Presentations

On the structure of asymmetric sigmoid functions in the approximation of percolation probability functions

Moskalev P.V.¹, Myagkov A.S.¹

Voronezh State Technical University, Russia, 394006, Voronezh, 20th Anniversary of October st., 84, E-mail: moskalefff@gmail.com;

¹Moscow State University of Technology “STANKIN”, Russia, 127994, Moscow, Vadkovsky lane, 1

Simple and clear interpretation of critical phenomena has largely determined the spread of lattice percolation models in applied research. A significant part of the conclusions in percolation theory are obtained under the condition that the size of the percolation lattice increases without limit (for $x \to \infty$), while the main method of modeling percolation on limited lattices is the Monte Carlo method (for $x < \infty$).

To resolve the above contradiction on bounded lattices, the problem of approximating statistical estimates of the effective characteristics of percolation clusters is posed. In our works [1, 2], hypotheses were formulated that the construction of such approximations for percolation probability functions can be based on the product of the cumulative distribution function $F_0(p)$ weighing the percolation lattice of the random variable $S$, and a sufficiently arbitrary sigmoid function $F(p)$, which, as the authors' computational experiments show, in the general case has an explicitly asymmetric form.

It is quite clear that various options for constructing asymmetric sigmoid functions are possible, and one of them is the product of two logistic functions $F(p) = F_1(p) F_2(p)$ with different shift $a$ and scale $b$ parameters [2]

$$F(p) = \frac{\frac{2}{1 + \exp(-(p - a_2)/b_2)} - 1}{1 + \exp(-(p - a_1)/b_1)}.$$

An analysis of this formula shows that the $F_2(p)$ component can be simplified by replacing it with an analogue of the cumulative function of the exponential distribution. Then the expression for the asymmetric sigmoid will take a simpler form, and will also demonstrate a slightly better quality of approximation if we focus on the residual standard error (RSE) as a quality metric

$$F(p) = \frac{1 - \exp(-(p - a_2)/b_2)}{1 + \exp(-(p - a_1)/b_1)}.$$

This research was funded by the Russian Science Foundation (project No. 23-21-00376).

References

1. Moskalev P.V. Convergence of percolation probability functions to cumulative distribution functions on square lattices with (1, 0)-neighborhood // Physica A. V. 553, 2020, P. 124657. – DOI: https://doi.org/10.1016/j.physa.2020.124657.

2. Moskalev P.V., Myagkov A.S. Bilogistic approximation of percolation probability functions on bounded non-uniformly weighted square lattices with (1, 0)-neighborhood // Preprints.ru, 2024. – DOI: https://doi.org/10.24108/preprints-3113167 [in Russian].

• abstract in Russian (PDF)