Multifractal analysis of the branch structure of diffusion-limited aggregates

Hanan, W. G.; Heffernan, D.

doi:10.1103/physreve.85.021407

Cited by 15 publications

(10 citation statements)

References 42 publications

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“…The crucial impact of the lattice anisotropy on the DLA growth naturally disappears for the intrinsically isotropic off-lattice DLA that had also been extensively investigated (see [38][39][40][41][42][43][44] and references therein). This rises an important question: how to explain the qualitative difference between the on-lattice and off-lattice DLA?…”

Section: Discussionmentioning

confidence: 99%

How anisotropy beats fractality in two-dimensional on-lattice diffusion-limited-aggregation growth

Beliaev

2017

Phys. Rev. E

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We study the fractal structure of diffusion-limited aggregation (DLA) clusters on a square lattice by extensive numerical simulations (with clusters having up to 10^{8} particles). We observe that DLA clusters undergo strongly anisotropic growth, with the maximal growth rate along the axes. The naive scaling limit of a DLA cluster by its diameter is thus deterministic and one-dimensional. At the same time, on all scales from the particle size to the size of the entire cluster it has a nontrivial box-counting fractal dimension which corresponds to the overall growth rate, which, in turn, is smaller than the growth rate along the axes. This suggests that the fractal nature of the lattice DLA should be understood in terms of fluctuations around the one-dimensional backbone of the cluster.

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Section: Discussionmentioning

confidence: 99%

How anisotropy beats fractality in two-dimensional on-lattice diffusion-limited-aggregation growth

Beliaev

2017

Phys. Rev. E

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“…We chose regular constructions because they can be associated with the mean DLA branching number 5, resulting in synthetic clusters that look somewhat similar to DLA clusters. curves obtained for mono and multifractal toy DLA synthetics In order to put our results in perspective, Figure 10 shows our obtained multifractal obtained in [32] and [33] for DLA clusters with similar mass. All three results are for DLA clusters of 1 million particles.…”

Section: -Application To the Diffusion-limited Aggregation (Dla) Promentioning

confidence: 99%

“…All three results are for DLA clusters of 1 million particles. In contrast to [32] random pivot points, [33] uses 5*10 4 random pivot points and square boxes for size sandbox method. Also, the R interval for this study was wider we conducted the following test: we (log(6)/log(3)) and confirmed that the sample sizes we created a synthetic multifractal having a D q…”

Section: -Application To the Diffusion-limited Aggregation (Dla) Promentioning

confidence: 99%

Barycentric fixed-mass method for multifractal analysis

Kamer

Ouillon²,

Sornette

2013

Phys. Rev. E

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Abstract:We present a novel method to estimate the multifractal spectrum of point distributions. The method incorporates two motivated criteria (barycentric pivot point selection and non-overlapping coverage) in order to reduce edge effects, improve precision and reduce computation time. Implementation of the method on synthetic benchmarks demonstrates the superior performance of the proposed method compared with existing alternatives routinely used in the literature. Finally, we use the method to estimate the multifractal properties of the widely studied growth process of Diffusion Limited Aggregation and compare our results with recent and earlier studies. Our tests support the conclusion of a genuine but weak multifractality of the central core of DLA clusters, with D q decreasing from 1.75±0.01 for q=-10 to 1.65±0.01 for q=+10. 1-IntroductionSince their popularization by Mandlebrot [1], fractals and fractal geometry have been empirically observed and extensively studied in a wealth of natural and experimental physical phenomena. A common way to quantify the fractal or multifractal properties of a given set of data points is to calculate its generalized (Renyi) dimensions [2], given as:where ε is the scale of observation, p i (ε) is the fraction of data points (e.g, estimated measure) within box i of size ε, q is a real-valued moment order and the sum is performed over all boxes covering the data set under investigation.

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“…T hese syst ems include diff usion limit ed aggregat ion [3][4][5], fluid flow t hrough random porous media [6], at omic spect ra of rare-eart h element s [7], clust erclust er aggregat ion [8] and t urbulent flow [9]. In physiology, mult ifract al st ruct ures have been found in heart rat e variability [10] and brain dynamics [11], and mult ifract al analysis has been helpful in dist inguishing between healthy and pathological patients [12].…”

Section: Introductionmentioning

confidence: 99%

The origins of multifractality in financial time series and the effect of extreme events

2014

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Abst ract . T his paper present s t he result s of mult ifract al t est ing of two set s of financial dat a: daily dat a of t he Dow J ones Indust rial Average (DJ IA) index and minut ely dat a of t he Euro St oxx 50 index. W here mult ifract al scaling is found, t he spect rum of scaling exponent s is calculat ed via Mult ifract al Det rended F luct uat ion Analysis. In bot h cases, furt her invest igat ions reveal t hat t he t emporal correlat ions in t he dat a are a more significant source of t he mult ifract al scaling t han are t he dist ribut ions of t he ret urns. It is also shown t hat t he ext reme event s which make up t he heavy t ails of t he dist ribut ion of t he Euro St oxx 50 log ret urns dist ort t he scaling in t he dat a set . T he most ext reme event s are inimical t o t he scaling regime. T his result is in cont rast t o previous findings t hat ext reme event s cont ribut e t o mult ifract ality. Int roduct ionMult ifract al analysis has proved t o be a valuable met hod of capt uring t he underlying scaling st ruct ure present in many types of syst ems via generalised dimensions [1] and f (α) spect ra [2]. T hese syst ems include diff usion limit ed aggregat ion [3][4][5], fluid flow t hrough random porous media [6], at omic spect ra of rare-eart h element s [7], clust erclust er aggregat ion [8] and t urbulent flow [9]. In physiology, mult ifract al st ruct ures have been found in heart rat e variability [10] and brain dynamics [11], and mult ifract al analysis has been helpful in dist inguishing between healthy and pathological patients [12]. Mult ifract al measures have also been found in man-made phenomena such as t he Int ernet [13], art [14] and the stock market [15][16][17].T he concept of mult ifract ality was first int roduced in t he cont ext of t urbulence. It was soon applied t o finance because of it s heavy t ails and long-t erm dependence. T hese two feat ures are also argued t o be present in financial dat a [18,19]. P erforming mult ifract al analysis helps t o increase our knowledge about t he financial syst em and furt her charact erise it . Many st udies have found mult ifract al scaling in financial dat a [20][21][22][23]. An underst anding of t his mult ifract al st ruct ure can enable deeper underst anding of t he dynamics of financial market s. If it is found t o be a universal feat ure of financial dat a, it provides an addit ional benchmark by which t o measure t he fitness of financial models. This in turn can help in the design of well performing port folios and in risk management [17]. a e-mail: elena.s.green@nuim.ieThe Multifractal Model of Asset Returns (MMAR) was int roduced by Mandelbrot et al. [24] as an explanat ion of t he volat ility clust ers in financial dat a and t o include "out liers", large deviat ions which make up t he fat t ails of t he ret urn dist ribut ion. T he MMAR was present ed as an alt ernat ive t o Aut oregressive Condit ional Het eroscedast icity (ARCH) models which were int roduced by Engle [25] t o account for volat ility clust er...

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Multifractal analysis of the branch structure of diffusion-limited aggregates

Cited by 15 publications

References 42 publications

How anisotropy beats fractality in two-dimensional on-lattice diffusion-limited-aggregation growth

How anisotropy beats fractality in two-dimensional on-lattice diffusion-limited-aggregation growth

Barycentric fixed-mass method for multifractal analysis

The origins of multifractality in financial time series and the effect of extreme events

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