Network Fractal Dimension Research Articles

Repeatability and Reproducibility of Retinal Fractal Dimension Measured with Swept-Source Optical Coherence Tomography Angiography in Healthy Eyes: A Proof-of-Concept Study

The retinal vascular network fractal dimension (FD) could be a promising imaging biomarker. Our objective was to evaluate its repeatability and reproducibility in healthy eyes. A cross-sectional study was undertaken with young, healthy volunteers who had no reported cardiac risk factors or ocular disease history. For each participant, three SS-OCTA images (12 × 12 mm) were acquired using the Plex Elite 9000 (Carl Zeiss Meditec AG, Jena, Germany) by two ophthalmologists. Automated segmentation was obtained from both the superficial and deep capillary plexuses. FD was estimated by box counting. The intraclass correlation coefficients (ICC) were used as measures for repeatability and reproducibility. A total of 43 eyes of healthy volunteers were included. The mean ± standard deviation (SD) age was 30 ± 6.2 years. The results show good repeatability. The ICC was 0.722 (95% CI, 0.541–0.839) in the superficial capillary plexus and 0.828 (95% CI, 0.705–0.903) in the deep capillary plexus. For reproducibility, the ICC was 0.651 (95% CI, 0.439–0.795) and 0.363 (95% CI, 0.073–0.596) at the superficial and deep capillary plexus, respectively. In this study, the FD of the vascular network measured via SS-OCTA showed good repeatability and reproducibility in healthy participants.

Centre including eccentricity algorithm for complex networks

In this work, the authors propose a new centre including eccentricity algorithm, to define the fractal dimension of networks. The authors did the fractal analysis of the real Escherichia coli network and a model UV-flower network and confirmed that these networks are fractals . The fractal dimensions ( D $D$ ) of these networks are calculated and D $D$ = 2.485 for the real E. coli network and D $D$ = 2.1 for the UV-flower network is obtained. Also, the authors defined the fractal dimensions of real social networks and compared their method with Song's, Zhang's and Zheng's methods. Furthermore, the authors’ algorithm can solve situations with single-node boxes at the edges of the network and can cover networks with minimum number of boxes. The authors believe that centre including eccentricity algorithm is competitive among existing algorithms and can be used to evaluate fractal properties of complex networks.

Comprehensive identification of statistical homogeneity of fractured rock masses for a candidate HLW repository site, China

Determining the statistically homogeneous regions is important for engineering geological modeling and promoting systematic research on regional rock mass characteristics and rock engineering designs. This study proposes a comprehensive framework considering multiple discontinuity geometry parameters (e.g., orientation, trace length, density, and cluster characteristics.) to distinguish statistical homogeneity of fractured rock masses. First, based on the equal-area plot method (Schmidt net), the quantitative judgment ability of the Miller method for determining the similarity between two discontinuity orientation populations was improved by optimizing block network division scheme and blank area merging principle. Then, considering the distribution characteristics of discontinuity in rock mass space and the variation features of discontinuity at different geographical locations, a three-dimensional fracture network model, established using the discontinuity geometry parameters obtained from field investigation, was combined with the fractal dimension method (called the network fractal dimension method; NFD) to further evaluate the statistical homogeneity of fractured rock mass. The proposed approaches were subsequently applied to the identification of the statistical homogeneity of several typical granite fracture outcrops in the Beishan area, which is a candidate site for high-level radioactive waste (HLW) repository of China. The case study reveals that the optimized Miller method is more flexible and applicable than the conventional orientation-based methods, especially in dealing with the complex situation of the coexistence of gentle dip angle and steep dip angle as well as clustering characteristics of discontinuity. The NFD method was used to unify the influence of the main discontinuity characteristic parameters for a logical and unbiased evaluation of the statistical homogeneity. The comprehensive framework developed herein integrates the advantages of the two methods and reduces the interference of subjective evaluation. Therefore, it can be employed to efficiently determine the statistical homogeneity of rock masses at different locations at a large scale without concern over the deviation caused by different directions and size of the compared areas.

Analysis of blood flow characteristics in fractal vascular network based on the time fractional order

Fractional calculus has shown good adaptability in describing the mechanical properties of viscoelastic materials. Based on fractional order theory, the characteristics of blood flow in the fractal network of blood vessels are analyzed, and experiments are carried out by using a mixed solution to prove the results of simulation analysis that contains glycerol, gelatin, sodium chloride, etc. Under different time of fractional order conditions, the effects of Reynolds number, vascular network fractal dimension, and bifurcation series conditions, the blood flow characteristics in fractal vascular network are analyzed. The analysis results show that the blood flow increases with the increase in time fractional order and the flow resistance decreases with the increase in time fractional order. The blood flow law curve based on the equivalent Casson fluid theory has the same trend between the orders α = 0.9 and α = 1.0 of the time fractional order curve, so the order of time fractional order can describe the blood flow characteristics of equivalent Casson fluid. The experimental data are distributed on both sides of the theoretical calculation curve and the relative error is small, which is in good agreement with the effect of vascular fractal parameters obtained by integer order on blood.

Fractal analysis of recurrence networks constructed from the two-dimensional fractional Brownian motions.

In this study, we focus on the fractal property of recurrence networks constructed from the two-dimensional fractional Brownian motion (2D fBm), i.e., the inter-system recurrence network, the joint recurrence network, the cross-joint recurrence network, and the multidimensional recurrence network, which are the variants of classic recurrence networks extended for multiple time series. Generally, the fractal dimension of these recurrence networks can only be estimated numerically. The numerical analysis identifies the existence of fractality in these constructed recurrence networks. Furthermore, it is found that the numerically estimated fractal dimension of these networks can be connected to the theoretical fractal dimension of the 2D fBm graphs, because both fractal dimensions are piecewisely associated with the Hurst exponent H in a highly similar pattern, i.e., a linear decrease (if H varies from 0 to 0.5) followed by an inversely proportional-like decay (if H changes from 0.5 to 1). Although their fractal dimensions are not exactly identical, their difference can actually be deciphered by one single parameter with the value around 1. Therefore, it can be concluded that these recurrence networks constructed from the 2D fBms must inherit some fractal properties of its associated 2D fBms with respect to the fBm graphs.

A new Systemic Taxonomy of Cyber Criminal activity

Cybercrime commonly refers to a broad range of different criminal activities that involve computers and information systems, either as primary tools or as primary targets. Cybercrime Science combines the methodology of Crime Science with the technology of Information Security. The few existing taxonomies of Cybercrime provide only general insights into the benefits of information structures; they are neither complete nor elaborated in a systemic manner to provide a proper framework guided by real system-principles. The main problem with such taxonomies is the inability to dynamically upgrade, which is why there is no timely cybersecurity actions. The current and past approaches were based mainly on the technical nature of cyberattacks and such approaches classified the impact of the activities from a criminological perspective. In this article, we present a systemic taxonomy of Cybercrime, based on definitions of the field items and the related data specifications. We develop a new method for estimating the fractal dimension of networks to explore a new taxonomy of Cybercrime activity. This method can serve to dynamically upgrade taxonomy and thus accelerate the prevention of cybercrime.

Path optimization of box-covering based routing to minimize average packet delay in software defined network

A routing algorithm plays a major role in Data communication across an inter-network. Software-Defined Networking (SDN) is a new era of computer networking that separates Data plane from Control plane which controls, changes and manages the network behavior dynamically via open interfaces with the help of SDN Controllers. In SDN, the networking devices route packets in the form of flows with the help of flow rules given by the controllers and the flow rules are stored in the corresponding flow tables of the networking devices. The Box-Covering (BC) algorithm is an existing algorithm used in SDN for achieving renormalization of networks by calculating the fractal dimension of networks and covering the network with the minimum possible number of boxes and Dijkstra’s algorithm is used for calculating shortest paths. The proposed work focuses on Path Optimization in the Box-Covering-Based Routing (BCR) algorithm, which is an existing algorithm used in large-scale SDN. In the proposed work, a Link-Weight system has been used for bounding the Link Utilization of the shortest path between the Source and the Destination to minimize the Average Packet Delay in the Network. The results show that the proposed work reduces the Average Packet Delay compared to the existing algorithms such as Dijkstra’s algorithm, Graph Compression algorithm and BCR algorithm and also improves the performance of the network.

A Fractal Model for Characterizing Hydraulic Properties of Fractured Rock Mass under Mining Influence

In this paper, two basic assumptions are introduced: (1) The number and length distribution of fractures in fractured rock mass are in accordance with the fractal law. (2) Fluid seepage in the fractures satisfies the cubic law. Based on these two assumptions, the fractal model of parallel seepage and radial seepage in fractured rock mass is established, and the seepage tensor of fracture network which reflects the geometric characteristics and fractal characteristics of fracture network under two kinds of seepage is derived. The influence of fracture geometry and fractal characteristics on permeability is analyzed, and the validity and accuracy of the model are verified by comparing the calculated results of the theoretical model and physical model test. The results show that the permeability coefficient K of fracture network is a function of the geometric (maximum crack length Lmax, fractured horizontal projection length L0, diameter calculation section porosity Φ, fracture strike α, and fracture angle θ) and fractal characteristics (fracture network fractal dimension Df and seepage flow fractal dimension DT). With the increase of fractal dimension Df, the permeability coefficient increases. With the increase of DT, the permeability coefficient decreases rapidly. And the larger the Df (Df>1.5), the greater the change of permeability coefficient K with DT.

Structuration, elastic properties scaling, and mechanical reversibility of candelilla wax oleogels with and without emulsifiers.

The crystal network development, elastic properties scaling behavior, and mechanical reversibility of candelilla wax (CW) oleogels with and without emulsifiers were studied. Saturated monoglycerides (MG) and polyglycerol polyricinoleate (PGPR) were added at 1 or 2 times the critical micelle concentration. Although the micelles of both emulsifiers act as nucleation sites for the mixture of aliphatic acids and alcohols of CW, they did not affect the oleogel's thermodynamic stability. It was established that the crystal network of CW consists of at least two types of crystals, one rich in n-hentriacontane and other rich in aliphatic acids. Both crystals species contributed significantly to the oleogel elasticity. The elastic properties scaling behavior of CW oleogels fitted the fractal model within the weak-link regime. The setting temperature and added emulsifier modified the crystal network fractal dimension. During shearing, oleogels had massive breaking of junction zones, causing the loss of fractality in the crystal network, which in turn decreased the system's elasticity.

Hydrologic Impact of Urbanization on Catchment and River System Downstream from Taihu Lake

Gao, Y.; Wang, H.; Lu, X.; Xu, Y.; Zhang, Z., and Schmidt, A.R., 2018. Hydrologic impact of urbanization on catchment and river system downstream from Taihu Lake. In: Wang, D. and Guido-Aldana, P.A. (eds.), Select Proceedings from the 3rd International Conference on Water Resource and Environment (WRE2017). Journal of Coastal Research, Special Issue No. 84, pp. 82–88. Coconut Creek (Florida), ISSN 0749-0208.This study examined the hydrologic impacts from rapid urbanization of a river valley and associated floodplain. The catchment downstream of the Taihu Lake was selected as the study region. Remote sensing images and topographic data for the period of post-1990 were employed to estimate hydrologic indicators for analyzing evolutionary features of the river system. Time series methods were adopted to analyze the hydrological effects on the watershed and river network resulting from rapid urbanization over the past 20 years. The results demonstrated that: (1) the number, length, and drainage density of rivers decreased from 1991 to 2006. The overall number of rivers decreased by 36.3%, and the overall river length was reduced by 25.5%; (2) the relationship between river length, river number, and stream order showed a geometric series over time, indicating a self-similarity in the river network that persisted even as the network changed due to urbanization; (3) the river network fractal dimension and complexity showed a decreasing trend over time, which indicated a simplification of the river system. In 1991, 2001, and 2006, the river network fractal dimension values were 2.24, 2.12, and 1.9, respectively; (4) excluding the influence of meteorological factors, there were at different levels increases in regional and seasonal precipitation. In addition, the difference in precipitation between urban and suburban areas increased gradually. Water levels also rose when the influence of precipitation was excluded.

A Route toward Ultrasensitive Layered Carbon Based Piezoresistive Sensors through Hierarchical Contact Design.

Ultrahigh sensitive piezoresistive sensors at small deformation are highly desired in many applications. Here, we propose a hierarchical contact design concept and implement it through a direct laser writing technique for fabricating layered carbon piezoresistive sensors with ultrahigh sensitivity. Sensors with unprecedented gauge factors (∼5000-10 000) at small deformation (ε < 0.1%) were successfully fabricated and demonstrated for their use in sensing both static and high-frequency (20-30 kHz) dynamic mechanical loads. A simple basic structure unit (BSU) contact network model was developed for understanding the importance of the BSU/BSU contact strength and network fractal dimension in dictating the piezoresistive sensitivity of the layered carbon piezoresistive sensors with designed hierarchical contact structures. The hierarchical contact design concept and the contact network model proposed in our work could open a general route for developing ultrasensitive piezoresistive sensors based on granular matter and composite materials.

Phase Behavior and Micro‐Structure of Fat–Oil Mixtures: Engineering the Shape of Fat Clusters

AbstractThe mechanical properties of fat–oil mixtures are dictated by the morphology of the fat crystal network, which in turn is strongly affected by the processing conditions. In this work the effect of uniform (linear) cooling rates on the size and shape of tribehenin (fat) clusters in isopropyl myristate (oil) was studied. The state of the fat–oil mixtures was comprehensively analyzed by establishing pseudo‐phase diagrams via visual observations, brightfield and polarization microscopy at various cooling rates and fat concentrations. The microstructure of the constituent fat clusters undergo a shape transition from bundles of needles at low cooling rates to spherical clusters at high cooling rates. The network structures formed with these different shaped fat clusters were examined using rheology. At a given cooling rate, the storage modulus (Gʹ) of fat–oil mixtures versus fat concentration followed a power law relationship. For a given fat concentration, the fat crystal networks comprising of bundles of needles showed higher Gʹ values as compared to fat networks made of spherical clusters. While the fractal dimension of networks cooled at different cooling rates varied over a small range of 2.78–2.86, considerable change in the pre‐exponential factor (γ) was observed. The results show the critical importance of processing conditions on the shape and size of fat clusters and its impact on the rheological properties of the fat crystal network.

Modeling the self-similarity in complex networks based on Coulomb’s law

Recently, self-similarity of complex networks have attracted much attention. Fractal dimension of complex network is an open issue. Hub repulsion plays an important role in fractal topologies. This paper models the repulsion among the nodes in the complex networks in calculation of the fractal dimension of the networks. Coulomb’s law is adopted to represent the repulse between two nodes of the network quantitatively. A new method to calculate the fractal dimension of complex networks is proposed. The Sierpinski triangle network and some real complex networks are investigated. The results are illustrated to show that the new model of self-similarity of complex networks is reasonable and efficient.

The Use of Cooling Rate to Engineer the Microstructure and Oil Binding Capacity of Wax Crystal Networks

The effects of cooling rate during melt crystallization of rice bran wax, sunflower wax, candelilla wax and a commercial peanut butter stabilizer in peanut oil were evaluated and correlated to their oil binding capacity. Rapid cooling upon crystallization decreased crystal length, decreased network pore area fraction, increased the fractal dimension of the crystal network, and lead to an increase in the oil binding capacity. Oleogels structured using 1 % sunflower wax exhibited the highest oil binding capacity, followed by candelilla wax and rice bran wax gels. The oil binding capacity of the commercial stabilizer was the lowest, but could be improved by crystallizing the material under high cooling rates. Linear correlation analysis revealed that the network fractal dimension decreases as pore area fraction increases, which was correlated to a greater oil loss. In general, the oil binding capacity of a gel can be significantly improved if the pore area fraction is decreased below 96 %, demonstrating how the fractal dimension and pore area fraction of a gel network can be modulated to tailor oil binding capacity.

Permeability of cracked porous solids through percolation approach

This paper investigates the permeability of microcracked porous solids incorporating random crack networks in terms of continuum percolation theory. Main factors of permeability include the geometry of crack networks, permeability of porous matrix, and crack opening. For the two-dimensional random crack networks, a new connectivity factor is defined to take into consideration the spanning cluster of cracks, fractal dimension of networks, and the size of a finite domain. For an infinite domain, the connectivity factor around a percolation threshold observes the scaling law, so this definition of connectivity is proved to be consistent with the percolation concepts. Geometric analysis reveals that the local clustering will not necessarily contribute to the global connectivity of networks. It is also found that too strong a local clustering of cracks will decrease the probability of the global percolation, and this adverse aspect of the local clustering effect has never been reported in the literature. The percolation threshold changes with the crack pattern of networks and the scaling exponents of percolation are not constant but depend on the fractal dimension of the crack networks. On the basis of connectivity and tortuosity of crack networks, the scaling law for permeability is established, K=K0(Km,b)(-c), taking into consideration the geometris characteristics through (-c), the permeability of porous matrix Km, and the crack opening aperture b. Then the permeability of a solid incorporating random crack networks is solved by finite element methods: all the cracks are idealized as 2-node elements and the matrix is divided into 6-node triangle elements. The fluid is assumed to be incompressible and Newtonian. With these assumptions the effective permeability of numerical samples is evaluated through Darcy's law. The scaling exponents of the permeability obtained numerically are very near to the theoretical values, and the impact of crack opening is less important as the crack density is far below the percolation threshold and the effect of crack opening becomes significant only as the crack density approaches the percolation threshold. Influence of crack opening on the permeability is strongly dependent on the opening aperture of the cracks. Finite element simulation results show that K0 depends on b through a power law near the percolation threshold and this dependence disappears as the ratio between the local permeability of crack and the matrix permeability exceeds 106.

Quantifying tissue microvasculature with speckle variance optical coherence tomography

In this Letter, we demonstrate high resolution, three-dimensional optical imaging of in vivo blood vessel networks using speckle variance optical coherence tomography, and the quantification of these images through the development of biologically relevant metrics using image processing and segmentation techniques. Extracted three-dimensional metrics include vascular density, vessel tortuosity, vascular network fractal dimension, and tissue vascularity. We demonstrate the ability of this quantitative imaging approach to characterize normal and tumor vascular networks in a preclinical animal model and the potential for quantitative, longitudinal vascular treatment response monitoring.

Accuracy of the ball-covering approach for fractal dimensions of complex networks and a rank-driven algorithm

Minimum box-covering method is a basic tool to measure fractal dimension of a network but unfortunately belongs to a family of NP-hard problems. Finding more accurate approximation in a relative shorter time is an important problem and attracts considerable attention. In this paper, the accuracy of the ball-covering approach for fractal dimension is measured by the relative error eta to the true fractal dimension, and its upper bound epsilon is analytically deduced by the linear least-squares method. When the ball-covering algorithm is applied to real-world networks, the relative error eta always stays in a narrow range which is far below its theoretical upper bound epsilon . It indicates that the ball-covering approach provides a helpful method to approximate the true fractal dimension. Then a rank-driven ball-covering algorithm is presented. It is demonstrated that the ranking mechanism ensures the search steps of our algorithm for finding balls are obviously less than that of the original algorithm introduced by Kim [Chaos 17, 026116 (2007)] and our algorithm is much more efficient.

Characterization of complex networks by higher order neighborhood properties

A concept of higher order neighborhood in complex networks, introduced previously [Phys. Rev. E 73, 046101 (2006)], is systematically explored to investigate larger scale structures in complex networks. The basic idea is to consider each higher order neighborhood as a network in itself, represented by a corresponding adjacency matrix, and to settle a plenty of new parameters in order to obtain a best characterization of the whole network. Usual network indices are then used to evaluate the properties of each neighborhood. The identification of high order neighborhoods is also regarded as intermediary step towards the evaluation of global network properties, like the diameter, average shortest path between node, and network fractal dimension. Results for a large number of typical networks are presented and discussed.

Computer simulation of fractal dimensions of fat crystal networks

AbstractThe rheological properties of fat‐structured products are determined by the microstructure of their fat crystal networks, which can be quantified by using microscopical and rheological techniques. Of particular interest to this study is the quantification of the fractal dimension of the network using these two techniques. Fractal dimensions determined by polarized light microscopy include box‐counting, particle‐counting, and Fouriertransform fractal dimensions, whereas the fractal dimensions determined by small deformation dynamic rheology exploit the dependence of the storage modulus on the solids' volume fraction. This work reveals that different microscopy fractal dimensions are sensitive to different microstructural factors within the fat crystal network, and thus have different physical meanings. The boxcounting fractal dimension, Db, increases with increases in crystal size and area fraction of the fat crystals, whereas the particlecounting fractal dimension, Df, is sensitive to the radial distribution pattern of fat crystals; and the Fourier‐transform fractal dimension, DFT, decreases with increasing crystal size. In the studies on the macroscopic physical properties of fat crystal networks, it is necessary to find the determining structural characteristics and then use the fractal dimensions that are most closely related.

Self-affine nature of the stress-strain behavior of an elastic fractal network

The stress-strain behavior of fractal networks possesses self-affine scaling associated with statistical scale invariance of damage patterns. We found that the damage part of stress-strain curve of fractal network and the corresponding rupture line are characterized by the same scaling (Hurst) exponent, H, which is not universal, rather it depends on the network fractal dimension as H= D B−1. Furthermore, the same exponent governs the changes in the stress-strain behavior as the strain rate increases. These results were reproduced by Monte Carlo simulations using a fractal version of fiber bundle model.