Multiplicative cascade

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title: "Multiplicative cascade" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["fractals"] description: "Fractal distribution of random points" topic_path: "general/fractals" source: "https://en.wikipedia.org/wiki/Multiplicative_cascade" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Fractal distribution of random points ::

In mathematics, a multiplicative cascade is a fractal/multifractal distribution of points produced via an iterative and multiplicative random process.

Definition

The plots above are examples of multiplicative cascade multifractals.

To create these distributions there are a few steps to take. Firstly, we must create a lattice of cells which will be our underlying probability density field.

Secondly, an iterative process is followed to create multiple levels of the lattice: at each iteration the cells are split into four equal parts (cells). Each new cell is then assigned a probability randomly from the set \lbrace p_1,p_2,p_3,p_4 \rbrace without replacement, where p_i \in [0,1]. This process is continued to the Nth level. For example, in constructing such a model down to level 8 we produce a 48 array of cells.

Thirdly, the cells are filled as follows: We take the probability of a cell being occupied as the product of the cell's own p**i and those of all its parents (up to level 1). A Monte Carlo rejection scheme is used repeatedly until the desired cell population is obtained, as follows: x and y cell coordinates are chosen randomly, and a random number between 0 and 1 is assigned; the (x, y) cell is then populated depending on whether the assigned number is lesser than (outcome: not populated) or greater or equal to (outcome: populated) the cell's occupation probability.

Examples

::figure[src="https://upload.wikimedia.org/wikipedia/commons/7/7e/3fractals2.jpg" caption="Three multiplicative cascades.
Generators (left to right): \lbrace p_1,p_2,p_3,p_4 \rbrace = \lbrace 1,1,1,0 \rbrace, \lbrace p_1,p_2,p_3,p_4 \rbrace = \lbrace 1,0.75,0.75,0.5 \rbrace, \lbrace p_1,p_2,p_3,p_4 \rbrace = \lbrace 1,0.5,0.5,0.25 \rbrace"] ::

To produce the plots above, the probability density field is filled with 5,000 points in a space of 256 × 256.

An example of the probability density field:

::figure[src="https://upload.wikimedia.org/wikipedia/commons/2/22/Multifractal_density_field.jpg"] ::

The fractals are generally not scale-invariant and therefore cannot be considered standard fractals. They can however be considered multifractals. The Rényi (generalized) dimensions can be theoretically predicted. It can be shown that as N \rightarrow \infty,

: D_q=\frac{\log_2\left( f^q_1+f^q_2+f^q_3+f^q_4\right)}{1-q},

where N is the level of the grid refinement and,

: f_i=\frac{p_i}{\sum_i p_i}.

References

References

1. (September 1987). "Diffusion-limited aggregation on multifractal lattices: A model for fluid-fluid displacement in porous media". Physical Review A.
2. [https://arxiv.org/abs/0803.3212 Cristano G. Sabiu, Luis Teodoro, Martin Hendry, arXiv:0803.3212v1 ''Resolving the universe with multifractals'']
3. Martinez et al. ApJ 357 50M "Clustering Paradigms and Multifractal Measures" [http://adsabs.harvard.edu/abs/1990ApJ...357...50M]

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::