Fractal Connectivity Research Articles

Mapping physical problems on fractals onto boundary value problems within continuum framework

In this Letter, we emphasize that methods of fractal homogenization should take into account a loop structure of the fractal, as well as its connectivity and geodesic metric. The fractal attributes can be quantified by a set of dimension numbers. Accordingly, physical problems on fractals can be mapped onto the boundary values problems in the fractional-dimensional space with metric induced by the fractal topology. The solutions of these problems represent analytical envelopes of non-analytical functions defined on the fractal. Some examples are briefly discussed. The interplay between effects of fractal connectivity, loop structure, and mass distributions on electromagnetic fields in fractal media is highlighted. The effects of fractal connectivity, geodesic metric, and loop structure are outlined.

Multivariate Hadamard self-similarity: Testing fractal connectivity

While scale invariance is commonly observed in each component of real world multivariate signals, it is also often the case that the inter-component correlation structure is not fractally connected, i.e., its scaling behavior is not determined by that of the individual components. To model this situation in a versatile manner, we introduce a class of multivariate Gaussian stochastic processes called Hadamard fractional Brownian motion (HfBm). Its theoretical study sheds light on the issues raised by the joint requirement of entry-wise scaling and departures from fractal connectivity. An asymptotically normal wavelet-based estimator for its scaling parameter, called the Hurst matrix, is proposed, as well as asymptotically valid confidence intervals. The latter are accompanied by original finite sample procedures for computing confidence intervals and testing fractal connectivity from one single and finite size observation. Monte Carlo simulation studies are used to assess the estimation performance as a function of the (finite) sample size, and to quantify the impact of omitting wavelet cross-correlation terms. The simulation studies are shown to validate the use of approximate confidence intervals, together with the significance level and power of the fractal connectivity test. The test performance and properties are further studied as functions of the HfBm parameters.

Synchronization patterns: from network motifs to hierarchical networks.

We investigate complex synchronization patterns such as cluster synchronization and partial amplitude death in networks of coupled Stuart-Landau oscillators with fractal connectivities. The study of fractal or self-similar topology is motivated by the network of neurons in the brain. This fractal property is well represented in hierarchical networks, for which we present three different models. In addition, we introduce an analytical eigensolution method and provide a comprehensive picture of the interplay of network topology and the corresponding network dynamics, thus allowing us to predict the dynamics of arbitrarily large hierarchical networks simply by analysing small network motifs. We also show that oscillation death can be induced in these networks, even if the coupling is symmetric, contrary to previous understanding of oscillation death. Our results show that there is a direct correlation between topology and dynamics: hierarchical networks exhibit the corresponding hierarchical dynamics. This helps bridge the gap between mesoscale motifs and macroscopic networks.This article is part of the themed issue 'Horizons of cybernetical physics'.

Multi-chimera states and transitions in the Leaky Integrate-and-Fire model with nonlocal and hierarchical connectivity

The effects of nonlocal and fractal connectivity are investigated in a network of Leaky Integrate-and-Fire (LIF) elements. The idea of fractal coupling originates from the hierarchical topology of networks formed by neuronal axons, which transmit the electrical signals in the brain. If a number of LIF elements with finite refractory period are nonlocally coupled, multi-chimera states emerge whose multiplicity depends both on the coupling strength and on the refractory period. We provide evidence that the introduction of a hierarchical topology in the coupling induces novel complex spatial and temporal structures, such as nested chimera states and transitions between multi-chimera states with different multiplicities. These results demonstrate new complex patterns, as well as transitions between different multi-chimera states arising from the combination of nonlinear dynamics with the hierarchical coupling.

Analysis of Bone Quality on Panoramic Radiograph in Osteoporosis Research by Fractal Dimension

Objective: To Assess the correlation between different quality analysis parameters of trabecular pattern in digital panoramic radiographies and relations with forearm bone mass density (BMD) performed by DXA. Methods: The study was developed using panoramic and peripheral bone densitometry dual energy X-ray absorptiometry (DXA) of 68 patients, 9 males and 59 females (19 - 73 years old). In the panoramic radiographs, evaluation of the trabecular bone morphology through assessment of fractal dimension (FD), connectivity (C) and total number of “bright” pixels (ET) was performed. In DXA, the exam determines the bone mineral density of the forearm to identify who has a high risk of osteoporosis. Statistics analyzed the relationship of these exams and the contribution of dental radiographs in detecting patients at risk for osteoporosis. Results: The average age of subjects was 43.85. In the analysis of trabecular pattern, a significant correlation between the FD, ET and C factors in level of 5% (Pearson correlation test) was found. Correlation tests showed no significant correlation between DF and BMD. Conclusions: The analysis showed correlations with each other, detecting alterations in the trabecular pattern. It cannot be related to BMD with FD but should be taken into account that examining the bone or trabecular alveolar process, when, for example, diagnostic analysis of pre-implant bone quality, is required.

Interplay between functional connectivity and scale-free dynamics in intrinsic fMRI networks

Studies employing functional connectivity-type analyses have established that spontaneous fluctuations in functional magnetic resonance imaging (fMRI) signals are organized within large-scale brain networks. Meanwhile, fMRI signals have been shown to exhibit 1/f-type power spectra — a hallmark of scale-free dynamics. We studied the interplay between functional connectivity and scale-free dynamics in fMRI signals, utilizing the fractal connectivity framework — a multivariate extension of the univariate fractional Gaussian noise model, which relies on a wavelet formulation for robust parameter estimation. We applied this framework to fMRI data acquired from healthy young adults at rest and while performing a visual detection task. First, we found that scale-invariance existed beyond univariate dynamics, being present also in bivariate cross-temporal dynamics. Second, we observed that frequencies within the scale-free range do not contribute evenly to inter-regional connectivity, with a systematically stronger contribution of the lowest frequencies, both at rest and during task. Third, in addition to a decrease of the Hurst exponent and inter-regional correlations, task performance modified cross-temporal dynamics, inducing a larger contribution of the highest frequencies within the scale-free range to global correlation. Lastly, we found that across individuals, a weaker task modulation of the frequency contribution to inter-regional connectivity was associated with better task performance manifesting as shorter and less variable reaction times. These findings bring together two related fields that have hitherto been studied separately — resting-state networks and scale-free dynamics, and show that scale-free dynamics of human brain activity manifest in cross-regional interactions as well.

Beyond Scale-Free Small-World Networks: Cortical Columns for Quick Brains

We study to what extent cortical columns with their particular wiring boost neural computation. Upon a vast survey of columnar networks performing various real-world cognitive tasks, we detect no signs of enhancement. It is on a mesoscopic--intercolumnar--scale that the existence of columns, largely irrespective of their inner organization, enhances the speed of information transfer and minimizes the total wiring length required to bind distributed columnar computations towards spatiotemporally coherent results. We suggest that brain efficiency may be related to a doubly fractal connectivity law, resulting in networks with efficiency properties beyond those by scale-free networks.

Fractal dimension of structural inhomogeneities in granular YBCO superconductor in magnetic field

The influence of a fractal character of the structure of an YBCO type high-temperature superconductor on its current-voltage characteristic in a magnetic field has been studied. The fractal dimension and connectivity index of vortex transport channels are determined as functions of the magnetic field strength. It is established that the glassy state index corresponds to that of a vortex glass.

Self-organized stiffness in regular fractal polymer structures

We investigated membrane-like polymer structures of fractal connectivity such as Sierpinski gaskets and Sierpinski carpets applying the bond fluctuation model in three dimensions. Without excluded volume (phantom), both polymeric fractals obey Gaussian elasticity on larger scales determined by their spectral dimension. On the other hand, the swelling effect due to excluded volume is rather distinct between the two polymeric fractals: Self-avoiding Sierpinski gaskets can be described using a Flory-type mean-field argument. Sierpinski carpets having a spectral dimension closer to perfect membranes are significantly more strongly swollen than predicted. Based on our simulation results it cannot be excluded that Sierpinski carpets in athermal solvent show a flat phase on larger scales. We tested the self-consistency of Flory predictions using a virial expansion to higher orders. From this we conclude that the third virial coefficient contributes marginally to Sierpinski gaskets, but higher order virial coefficients are relevant for Sierpinski carpets.

In vitro structural changes in porous HA/β-TCP scaffolds in simulated body fluid

Porous scaffolds of biphasic calcium phosphate (hydroxyapatite/β-tricalcium phosphate (β-TCP)) have been fabricated and changes induced both in phase composition and porous architecture by immersion in simulated body fluid (SBF) under static and orbital stirring conditions have been studied. The starting porous scaffolds exhibit a low and randomized micro- and mesoporosity, an interconnected macroporosity centered at 100 and 0.6 μm, a fractal connectivity of D = 2.981 and total percent porosity of ca. 80%. After immersion for up to 60 days the micro- and mesoporosity increase slightly, which could be attributed to dissolution of the β-TCP phase confirmed by transmission electron microscopy. The effects of the change in the porous framework with SBF immersion time favor the bioactive behavior of the tested materials, inducing a nucleation and growth of a nanocrystalline apatite phase as the interconnected macroporosity centered at 0.6 μm is reduced. The macroporosity centered at 100 μm is still stable after 60 days in SBF. Therefore, these biphasic calcium phosphate porous scaffolds combine bioactive behavior with the stability of interconnected macroporosity over large periods of soaking time in SBF under static and orbital stirring conditions.

Elastic and structural properties of vanadium–lithium–borate glasses

The ternary xV2O5–(40 − x)Li2O–60B2O3 glass system, where x = 1, 2, 3, 4 and 6 mol%, was prepared by normal quenching. Ultrasonic velocities and attenuation were measured at room temperature using a pulse-echo technique. Various parameters, such as elastic moduli, micro-hardness, Poisson's ratio and Debye temperature, were determined from the measured densities and velocities. The composition dependence of these parameters, in addition to the glass-transition temperature, suggested that vanadium ions were incorporated into these glasses as a network modifier, resulting in the reconversion of BO4 tetrahedra to BO3 triangles by the breaking of B–O–B linkages and the formation of nonbridging oxygens (NBOs). The outcome was a reduction in network connectivity and rigidity with increasing V2O5 content. The results are explained quantitatively in terms of fractal bond connectivity, average atomic volume, network dimensionality, packing density, number of network bonds per unit volume, cross-link density and atomic ring size. The Makishima and Mackenzie model appears to be valid for the studied glasses when the fate of BO4 tetrahedra and creation of NBOs are taken into account.

Fractal connectivity of long-memory networks

Using the multivariate long memory (LM) model and Taylor expansions, we find the conditions for convergence of the wavelet correlations between two LM processes on an asymptotic value at low frequencies. These mathematical results, and a least squares estimator of LM parameters, are validated in simulations and applied to neurophysiological (human brain) and financial market time series. Both brain and market systems had multivariate LM properties including a "fractal connectivity" regime of scales over which wavelet correlations were invariantly close to their asymptotic value. This analysis provides efficient and unbiased estimation of long-term correlations in diverse dynamic networks.

Pressure dependence of elastic properties of low-silica calcium alumino-silicate glasses

The hydrostatic and uni-axial pressure dependence of elastic properties of a low-silica calcium alumino-silicate glass (LSCAS) is determined by ultrasonic pulse-echo techniques at room temperature. The experimental results are used to obtain the third-order elastic constants (TOECs) of these glasses. The pressure dependence of fractal bond connectivity of these glasses is discussed. The normal behavior of positive pressure dependence of ultrasonic velocities was observed for the glass. The pressure dependence of both shear modulus and bulk modulus are positive for these glasses.

Fractal planning for integral economic development

PurposeTo conceptualize a new approach to economic development that fully embraces its fractal complexity, providing a basis for sustained socioeconomic welfare within cultures that encourage collaborative democracy and social learning.Design/methodology/approachFollowing the premise that healthy development must follow the natural laws of growth that apply to all ecosystems, the paper examines fractal intricacy as the basis of economic systems that are able to sustain sufficient flows of energy and information to all sub‐systems. Two methodological approaches that emerge for planning are called hierarchical (fractal) coherence and fractal connectivity. The first refers to sufficient density and variety of nodes (firms, economic processes, customers, etc.) at all scales in the hierarchy of an economic system; and the second denotes multiple paths of connection between the nodes, to handle the necessary flows.FindingsThis approach highlights the fundamental importance of locally‐based entrepreneurship and democratic control, and also suggests new methods for measuring system interconnectedness at all scales, supplementing economic growth measures such as GDP.Research limitations/implicationsThe paper indicates the need for empirical research to calibrate and further refine the approach in real‐world settings.Practical implicationsThe paper outlines a fresh planning strategy for dealing with the geometrically worsening dimensions of uneven economic development and poverty, at all levels of scale from the local to the global.Originality/valueThe paper articulates a viable cybernetic alternative to prevailing economic development approaches that are based on the neo‐classical, neo‐liberal, and neo‐conservative models embodied in our present system of economic globalization.

Temperature dependence of elastic moduli of lanthanum gallogermanate glasses

Temperature dependence of sound velocities of a series of Lanthanum Gallogermanate glasses have been determined from ultrasonic pulse-echo and Brillouin scattering measurements ranging from room temperature up to and through the glass transition temperature. Both longitudinal and transverse velocities of these glasses are composition dependent. The density and index of refraction of the samples were also studied. The experimental results are used to obtain elastic moduli. The correlation of elastic stiffness, the crosslink density, and the fractal bond connectivity of the glass are discussed. The normal behavior of negative temperature dependence of elastic properties is observed in these glasses. A possible explanation of the observed discrepancy of high temperature sound velocity of these glasses from two different measurements is given.

Molecular theory of strain hardening of a polymer gel: Application to gelatin

The elasticity of gelatin gels at large deformation has been measured for various experimental conditions. The general pattern is that stress increases with strain in a nonlinear way up to the point where the gel fails. To interpret this nonlinear stress increase, we studied a number of molecular models by Monte Carlo simulation and by mean-field methods. The effect of finite polymer length is studied via the FENE model (finite extensible nonlinear polymer connections) and via the exact statistics of Kramers’ model (chains of freely rotating stiff rods) for a small number of elements per chain. To investigate the effect of fractal connections, the end-point distribution that comes forward from scaling theory has been generalized to arbitrary fractal dimension. Finally we studied a heterogeneous network model: connections formed by rods and coils. We also discuss the consequence of microphase separation. Combining experiment and theory we conclude the following: (i) The elastically active network connections in gelatin are most certainly not Gaussian. (ii) Strain hardening in gelatin can be attributed to either: (a) finite polymer length (the chain length between connection points should be some 2.5 times the persistence length), or (b) a fractal structure of the polymer strands (the fractal dimension should be roughly df=1.3–1.5), or (c) the presence of both stiff rods and flexible coils (the length of the rods should be 1.4–4.4 times the radius of gyration of the coils). (iii) Models b and c describe the experimental data significantly better than model a. From a single parameter (the fractal dimension) the fractal model correctly describes (1) the nonlinearity of the stress–strain curve, (2) the scaling of Young’s modulus with polymer concentration, (3) the scaling of neutron scattering intensity with wave number, and (4) it predicts the scaling exponent of the linear dynamic modulus with frequency in the glassy transition zone (no experimental data available). The experimental parameters found for the rod+coils model suggest a Rouse diffusion controlled growth mechanism for the rods. Although the theory presented here is applied to gelatin, its formulation is quite general, and its implications are also relevant for other strain hardening polymer gels.

Fractal Neural Network: Computational Performance as an Associative Memory

Associative memory with fractal connections was studied. We show numerically that the fractal neural network has a higher capability than randomly connected neural network for fractal pattern and fractally localized patterns, but not for random patterns. We discuss this result from a theoretical standpoint

Fractal Neural Network

Associative memory with fractal connections was studied. We show numerically that the fractal neural network has a higher capability than randomly connected neural network for fractal pattern and fractally localized patterns, but not for random patterns. We discuss this result from a theoretical standpoint.

Entropic elasticity of a regular fractal.

We apply the Monte Carlo method to investigate the entropic elasticity of a model structure with a regular fractal connectivity. We find deviations from Flory theory and from the simple scaling predictions used in polymer physics: unlike linear polymers, the system here maintains its shape on large length scales, and its relative fluctuations decrease with increasing system size, while the behavior of the elastic constants resembles that of energy-dominated models.

Flory theory of polymeric fractals - intersection, saturation and condensation

Abstract We consider swelling effects of polymeric fractals, recently introduced by Cates, by usual simple Flory arguments for the free energy. The Flory arguments can be formulated to give a unified view for all polymers, linear, branched, or percolation clusters, as long they are of fractal connectivity. If the size of solvent molecules, being fractals themselves, is comparable to the given cluster, new values of the fractal dimensions can be found. The upper critical dimension is reduced. This is due to usual screening of the excluded volume. By standing overlap and repulsive energies of fractals of different fractal dimensions we find condensation to non-fractal objects depending on the value of the fracton dimension. A melt of polymeric fractals of the same fractal dimension and the same size becomes compact if the spectral dimension exceeds a “critical” value. These considerations are of relevance concerning recent experiments, considering static and dynamic properties of mixtures of microgels and linear polymers of different or equal sizes.