The new face of multifractality: Multi-branchedness and the phase transitions in time series of mean inter-event times

TL;DR

This paper extends multifractal analysis to include multi-branched spectra and phase transition phenomena in time series of inter-event times, revealing complex thermodynamic behaviors.

Contribution

It introduces a modified multifractal analysis using the Legendre-Fenchel transform to uncover multi-branched spectra and phase transitions in empirical time series.

Findings

Discovery of multi-branched multifractal spectra.

Identification of phase transitions of first and second order.

Extension of multifractal analysis methodology.

Abstract

Empirical time series of inter-event or waiting times are investigated using a modified Multifractal Detrended Fluctuation Analysis operating on fluctuations of mean detrended dynamics. The core of the extended multifractal analysis is the non-monotonic behavior of the generalized Hurst exponent $h(q)$ -- the fundamental exponent in the study of multifractals. The consequence of this behavior is the non-monotonic behavior of the coarse H\"older exponent $\alpha (q)$ leading to multi-branchedness of the spectrum of dimensions. The Legendre-Fenchel transform is used instead of the routinely used canonical Legendre (single-branched) contact transform. Thermodynamic consequences of the multi-branched multifractality are revealed. These are directly expressed in the language of phase transitions between thermally stable, metastable, and unstable phases. These phase transitions are of the first and second orders according to Mandelbrot's modified Ehrenfest classification. The discovery of multi-branchedness is tantamount in significance to extending multifractal analysis.