# A Novel Measure Inspired by Lyapunov Exponents for the Characterization of Dynamics in State-Transition Networks

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## Abstract

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## 1. Introduction

## 2. Results

#### 2.1. State-Transition Networks

#### 2.2. Lyapunov Measure

#### 2.2.1. Properties of the Lyapunov Measure

#### 2.2.2. Analytical Study of the Lyapunov Measure

#### 2.3. Time Series Analysis with STNs

#### Construction of STNs

#### 2.4. Network Measures

## 3. Materials and Methods

#### 3.1. Discrete- and Continuous-Time Dynamical Systems

#### 3.2. Phase-Space Discretization and Poincare Sections

#### 3.3. Ensemble Averages and Asymptotic Behavior

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Offset at c = 0 in the Lyapunov Measure According to the Random Network Model

## Appendix B. Location and Size of the Maximum in the Lyapunov Measure According to the Random Network Model

## Appendix C. Distribution of Uniform Random Numbers Normalized over Large Sets

**Figure A1.**Depiction of the essentials behind Equation (A5).

## Appendix D. Insight into the Emergence of Precursor Peaks

**Figure A2.**Semianalytic estimation of the cyclicity, c, and Lyapunov measure, $\mathsf{\Lambda}$, of the Lorenz STNs created according to the procedure described in Section 2.3. The values of the mean edge length, $\langle l\rangle $, are computed following the definition of the Kolmogorov–Sinai entropy for STNs expressed by Equation (9). The cyclicity parameter is estimated by inverting numerically the $\langle l\rangle \left(c\right)$ relationship in Equation (18). As a last step, the Lyapunov measure was approximated using the analytical expression from Equation (20). The random walk simulation counterpart of the image is presented in the bottom right panel of Figure 4. For more details, refer to Appendix D.

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**Figure 1.**Dependence of the Lyapunov measure defined in Equation (14) on the cyclicity parameter, c, for complete graphs of different sizes, N, edges associated with probabilities defined in Equation (15) and lengths expressed by Equation (3). (

**A**) Examples of graphs with $N=10$ nodes using different cyclicity parameters, c. The probabilities, ${p}_{ij}$, associated with edges are represented by their width. (

**B**) Dependence of the mean, $\langle L\rangle $, and variance, $\Delta L$, of trajectory length (see Equation (8)) on the number of steps, t, during simulated random walk on graphs similar to those in panel (

**A**). The size of the networks is $N=50$. Data points correspond to ensemble averages of 1000 independent trajectories. (

**C**) The Lyapunov measure as a function of the cyclicity parameter, c, for graphs similar to those in panel (

**A**). Noisy lines with markers represent simulation results of random walk on these graphs. The measure is calculated according to Equation (14) using ensemble averages of 1000 trajectories. Each trajectory includes 1000 steps. Starting nodes are chosen randomly with uniform probability and the first 500 “thermalization” steps are ignored when estimating trajectory lengths. Dashed lines represent simulation results on similar graphs by estimating the variance of the edge length according to Equation (13) with $\langle C\rangle =0$. Continuous lines represent the analytical model (see Equation (20)).

**Figure 2.**Constructing state-transition networks for the Henon-map.

**Top row:**Chaotic and periodic attractors of the Henon-map in the $({x}_{n},{y}_{n})$ plane for $a=1.220/1.2265/1.24/1.27$ (orange/red/blue/green), respectively.

**Bottom row:**The corresponding state-transition networks with a spanning layout respecting the $({x}_{n},{y}_{n})$ positions of the nodes in the phase plane of the system. For illustration, the widths of edges are set proportionally to the occurrence of the respective transitions during the dynamics. To construct the STN ${n}_{\mathrm{max}}={10}^{4}$ steps long trajectories are started from randomly chosen initial conditions while discarding the first ${n}_{\mathrm{trans}}={10}^{3}$ transient steps. For discretization of the effective phase plane a $20\times 20$ mesh is constructed.

**Figure 3.**Constructing state-transition networks for the Lorenz system.

**Top row:**Chaotic and periodic attractors of the Lorenz in the $(z,y)$ plane for $\rho =180.10/180.70/180.78/181.10$ (orange/red/green/blue), respectively.

**Middle row:**The $x=15$, $\dot{x}<0$ Poincare sections of the attractors (see the top panels).

**Bottom row:**The corresponding state-transition networks with a spanning layout respecting the $(z,y)$ positions of the nodes in the Poincare sections of the system. For illustration, the widths of edges are set proportionally to the occurrence of the respective transitions during the dynamics. For illustrating the Poincare maps ${t}_{\mathrm{max}}=5000$ time unit long trajectories are used, discarding ${t}_{\mathrm{trans}}=300$ long transients. For discretization of the effective phase plane a $20\times 20$ mesh is constructed.

**Figure 4.**Network measures for STNs generated from time series of discrete- and continuous-time dynamical systems. (

**Left**) Results for the Henon-map as function of the $a\in [1,1.4]$ control parameter. (

**Right**) Results for the Lorenz system, for the $\rho \in [180,182]$ parameter interval. From top to bottom: Bifurcation diagrams, largest Lyapunov exponents ${\lambda}_{m}$, Kolmogorov–Sinai entropy ${S}_{KS}=\langle l\rangle $, and Lyapunov-type network measure $\mathsf{\Lambda}=\Delta L/t$ for the same parameter intervals as for the bifurcation diagrams. The colored dots denote the parameter values for which the attractors, Poincare maps and STNs are shown, respectively, in Figure 2 and Figure 3. To construct the STNs the Henon-map is iterated for 3 × 10${}^{4}$ timesteps omitting the initial 1000 transient steps, respectively for the Lorenz system 5000 timeunits were used ignoring the first 500 units long transients. The Kolmogorov–Sinai entropy is computed both using the adjacency matrix of global transition probabilities ${q}_{ij}^{*}$ (compare Equations (9) and (10)) and by generating and ensemble of random paths using the same number of steps as for the Lyapunov measure. The Lyapunov network measure, defined by Equation (14), is calculated over an ensemble of 100 random trajectories of ${10}^{4}$ steps, neglecting the initial 5000 steps (see Section 3.3). As an example of boundary-crisis precursor, a single peak in the Lyapunov measure is colored in red for both systems.

**Figure 5.**Lyapunov measure: variance of path lengths for random trajectories using the STNs generated for the Lorenz system. (

**Left**) Averaged over an ensemble of n trajectories (see the legend) for the chaotic regime with $\rho =180.7$. (

**Right**) Averaged over $n={10}^{4}$ trajectories for chaotic and periodic dynamics using $\rho =180.10/180.70/180.78/181.10$, respectively (orange/red/green/blue).

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**MDPI and ACS Style**

Sándor, B.; Schneider, B.; Lázár, Z.I.; Ercsey-Ravasz, M.
A Novel Measure Inspired by Lyapunov Exponents for the Characterization of Dynamics in State-Transition Networks. *Entropy* **2021**, *23*, 103.
https://doi.org/10.3390/e23010103

**AMA Style**

Sándor B, Schneider B, Lázár ZI, Ercsey-Ravasz M.
A Novel Measure Inspired by Lyapunov Exponents for the Characterization of Dynamics in State-Transition Networks. *Entropy*. 2021; 23(1):103.
https://doi.org/10.3390/e23010103

**Chicago/Turabian Style**

Sándor, Bulcsú, Bence Schneider, Zsolt I. Lázár, and Mária Ercsey-Ravasz.
2021. "A Novel Measure Inspired by Lyapunov Exponents for the Characterization of Dynamics in State-Transition Networks" *Entropy* 23, no. 1: 103.
https://doi.org/10.3390/e23010103