sun

2024

Preprint

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Criticality and symmetry, studied by the renormalization groups, lie at the heart of modern physics theories of matters and complex systems. However, surveying these properties with massive experimental data is bottlenecked by the intolerable costs of computing renormalization groups on real systems. Here, we develop a time- and memory-efficient framework, termed as the random renormalization group, for renormalizing ultra-large systems (e.g., with millions of units) within minutes. This framework is based on random projections, hashing techniques, and kernel representations, which support the renormalization governed by linear and non-linear correlations. For system structures, it exploits the correlations among local topology in kernel spaces to unfold the connectivity of units, identify intrinsic system scales, and verify the existences of symmetries under scale transformation. For system dynamics, it renormalizes units into correlated clusters to analyze scaling behaviours, validate scaling relations, and investigate potential criticality. Benefiting from hashing-function-based designs, our framework significantly reduces computational complexity compared with classic renormalization groups, realizing a single-step acceleration of two orders of magnitude. Meanwhile, the efficient representation of different kinds of correlations in kernel spaces realized by random projections ensures the capacity of our framework to capture diverse unit relations. As shown by our experiments, the random renormalization group helps identify non-equilibrium phase transitions, criticality, and symmetry in diverse large-scale genetic, neural, material, social, and cosmological systems.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

mentioning

confidence: 99%

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Fast renormalizing the structures and dynamics of ultra-large systems via random renormalization group

Xu,

Tian,

Sun

2024

Preprint

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“…Recently, Wei et al (2019) proposed a novel way to tile a network, that could be considered as an "advanced version" and extension of the overlapping-box-covering algorithm, however, the authors did not compare it to the OBCA algorithm. Note that this proposal of Wei et al is rather a framework than a particular algorithm.…”

Section: Sampling-based Methodsmentioning

confidence: 99%

Comparative analysis of box-covering algorithms for fractal networks

2021

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Research on fractal networks is a dynamically growing field of network science. A central issue is to analyze the fractality with the so-called box-covering method. As this problem is known to be NP-hard, a plethora of approximating algorithms have been proposed throughout the years. This study aims to establish a unified framework for comparing approximating box-covering algorithms by collecting, implementing, and evaluating these methods in various aspects including running time and approximation ability. This work might also serve as a reference for both researchers and practitioners, allowing fast selection from a rich collection of box-covering algorithms with a publicly available codebase.

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“…Since Song et al [1] revealed the existence of the self-similarity in complex networks, the fractality of complex networks has attracted much attention with extensive investigations. This is due to its potential applications in various areas of science and technology [2][3][4][5][6][7][8][9][10][11][12][13]. For some complex systems with an inhomogeneous distribution of local density of their certain physical quantities, however, a unique fractal dimension is not sufficient to characterize their complexity.…”

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confidence: 99%

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

Ding,

Liu,

et al. 2020

Preprint

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The fractality of complex networks has attracted much attention with extensive investigations over the last 15 years. As a generalization of fractal analysis, 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 uses the minimum number of covering boxes to calculate the fractal dimension of complex networks, and 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. Thus, so far it has only been applied to small-scale complex networks of the size of about tens of thousands of nodes. 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. Distinct from the existing sandbox algorithm that uses the shortest-path distance matrix to obtain the required information for MFA of complex 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. The CESA is demonstrated to be effective, efficient and feasible through the MFA results of (u,v)-flower model networks from the 5th to the 12th generations. It enables us to study the multifractality of networks of the size of about 11 million nodes with a normal desktop computer. Furthermore, we have also found that increasing the size of (u,v)-flower model network does improve the accuracy of MFA results. Finally, our CESA is applied to a few typical real-world networks of large scale.

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Sampling-based box-covering algorithm for renormalization of networks

Cited by 9 publications

References 19 publications

Fast renormalizing the structures and dynamics of ultra-large systems via random renormalization group

Fast renormalizing the structures and dynamics of ultra-large systems via random renormalization group

Comparative analysis of box-covering algorithms for fractal networks

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

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