Sandbox Algorithm Research Articles

An approach to complex network analysis on pp collisions at LHC energies

We have introduced the complex network analysis techniques such as Visibility Graph (VG), Horizontal Visibility Graph (HVG) and Sandbox (SB) algorithm as a mapping between complex network and time series measurements. This study explores the application of complex network-based analysis techniques to proton–proton [Formula: see text] collisions data at [Formula: see text], 7, 5.02 and 2.76[Formula: see text][Formula: see text]TeV to unravel the underlying dynamics of the particles and their interactions. To ensure the robustness and comprehensive analysis, we have incorporated UrQMD-, PYTHIA- and AMPT-simulated data sets of [Formula: see text] interactions at LHC energies that enable a thorough investigation of the underlying phenomena of the interactions. We have analyzed the fluctuations in pseudorapidity densities of singly charged particles created in [Formula: see text] interactions at LHC energies. We have also investigated some properties like symmetry scaling, correlation, clustering and scale-freeness among the produced particles. The findings from this study contribute to our understanding of the complex dynamics within [Formula: see text] collisions and have implications for the field of high-energy physics and particle phenomenology.

Multifractality of Complex Networks Is Also Due to Geometry: A Geometric Sandbox Algorithm.

Over the past three decades, describing the reality surrounding us using the language of complex networks has become very useful and therefore popular. One of the most important features, especially of real networks, is their complexity, which often manifests itself in a fractal or even multifractal structure. As a generalization of fractal analysis, the multifractal analysis of complex networks is a useful tool for identifying and quantitatively describing the spatial hierarchy of both theoretical and numerical fractal patterns. Nowadays, there are many methods of multifractal analysis. However, all these methods take into account only the fact of connection between nodes (and eventually the weight of edges) and do not take into account the real positions (coordinates) of nodes in space. However, intuition suggests that the geometry of network nodes' position should have a significant impact on the true fractal structure. Many networks identified in nature (e.g., air connection networks, energy networks, social networks, mountain ridge networks, networks of neurones in the brain, and street networks) have their own often unique and characteristic geometry, which is not taken into account in the identification process of multifractality in commonly used methods. In this paper, we propose a multifractal network analysis method that takes into account both connections between nodes and the location coordinates of nodes (network geometry). We show the results for different geometrical variants of the same network and reveal that this method, contrary to the commonly used method, is sensitive to changes in network geometry. We also carry out tests for synthetic as well as real-world networks.

Sandbox edge-based algorithm for multifractal analysis of complex networks

Multifractal analysis of complex networks has attracted increasing interest in various fields of science, because it can help us unfold the intricate structure of a variety of networks for better understanding the underlying mechanism of these networks. In this study, we propose a sandbox edge-based (SBEB) multifractal analysis algorithm, which introduces the idea of the sandbox algorithm into the existing edge-based box-counting (EBBC) algorithm. Instead of using boxes to cover the nodes of a given network, the SBEB algorithm uses boxes to cover the entries ‘1’ in the adjacency matrix of the network, where these entries ‘1’ represent the edges of the network. We apply the SBEB algorithm to model networks to verify its effectiveness and feasibility. The results show that the SBEB algorithm can effectively capture the multifractal nature of these model networks. Compared with the existing EBBC algorithm, our algorithm greatly improves the linearity and smoothness in estimating the mass exponents τ(q) and the generalized fractal dimensions D(q), thereby resulting in more accurate estimates. In addition, the SBEB algorithm inherits the advantage of the existing EBBC algorithm on studying small-diameter networks which are difficult to be dealt with by the traditional node-based box-covering algorithms. In application, we apply the SBEB algorithm to several real-world networks, and results indicate that this algorithm can uncover their genuine multifractal behaviors.

A Hybrid MPI-OpenMP Parallel Algorithm for the Assessment of the Multifractal Spectrum of River Networks

The possibility to create a flood wave in a river network depends on the geometric properties of the river basin. Among the models that try to forecast the Instantaneous Unit Hydrograph (IUH) of rainfall precipitation, the so-called Multifractal Instantaneous Unit Hydrograph (MIUH) rather successfully connects the multifractal properties of the river basin to the observed IUH. Such properties can be assessed through different types of analysis (fixed-size algorithm, correlation integral, fixed-mass algorithm, sandbox algorithm, and so on). The fixed-mass algorithm is the one that produces the most precise estimate of the properties of the multifractal spectrum that are relevant for the MIUH model. However, a disadvantage of this method is that it requires very long computational times to produce the best possible results. In a previous work, we proposed a parallel version of the fixed-mass algorithm, which drastically reduced the computational times almost proportionally to the number of Central Processing Unit (CPU) cores available on the computational machine by using the Message Passing Interface (MPI), which is a standard for distributed memory clusters. In the present work, we further improved the code in order to include the use of the Open Multi-Processing (OpenMP) paradigm to facilitate the execution and improve the computational speed-up on single processor, multi-core workstations, which are much more common than multi-node clusters. Moreover, the assessment of the multifractal spectrum has also been improved through a direct computation method. Currently, to the best of our knowledge, this code represents the state-of-the-art for a fast evaluation of the multifractal properties of a river basin, and it opens up a new scenario for an effective flood forecast in reasonable computational times.

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

Among various algorithms of multifractal analysis (MFA) for complex networks, the sandbox MFA algorithm behaves with the best computational efficiency. However, the existing sandbox algorithm is still computationally expensive for MFA of large-scale networks with tens of millions of nodes. It is also not clear whether 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 networks, our CESA employs the compressed sparse row format of the adjacency matrix and the breadth-first search technique to directly search the neighbor nodes of each layer of center nodes, and then to retrieve the required information. 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. Then the CESA is demonstrated to be effective, efficient, and feasible through the MFA results of (u,v)-flower model networks from the fifth 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.

An approach to explore exotic hadronic states in $$^{24}$$Mg-Ag/Br interactions at 4.5 A GeV/c in framework of complex network analysis

The visibility graph (VG) and horizontal visibility graph (HVG) have been recently introduced as a mapping between the time series and the complex networks. Here, the relativistic nucleus–nucleus interaction data, $$^{24}$$ Mg-Ag/Br at 4.5 A GeV/c are analyzed and used to characterizing the multifractal properties by extending VG, HVG and the sandbox algorithm (SB). In this work, the HVG–SB techniques have been applied to confirm the scale-freeness and the presence of fractal properties in the process of multiplicity fluctuation. We try to find deterministic information by means of precise determination of topological parameter from a complex network perspective. We have analyzed the degree distributions which are associated with event-wise pseudorapidity distributions, typical of any multifractality measurement, different patterns of symmetry scaling, scale-freeness, correlation and clustering among the produced particles. We try to find deterministic information by means of precise determination of topological parameter from a complex network perspective. It has been also observed that the multifractal variables were derived by SB algorithm for the experimental data sets to some extent distinct from the respective Monte Carlo simulated results.

Pronounced fluctuations of pions in ring-like events in 16O–Ag/Br interactions at 60AGeV/c in the framework of complex network analysis

Visibility graphs (VGs), horizontal visibility graphs (HVGs) and the sandbox algorithm (SB) are applied for multifractal characterization of complex network systems that are converted from time series measurements, are used to characterize the fluctuations in pseudorapidity densities of singly charged particles produced in [Formula: see text]O–AgBr interactions at 60[Formula: see text]A[Formula: see text]GeV. The work presents the analysis of ring-like and jet-like events in terms of multifractality characterization of [Formula: see text]O–AgBr interactions at 60[Formula: see text]A[Formula: see text]GeV. We systematically compared the experimental events of both ring- and jet-like events with Monte Carlo (MC) simulated events. The investigation reveals that the multifractal spectrum for the jet-like events and ring-like events is different and can be distinguished. The ring-like events have its parameters slightly higher than that of jet-like events. Further analysis shows that the strength of the nonstatistical fluctuations is larger for ring-like events than those of jet-like events. The SB method presented here appears to be more useful than the conventional methods used for multifractal characterization.

Multifractal approach for comparing road transport network geometry: The case of Spain

Spain is a highly decentralized country composed of 17 autonomous communities (regional political and administrative territories), where the power is divided between the central government and a limited self-government of the communities. In the case of the road network, the competences are not completely delegated but the management and planning of its roads are shared between the central state and the communities. This dual management has led to asymmetries in the road networks of the different autonomous communities that have not been characterized yet. Its complex geometry and the different size of the territories make difficult both, the characterization of each network, as well as the comparison between the road networks of the different territories. Multifractal analysis is a robust tool that allows to summarizing characteristic parameters of the network structures. One of its main features is that the results obtained from this analysis are independent of the scale. The sandbox algorithm has been extensively used for the estimation of the dimension function, D (q), because of the adequate construction of the entirely generalized dimension function and for avoiding the undesirable “border effects”. In this work, we are applying the sandbox algorithm for characterizing the Spanish road network. This study shows that road networks exhibit a multifractal behaviour depending on the contribution of underlying subnetworks. In order to measure the contribution of these subnetworks we have also performed a multifractal analysis taking into account two different approaches: the ownership and the capacity of the road. Results also suggest that Spanish road network has a remarkably tendency to fractality with a fractal dimension around 1.8. However, this behaviour is not identical in all autonomous communities. Considering that multifractal parameters are independent of the scale, we were able to group the road networks with similar multifractal structures in the autonomous communities’ networks, through a cluster analysis.

From standard alpha-stable Lévy motions to horizontal visibility networks: dependence of multifractal and Laplacian spectrum

In recent years, researchers have proposed several methods to transform time series (such as those of fractional Brownian motion) into complex networks. In this paper, we construct horizontal visibility networks (HVNs) based on the -stable Lévy motion. We aim to study the relations of multifractal and Laplacian spectrum of transformed networks on the parameters and of the -stable Lévy motion. First, we employ the sandbox algorithm to compute the mass exponents and multifractal spectrum to investigate the multifractality of these HVNs. Then we perform least squares fits to find possible relations of the average fractal dimension , the average information dimension and the average correlation dimension against using several methods of model selection. We also investigate possible dependence relations of eigenvalues and energy on , calculated from the Laplacian and normalized Laplacian operators of the constructed HVNs. All of these constructions and estimates will help us to evaluate the validity and usefulness of the mappings between time series and networks, especially between time series of -stable Lévy motions and HVNs.

Fractal and multifractal analysis of complex networks: Estonian network of payments

Complex networks have gained much attention from different areas of knowledge in recent years. Particularly, the structures and dynamics of such systems have attracted considerable interest. Complex networks may have characteristics of multifractality. In this study, we analyze fractal and multifractal properties of a novel network: the large scale economic network of payments of Estonia, where companies are represented by nodes and the payments done between companies are represented by links. We present a fractal scaling analysis and examine the multifractal behavior of this network by using a sandbox algorithm. Our results indicate the existence of multifractality in this network and consequently, the existence of multifractality in the Estonian economy. To the best of our knowledge, this is the first study that analyzes multifractality of a complex network of payments.

Multifractal analysis of multiparticle emission data in the framework of visibility graph and sandbox algorithm

Multiparticle emission data in nucleus–nucleus collisions are studied in a graph theoretical approach. The sandbox algorithm used to analyze complex networks is employed to characterize the multifractal properties of the visibility graphs associated with the pseudorapidity distribution of charged particles produced in high-energy heavy-ion collisions. Experimental data on 28Si+Ag/Br interaction at laboratory energy Elab=14.5A GeV, and 16O+Ag/Br and 32S+Ag/Br interactions both at Elab=200A GeV, are used in this analysis. We observe a scale free nature of the degree distributions of the visibility and horizontal visibility graphs associated with the event-wise pseudorapidity distributions. Equivalent event samples simulated by ultra-relativistic quantum molecular dynamics, produce degree distributions that are almost identical to the respective experiment. However, the multifractal variables obtained by using sandbox algorithm for the experiment to some extent differ from the respective simulated results.

Multifractal analysis of charged particle distributions using horizontal visibility graph and sandbox algorithm

Horizontal visibility graphs (HVGs) and the sandbox (SB) algorithm usually applied for multifractal characterization of complex network systems that are converted from time series measurements, are used to characterize the fluctuations in pseudorapidity densities of singly charged particles produced in high-energy nucleus–nucleus collisions. Besides obtaining the degree distribution associated with event-wise pseudorapidity distributions, the common set of observables, typical of any multifractality measurement, are studied in [Formula: see text]O-Ag/Br and [Formula: see text]S-Ag/Br interactions, each at an incident laboratory energy of 200 GeV/nucleon. For a better understanding, we systematically compare the experiment with a Monte Carlo model simulation based on the Ultra-relativistic Quantum Molecular Dynamics (UrQMD). Our results suggest that the HVG-SB technique is an efficient tool that can characterize multifractality in multiparticle emission data, and in some cases, it is even superior to other methods more commonly used in this regard.

Multifractality and Laplace spectrum of horizontal visibility graphs constructed from fractional Brownian motions

Many studies have shown that additional information can be gained on time series by investigating their associated complex networks. In this work, we investigate the multifractal property and Laplace spectrum of the horizontal visibility graphs (HVGs) constructed from fractional Brownian motions. We aim to identify via simulation and curve fitting the form of these properties in terms of the Hurst index H. First, we use the sandbox algorithm to study the multifractality of these HVGs. It is found that multifractality exists in these HVGs. We find that the average fractal dimension of HVGs approximately satisfies the prominent linear formula ; while the average information dimension and average correlation dimension are all approximately bi-linear functions of H when . Then, we calculate the spectrum and energy for the general Laplacian operator and normalized Laplacian operator of these HVGs. We find that, for the general Laplacian operator, the average logarithm of second-smallest eigenvalue , the average logarithm of third-smallest eigenvalue , and the average logarithm of maximum eigenvalue of these HVGs are approximately linear functions of H; while the average Laplacian energy is approximately a quadratic polynomial function of H. For the normalized Laplacian operator, and of these HVGs approximately satisfy linear functions of H; while and are approximately a 4th and cubic polynomial function of H respectively.

Multifractal analysis of weighted networks by a modified sandbox algorithm.

Complex networks have attracted growing attention in many fields. As a generalization of fractal analysis, multifractal analysis (MFA) is a useful way to systematically describe the spatial heterogeneity of both theoretical and experimental fractal patterns. Some algorithms for MFA of unweighted complex networks have been proposed in the past a few years, including the sandbox (SB) algorithm recently employed by our group. In this paper, a modified SB algorithm (we call it SBw algorithm) is proposed for MFA of weighted networks. First, we use the SBw algorithm to study the multifractal property of two families of weighted fractal networks (WFNs): “Sierpinski” WFNs and “Cantor dust” WFNs. We also discuss how the fractal dimension and generalized fractal dimensions change with the edge-weights of the WFN. From the comparison between the theoretical and numerical fractal dimensions of these networks, we can find that the proposed SBw algorithm is efficient and feasible for MFA of weighted networks. Then, we apply the SBw algorithm to study multifractal properties of some real weighted networks — collaboration networks. It is found that the multifractality exists in these weighted networks, and is affected by their edge-weights.

Determination of multifractal dimensions of complex networks by means of the sandbox algorithm.

Complex networks have attracted much attention in diverse areas of science and technology. Multifractal analysis (MFA) is a useful way to systematically describe the spatial heterogeneity of both theoretical and experimental fractal patterns. In this paper, we employ the sandbox (SB) algorithm proposed by Tél et al. (Physica A 159, 155-166 (1989)), for MFA of complex networks. First, we compare the SB algorithm with two existing algorithms of MFA for complex networks: the compact-box-burning algorithm proposed by Furuya and Yakubo (Phys. Rev. E 84, 036118 (2011)), and the improved box-counting algorithm proposed by Li et al. (J. Stat. Mech.: Theor. Exp. 2014, P02020 (2014)) by calculating the mass exponents τ(q) of some deterministic model networks. We make a detailed comparison between the numerical and theoretical results of these model networks. The comparison results show that the SB algorithm is the most effective and feasible algorithm to calculate the mass exponents τ(q) and to explore the multifractal behavior of complex networks. Then, we apply the SB algorithm to study the multifractal property of some classic model networks, such as scale-free networks, small-world networks, and random networks. Our results show that multifractality exists in scale-free networks, that of small-world networks is not obvious, and it almost does not exist in random networks.

Influence of image resolution and evaluation algorithm on estimates of the lacunarity of porous media

In recent years, experience has demonstrated that the classical fractal dimensions are not sufficient to describe uniquely the interstitial geometry of porous media. At least one additional index or dimension is necessary. Lacunarity, a measure of the degree to which a data set is translationally invariant, is a possible candidate. Unfortunately, several approaches exist to evaluate it on the basis of binary images of the object under study, and it is unclear to what extent the lacunarity estimates that these methods produce are dependent on the resolution of the images used. In the present work, the gliding-box algorithm of Allain and Cloitre [Phys. Rev. A 44, 3552 (1991)] and two variants of the sandbox algorithm of Chappard et al. [J. Pathol. 195, 515 (2001)], along with three additional algorithms, are used to evaluate the lacunarity of images of a textbook fractal, the Sierpinski carpet, of scanning electron micrographs of a thin section of a European soil, and of light transmission photographs of a Togolese soil. The results suggest that lacunarity estimates, as well as the ranking of the three tested systems according to their lacunarity, are affected strongly by the algorithm used, by the resolution of the images to which these algorithms are applied, and, at least for three of the algorithms (producing scale-dependent lacunarity estimates), by the scale at which the images are observed. Depending on the conditions under which the estimation of the lacunarity is carried out, lacunarity values range from 1.02 to 2.14 for the three systems tested, and all three of the systems used can be viewed alternatively as the most or the least "lacunar." Some of this indeterminacy and dependence on image resolution is alleviated in the averaged lacunarity estimates yielded by Chappard et al.'s algorithm. Further research will be needed to determine if these lacunarity estimates allow an improved, unique characterization of porous media.