On the Dynamical Foundation of Multifractality

TL;DR

This paper uncovers the underlying renewal events in multifractal dynamics, linking complexity matching to information transfer and demonstrating how multifractality propagates from more to less complex systems.

Contribution

It reveals the hidden renewal events in multifractal systems and clarifies the dual measures of complexity matching, advancing understanding of information transfer in complex networks.

Findings

Renewal events enable replacing strong anticipation with time delayed cross-correlation.

Complex systems transfer multifractality to less complex systems, but not vice versa.

Cross-correlation at delay captures information transfer in non-ergodic regimes.

Abstract

The crucial aspect of this demonstration is the discovery of renewal events, hidden in the computed dynamics of a multifractal metronome, which enables the replacement of the phenomenon of strong anticipation with a time delayed cross-correlation between the driven and the driving metronome. We establish that the phenomenon of complexity matching, which is the theme of an increasing number of research groups, has two distinct measures. One measure is the sensitivity of a complex system to environmental multi-fractality; another is the level of information transfer, between two complex networks at criticality. The cross-correlation function is evaluated in the ergodic long-time limit, but its delayed maximal value is the signature of information transfer occurring in the non ergodic short-time regime. It is shown that a more complex system transfers its multifractality to a less complex system while the reverse case is not possible.