Slawomir J. Nasuto | Searcharxiv

arXiv2026

Pairwise Distance-Diffusion Analysis (PDDA): A Geometric Framework for Estimating Hurst Exponents in Multivariate Long-Memory Processes

We introduce Pairwise Distance-Diffusion Analysis (PDDA), a geometric framework that connects the Hurst exponent to the scaling of pairwise distances in long-memory stochastic processes. From a single distance-plot representation, PDDA yields three complementary routes to persistence: R/S-PDDA, based on the growth of geometric extrema; MSD-PDDA, based on the scaling of the second moment of lagged distances; and recurrence-volume scaling, in which the decay of close returns with increasing temporal separation reflects the expansion of the lagged displacement cloud. The framework extends naturally to multivariate isotropic and anisotropic processes, where the local spatial dimension specifies the available degrees of freedom while the Hurst exponents govern their temporal expansion. These results establish the distance plot as a common geometric representation of persistence, providing a unified distance-based foundation for Hurst analysis.

stat.ME↗

arXiv2002

Steady State Resource Allocation Analysis of the Stochastic Diffusion Search

This article presents the long-term behaviour analysis of Stochastic Diffusion Search (SDS), a distributed agent-based system for best-fit pattern matching. SDS operates by allocating simple agents into different regions of the search space. Agents independently pose hypotheses about the presence of the pattern in the search space and its potential distortion. Assuming a compositional structure of hypotheses about pattern matching agents perform an inference on the basis of partial evidence from the hypothesised solution. Agents posing mutually consistent hypotheses about the pattern support each other and inhibit agents with inconsistent hypotheses. This results in the emergence of a stable agent population identifying the desired solution. Positive feedback via diffusion of information between the agents significantly contributes to the speed with which the solution population is formed. The formulation of the SDS model in terms of interacting Markov Chains enables its characterisation in terms of the allocation of agents, or computational resources. The analysis characterises the stationary probability distribution of the activity of agents, which leads to the characterisation of the solution population in terms of its similarity to the target pattern.

cs.AI↗