# Minimising the Kullback–Leibler Divergence for Model Selection in Distributed Nonlinear Systems

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. Background

#### 3.1. Notation

#### 3.2. Representing Distributed Dynamical Systems as Probabilistic Graphical Models

**Definition**

**1.**(Synchronous GDS)

#### 3.3. Network Scoring Functions

## 4. Computing Conditional KL Divergence

#### 4.1. A Tractable Expression via Embedding Theory

**Lemma**

**1**(Cliff et al. [12]).

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

#### 4.2. Information-Theoretic Interpretation

**Theorem**

**4.**

**Proof.**

**Corollary**

**1.**

**Proof.**

## 5. Application to Structure Learning

#### 5.1. Penalising Transfer Entropy by Independence Tests

#### 5.2. Implementation Details and Algorithm Analysis

## 6. Experimental Validation

#### 6.1. Distributed Lorenz and Rössler Attractors

#### 6.2. Case Study: Coupled Lorenz–Rössler System

#### 6.3. Case Study: Network of Lorenz Attractors

## 7. Discussion and Future Work

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Embedding Theory

**Theorem**

**A1**(Delay Embedding Theorem for Diffeomorphisms [31]).

**Theorem**

**A2**(Delay Embedding Theorem for Endomorphisms [71]).

**Theorem**

**A3**(Bundle Delay Embedding Theorem [44]).

**Theorem**

**A4**(Delay Embedding Theorem for Multivariate Observation Functions [50]).

## Appendix B. Information Theory

## Appendix C. Extended Results

**Table A1.**Classification results for three-node ($M=3$) networks for $N=5000$ samples. We present the precision (P), recall (R), fallout (F), and ${F}_{1}$-score for the eight arbitrary topologies of coupled Lorenz systems represented by Figure 4.

Graph | p-Value | ∞ | $0.01$ | $0.001$ | $0.0001$ | ||||
---|---|---|---|---|---|---|---|---|---|

${\mathbf{\sigma}}_{\mathbf{\psi}}$ | 1 | 10 | 1 | 10 | 1 | 10 | 1 | 10 | |

${G}^{1}$ | R | - | - | - | - | - | - | - | - |

F | 0.33 | 0.22 | 0.33 | 0.22 | 0.22 | 0.33 | 0.33 | 0.22 | |

P | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

${F}_{1}$ | - | - | - | - | - | - | - | - | |

${G}^{2}$ | R | 1 | 0.5 | 1 | 0.5 | 1 | 0.5 | 1 | 0.5 |

F | 0.14 | 0.14 | 0.14 | 0.14 | 0.14 | 0.14 | 0.14 | 0.14 | |

P | 0.67 | 0.5 | 0.67 | 0.5 | 0.67 | 0.5 | 0.67 | 0.5 | |

${F}_{1}$ | 0.8 | 0.5 | 0.8 | 0.5 | 0.8 | 0.5 | 0.8 | 0.5 | |

${G}^{3}$ | R | 1 | 0.5 | 1 | 1 | 1 | 1 | 1 | 0.5 |

F | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

P | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |

${F}_{1}$ | 1 | 0.67 | 1 | 1 | 1 | 1 | 1 | 0.67 | |

${G}^{4}$ | R | 1 | 0 | 1 | 1 | 1 | 0.5 | 1 | 0 |

F | 0.14 | 0.43 | 0.14 | 0.14 | 0.14 | 0.14 | 0.14 | 0.43 | |

P | 0.67 | 0 | 0.67 | 0.67 | 0.67 | 0.5 | 0.67 | 0 | |

${F}_{1}$ | 0.8 | - | 0.8 | 0.8 | 0.8 | 0.5 | 0.8 | - |

**Table A2.**Classification results for four-node ($M=4$) networks for $N=5000$ samples. We present the precision (P), recall (R), fallout (F), and ${F}_{1}$-score for the eight arbitrary topologies of coupled Lorenz systems represented by Figure 4.

Graph | p-Value | ∞ | $0.01$ | $0.001$ | $0.0001$ | ||||
---|---|---|---|---|---|---|---|---|---|

${\mathbf{\sigma}}_{\mathbf{\psi}}$ | 1 | 10 | 1 | 10 | 1 | 10 | 1 | 10 | |

${G}^{5}$ | R | - | - | - | - | - | - | - | - |

F | 0.31 | 0.25 | 0.31 | 0.19 | 0.31 | 0.25 | 0.31 | 0.19 | |

P | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

${F}_{1}$ | - | - | - | - | - | - | - | - | |

${G}^{6}$ | R | 0.67 | 0.67 | 0.67 | 0.33 | 0.67 | 0.33 | 0.67 | 0 |

F | 0.15 | 0.23 | 0.15 | 0.23 | 0.15 | 0.23 | 0.15 | 0.31 | |

P | 0.5 | 0.4 | 0.5 | 0.25 | 0.5 | 0.25 | 0.5 | 0 | |

${F}_{1}$ | 0.57 | 0.5 | 0.57 | 0.29 | 0.57 | 0.29 | 0.57 | - | |

${G}^{7}$ | R | 1 | 0.25 | 1 | 0.25 | 0.75 | 0.25 | 0.75 | 0.5 |

F | 0 | 0.25 | 0 | 0.17 | 0.083 | 0.25 | 0.083 | 0.083 | |

P | 1 | 0.25 | 1 | 0.33 | 0.75 | 0.25 | 0.75 | 0.67 | |

${F}_{1}$ | 1 | 0.25 | 1 | 0.29 | 0.75 | 0.25 | 0.75 | 0.57 | |

${G}^{8}$ | R | 1 | 0.25 | 1 | 0.5 | 1 | 0.75 | 1 | 0.25 |

F | 0 | 0.25 | 0 | 0.083 | 0 | 0.083 | 0 | 0.25 | |

P | 1 | 0.25 | 1 | 0.67 | 1 | 0.75 | 1 | 0.25 | |

${F}_{1}$ | 1 | 0.25 | 1 | 0.57 | 1 | 0.75 | 1 | 0.25 |

**Table A3.**Classification results for three-node ($M=3$) networks for $N=$ 10,000 samples. We present the precision (P), recall (R), fallout (F), and ${F}_{1}$-score for the eight arbitrary topologies of coupled Lorenz systems represented by Figure 4.

Graph | p-Value | ∞ | $0.01$ | $0.001$ | $0.0001$ | ||||
---|---|---|---|---|---|---|---|---|---|

${\mathbf{\sigma}}_{\mathbf{\psi}}$ | 1 | 10 | 1 | 10 | 1 | 10 | 1 | 10 | |

${G}^{1}$ | R | - | - | - | - | - | - | - | - |

F | 0.22 | 0.11 | 0.22 | 0.11 | 0.22 | 0.22 | 0.22 | 0.11 | |

P | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

${F}_{1}$ | - | - | - | - | - | - | - | - | |

${G}^{2}$ | R | 1 | 0.5 | 1 | 0.5 | 1 | 0.5 | 1 | 0.5 |

F | 0 | 0.14 | 0 | 0.14 | 0 | 0.14 | 0 | 0.14 | |

P | 1 | 0.5 | 1 | 0.5 | 1 | 0.5 | 1 | 0.5 | |

${F}_{1}$ | 1 | 0.5 | 1 | 0.5 | 1 | 0.5 | 1 | 0.5 | |

${G}^{3}$ | R | 1 | 0.5 | 1 | 1 | 1 | 0 | 1 | 0.5 |

F | 0 | 0.14 | 0 | 0 | 0 | 0.29 | 0 | 0.14 | |

P | 1 | 0.5 | 1 | 1 | 1 | 0 | 1 | 0.5 | |

${F}_{1}$ | 1 | 0.5 | 1 | 1 | 1 | - | 1 | 0.5 | |

${G}^{4}$ | R | 1 | 1 | 1 | 0.5 | 1 | 0.5 | 1 | 1 |

F | 0.14 | 0.14 | 0 | 0 | 0.14 | 0.14 | 0.14 | 0.14 | |

P | 0.67 | 0.67 | 1 | 1 | 0.67 | 0.5 | 0.67 | 0.67 | |

${F}_{1}$ | 0.8 | 0.8 | 1 | 0.67 | 0.8 | 0.5 | 0.8 | 0.8 |

**Table A4.**Classification results for four-node ($M=4$) networks for $N=$ 10,000 samples. We present the precision (P), recall (R), fallout (F), and ${F}_{1}$-score for the eight arbitrary topologies of coupled Lorenz systems represented by Figure 4.

Graph | p-Value | ∞ | $0.01$ | $0.001$ | $0.0001$ | ||||
---|---|---|---|---|---|---|---|---|---|

${\mathbf{\sigma}}_{\mathbf{\psi}}$ | 1 | 10 | 1 | 10 | 1 | 10 | 1 | 10 | |

${G}^{5}$ | R | - | - | - | - | - | - | - | - |

F | 0.31 | 0.25 | 0.31 | 0.19 | 0.31 | 0.19 | 0.31 | 0.25 | |

P | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

${F}_{1}$ | - | - | - | - | - | - | - | - | |

${G}^{6}$ | R | 0.67 | 0.33 | 0.67 | 0 | 1 | 1 | 0.67 | 0.33 |

F | 0.15 | 0.15 | 0.15 | 0.15 | 0.15 | 0.15 | 0.15 | 0.15 | |

P | 0.5 | 0.33 | 0.5 | 0 | 0.6 | 0.6 | 0.5 | 0.33 | |

${F}_{1}$ | 0.57 | 0.33 | 0.57 | - | 0.75 | 0.75 | 0.57 | 0.33 | |

${G}^{7}$ | R | 0.75 | 0.5 | 1 | 0.5 | 1 | 0.25 | 0.75 | 0.5 |

F | 0.083 | 0.083 | 0 | 0.083 | 0 | 0.17 | 0.083 | 0.083 | |

P | 0.75 | 0.67 | 1 | 0.67 | 1 | 0.33 | 0.75 | 0.67 | |

${F}_{1}$ | 0.75 | 0.57 | 1 | 0.57 | 1 | 0.29 | 0.75 | 0.57 | |

${G}^{8}$ | R | 1 | 0.25 | 1 | 0.25 | 1 | 0 | 1 | 0.25 |

F | 0 | 0.17 | 0 | 0.17 | 0 | 0.25 | 0 | 0.17 | |

P | 1 | 0.33 | 1 | 0.33 | 1 | 0 | 1 | 0.33 | |

${F}_{1}$ | 1 | 0.29 | 1 | 0.29 | 1 | - | 1 | 0.29 |

**Table A5.**Classification results for three-node ($M=3$) networks for $N=$ 25,000 samples. We present the precision (P), recall (R), fallout (F), and ${F}_{1}$-score for the eight arbitrary topologies of coupled Lorenz systems represented by Figure 4.

Graph | p-Value | ∞ | $0.01$ | $0.001$ | $0.0001$ | ||||
---|---|---|---|---|---|---|---|---|---|

${\mathbf{\sigma}}_{\mathbf{\psi}}$ | 1 | 10 | 1 | 10 | 1 | 10 | 1 | 10 | |

${G}^{1}$ | R | - | - | - | - | - | - | - | - |

F | 0.22 | 0.11 | 0.22 | 0.11 | 0.22 | 0.22 | 0.22 | 0.11 | |

P | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

${F}_{1}$ | - | - | - | - | - | - | - | - | |

${G}^{2}$ | R | 1 | 1 | 1 | 0.5 | 1 | 0.5 | 1 | 1 |

F | 0 | 0.14 | 0 | 0.14 | 0 | 0.14 | 0 | 0.14 | |

P | 1 | 0.67 | 1 | 0.5 | 1 | 0.5 | 1 | 0.67 | |

${F}_{1}$ | 1 | 0.8 | 1 | 0.5 | 1 | 0.5 | 1 | 0.8 | |

${G}^{3}$ | R | 1 | 1 | 1 | 0.5 | 1 | 1 | 1 | 1 |

F | 0 | 0 | 0 | 0.14 | 0 | 0 | 0 | 0 | |

P | 1 | 1 | 1 | 0.5 | 1 | 1 | 1 | 1 | |

${F}_{1}$ | 1 | 1 | 1 | 0.5 | 1 | 1 | 1 | 1 | |

${G}^{4}$ | R | 1 | 1 | 1 | 1 | 1 | 0.5 | 1 | 1 |

F | 0 | 0 | 0 | 0 | 0 | 0.14 | 0 | 0 | |

P | 1 | 1 | 1 | 1 | 1 | 0.5 | 1 | 1 | |

${F}_{1}$ | 1 | 1 | 1 | 1 | 1 | 0.5 | 1 | 1 |

**Table A6.**Classification results for four-node ($M=4$) networks for $N=$ 25,000 samples. We present the precision (P), recall (R), fallout (F), and ${F}_{1}$-score for the eight arbitrary topologies of coupled Lorenz systems represented by Figure 4.

Graph | p-Value | ∞ | $0.01$ | $0.001$ | $0.0001$ | ||||
---|---|---|---|---|---|---|---|---|---|

${\mathbf{\sigma}}_{\mathbf{\psi}}$ | 1 | 10 | 1 | 10 | 1 | 10 | 1 | 10 | |

${G}^{5}$ | R | - | - | - | - | - | - | - | - |

F | 0.31 | 0.19 | 0.31 | 0.19 | 0.31 | 0.19 | 0.31 | 0.19 | |

P | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

${F}_{1}$ | - | - | - | - | - | - | - | - | |

${G}^{6}$ | R | 1 | 0.33 | 1 | 0.33 | 1 | 0.33 | 1 | 0.33 |

F | 0.15 | 0.15 | 0.15 | 0.15 | 0.15 | 0.23 | 0.15 | 0.15 | |

P | 0.6 | 0.33 | 0.6 | 0.33 | 0.6 | 0.25 | 0.6 | 0.33 | |

${F}_{1}$ | 0.75 | 0.33 | 0.75 | 0.33 | 0.75 | 0.29 | 0.75 | 0.33 | |

${G}^{7}$ | R | 1 | 0.5 | 1 | 0.75 | 1 | 0.75 | 1 | 0.5 |

F | 0 | 0.17 | 0 | 0 | 0 | 0 | 0 | 0.17 | |

P | 1 | 0.5 | 1 | 1 | 1 | 1 | 1 | 0.5 | |

${F}_{1}$ | 1 | 0.5 | 1 | 0.86 | 1 | 0.86 | 1 | 0.5 | |

${G}^{8}$ | R | 1 | 0.75 | 1 | 0.75 | 1 | 0.75 | 1 | 0.75 |

F | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

P | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |

${F}_{1}$ | 1 | 0.86 | 1 | 0.86 | 1 | 0.86 | 1 | 0.86 |

**Table A7.**Classification results for three-node ($M=3$) networks with $N=$ 50,000 samples. We present the precision (P), recall (R), fallout (F), and ${F}_{1}$-score for the eight arbitrary topologies of coupled Lorenz systems represented by Figure 4.

Graph | p-Value | ∞ | $0.01$ | $0.001$ | $0.0001$ | ||||
---|---|---|---|---|---|---|---|---|---|

${\mathbf{\sigma}}_{\mathbf{\psi}}$ | 1 | 10 | 1 | 10 | 1 | 10 | 1 | 10 | |

${G}^{1}$ | R | - | - | - | - | - | - | - | - |

F | 0 | 0.11 | 0 | 0 | 0 | 0.11 | 0 | 0.22 | |

P | - | 0 | - | - | - | 0 | - | 0 | |

${F}_{1}$ | - | - | - | - | - | - | - | - | |

${G}^{2}$ | R | 1 | 0.5 | 1 | 0.5 | 1 | 0.5 | 1 | 0.5 |

F | 0 | 0.14 | 0 | 0.14 | 0 | 0.14 | 0 | 0.14 | |

P | 1 | 0.5 | 1 | 0.5 | 1 | 0.5 | 1 | 0.5 | |

${F}_{1}$ | 1 | 0.5 | 1 | 0.5 | 1 | 0.5 | 1 | 0.5 | |

${G}^{3}$ | R | 1 | 1 | 1 | 0.5 | 1 | 1 | 1 | 1 |

F | 0 | 0.14 | 0 | 0.14 | 0 | 0.14 | 0 | 0 | |

P | 1 | 0.67 | 1 | 0.5 | 1 | 0.67 | 1 | 1 | |

${F}_{1}$ | 1 | 0.8 | 1 | 0.5 | 1 | 0.8 | 1 | 1 | |

${G}^{4}$ | R | 1 | 0.5 | 1 | 1 | 1 | 0.5 | 1 | 1 |

F | 0 | 0.14 | 0 | 0 | 0 | 0.14 | 0 | 0 | |

P | 1 | 0.5 | 1 | 1 | 1 | 0.5 | 1 | 1 | |

${F}_{1}$ | 1 | 0.5 | 1 | 1 | 1 | 0.5 | 1 | 1 |

**Table A8.**Classification results for four-node ($M=4$) networks with $N=$ 50,000 samples. We present the precision (P), recall (R), fallout (F), and ${F}_{1}$-score for the eight arbitrary topologies of coupled Lorenz systems represented by Figure 4.

Graph | p-Value | ∞ | $0.01$ | $0.001$ | $0.0001$ | ||||
---|---|---|---|---|---|---|---|---|---|

${\mathbf{\sigma}}_{\mathbf{\psi}}$ | 1 | 10 | 1 | 10 | 1 | 10 | 1 | 10 | |

${G}^{5}$ | R | - | - | - | - | - | - | - | - |

F | 0.19 | 0.062 | 0.19 | 0.19 | 0.19 | 0.12 | 0.19 | 0.12 | |

P | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

${F}_{1}$ | - | - | - | - | - | - | - | - | |

${G}^{6}$ | R | 1 | 0.33 | 1 | 0 | 1 | 0.33 | 1 | 0.33 |

F | 0 | 0.15 | 0 | 0 | 0 | 0.23 | 0.15 | 0.15 | |

P | 1 | 0.33 | 1 | - | 1 | 0.25 | 0.6 | 0.33 | |

${F}_{1}$ | 1 | 0.33 | 1 | - | 1 | 0.29 | 0.75 | 0.33 | |

${G}^{7}$ | R | 1 | 0.75 | 1 | 0.5 | 1 | 0.5 | 1 | 0.75 |

F | 0 | 0 | 0 | 0.17 | 0 | 0.083 | 0 | 0 | |

P | 1 | 1 | 1 | 0.5 | 1 | 0.67 | 1 | 1 | |

${F}_{1}$ | 1 | 0.86 | 1 | 0.5 | 1 | 0.57 | 1 | 0.86 | |

${G}^{8}$ | R | 1 | 0.75 | 1 | 0.75 | 1 | 0.75 | 1 | 0.75 |

F | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

P | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |

${F}_{1}$ | 1 | 0.86 | 1 | 0.86 | 1 | 0.86 | 1 | 0.86 |

**Table A9.**Classification results for three-node ($M=3$) networks with $N=$ 100,000 samples. We present the precision (P), recall (R), fallout (F), and ${F}_{1}$-score for the eight arbitrary topologies of coupled Lorenz systems represented by Figure 4.

Graph | p-Value | ∞ | $0.01$ | $0.001$ | $0.0001$ | ||||
---|---|---|---|---|---|---|---|---|---|

${\mathbf{\sigma}}_{\mathbf{\psi}}$ | 1 | 10 | 1 | 10 | 1 | 10 | 1 | 10 | |

${G}^{1}$ | R | - | - | - | - | - | - | - | - |

F | 0 | 0.22 | 0 | 0.11 | 0 | 0.22 | 0 | 0.11 | |

P | - | 0 | - | 0 | - | 0 | - | 0 | |

${F}_{1}$ | - | - | - | - | - | - | - | - | |

${G}^{2}$ | R | 1 | 0.5 | 1 | 1 | 1 | 1 | 1 | 1 |

F | 0 | 0.14 | 0 | 0 | 0 | 0 | 0 | 0.14 | |

P | 1 | 0.5 | 1 | 1 | 1 | 1 | 1 | 0.67 | |

${F}_{1}$ | 1 | 0.5 | 1 | 1 | 1 | 1 | 1 | 0.8 | |

${G}^{3}$ | R | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

F | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

P | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |

${F}_{1}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |

${G}^{4}$ | R | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

F | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

P | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |

${F}_{1}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

**Table A10.**Classification results for four-node ($M=4$) networks with $N=$ 100,000 samples. We present the precision (P), recall (R), fallout (F), and ${F}_{1}$-score for the eight arbitrary topologies of coupled Lorenz systems represented by Figure 4.

Graph | p-Value | ∞ | $0.01$ | $0.001$ | $0.0001$ | ||||
---|---|---|---|---|---|---|---|---|---|

${\mathbf{\sigma}}_{\mathbf{\psi}}$ | 1 | 10 | 1 | 10 | 1 | 10 | 1 | 10 | |

${G}^{5}$ | R | - | - | - | - | - | - | - | - |

F | 0.19 | 0.062 | 0.19 | 0.062 | 0.19 | 0.19 | 0.19 | 0.12 | |

P | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

${F}_{1}$ | - | - | - | - | - | - | - | - | |

${G}^{6}$ | R | 1 | 0.33 | 1 | 0.67 | 1 | 0.33 | 1 | 0.33 |

F | 0 | 0.15 | 0 | 0.15 | 0 | 0.077 | 0 | 0.15 | |

P | 1 | 0.33 | 1 | 0.5 | 1 | 0.5 | 1 | 0.33 | |

${F}_{1}$ | 1 | 0.33 | 1 | 0.57 | 1 | 0.4 | 1 | 0.33 | |

${G}^{7}$ | R | 1 | - | 1 | - | 1 | - | 1 | - |

F | 0 | - | 0 | - | 0 | - | 0 | - | |

P | 1 | - | 1 | - | 1 | - | 1 | - | |

${F}_{1}$ | 1 | - | 1 | - | 1 | - | 1 | - | |

${G}^{8}$ | R | 1 | 0.75 | 1 | 0.75 | 1 | 0.5 | 1 | 0.75 |

F | 0 | 0 | 0 | 0 | 0 | 0.083 | 0 | 0 | |

P | 1 | 1 | 1 | 1 | 1 | 0.67 | 1 | 1 | |

${F}_{1}$ | 1 | 0.86 | 1 | 0.86 | 1 | 0.57 | 1 | 0.86 |

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**Figure 1.**Trajectory of a pair of coupled Lorenz systems. Top row: original state of the subsystems. Bottom row: time-series measurements of the subsystems. In each figure, the black lines represent an uncoupled simulation ($\lambda =0$), and teal lines illustrate a simulation where the first (leftmost) subsystem was coupled to the second ($\lambda =10$). (

**a**) $\sigma =10,\beta =8/3,\rho =28$; (

**b**) $\sigma =10,\beta =8/3,\rho =90$.

**Figure 2.**Representation of (

**a**) the synchronous GDS with two vertices (${V}^{1}$ and ${V}^{2}$), and (

**b**) the rolled-out DBN of the equivalent structure. Subsystems ${V}^{1}$ and ${V}^{2}$ are coupled by virtue of the edge ${X}_{n}^{1}\to {X}_{n+1}^{2}$.

**Figure 3.**Distributions of the (

**a**) TEA penalty function (28) and the (

**b**) TEE penalty function (28). Both distributions were generated by observing the outcome of 1000 samples from two Gaussian variables with a correlation of $0.05$. The figures illustrate: the distribution as a set of 100 sampled points (black dots); the area considered independent (grey regions); the measured transfer entropy (black line); and the difference between measurement and penalty term (dark grey region). Both tests use a value of $\alpha =0.9$ (a p-value of $0.1$). The distribution in (

**a**) was estimated by assuming variables were linearly-coupled Gaussians, and the distribution in (

**b**) was computed via a kernal box method (computed by the Java Information Dynamics Toolkit (JIDT), see [52] for details).

**Figure 4.**The network topologies used in this paper. The top row (

**a**–

**d**) are four arbitrary networks with three nodes ($M=3$) and the bottom row (

**e**–

**h**) are four arbitrary networks with four nodes ($M=4$).

**Figure 5.**Transfer entropy as a function of the parameters of a coupled Lorenz–Rössler system. These components are: coupling strength $\lambda $ and embedding dimension $\kappa $ in the top row (

**a**–

**c**); coupling strength $\lambda $ and observation noise ${\sigma}_{\psi}$ in the middle row (

**d**–

**f**); and observation noise ${\sigma}_{\psi}$ and embedding dimension $\kappa $ in the bottom row (

**g**–

**i**).

**Table 1.**${F}_{1}$-scores for three-node ($M=3$) networks. We present the classification summary for the three arbitrary topologies of coupled Lorenz systems represented by Figure 4b–d (network ${G}^{1}$ has no edges and thus an undefined ${F}_{1}$-score). The p-value of the TEE score is given in the top row of each table, with ∞ signifying using no significance testing, i.e., score (27).

$\mathit{p}=\mathit{\infty}$ | $\mathit{p}=0.01$ | $\mathit{p}=0.001$ | $\mathit{p}=0.0001$ | ||||||
---|---|---|---|---|---|---|---|---|---|

Graph | N | ${\mathit{\sigma}}_{\mathit{\psi}}=1$ | ${\mathit{\sigma}}_{\mathit{\psi}}=10$ | ${\mathit{\sigma}}_{\mathit{\psi}}=1$ | ${\mathit{\sigma}}_{\mathit{\psi}}=10$ | ${\mathit{\sigma}}_{\mathit{\psi}}=1$ | ${\mathit{\sigma}}_{\mathit{\psi}}=10$ | ${\mathit{\sigma}}_{\mathit{\psi}}=1$ | ${\mathit{\sigma}}_{\mathit{\psi}}=10$ |

${G}^{2}$ | 5 K | 0.8 | 0.5 | 0.8 | 0.5 | 0.8 | 0.5 | 0.8 | 0.5 |

25 K | 1 | 0.8 | 1 | 0.5 | 1 | 0.5 | 1 | 0.8 | |

100 K | 1 | 0.5 | 1 | 1 | 1 | 1 | 1 | 0.8 | |

${G}^{3}$ | 5 K | 1 | 0.67 | 1 | 1 | 1 | 1 | 1 | 0.67 |

25 K | 1 | 1 | 1 | 0.5 | 1 | 1 | 1 | 1 | |

100 K | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |

${G}^{4}$ | 5 K | 0.8 | - | 0.8 | 0.8 | 0.8 | 0.5 | 0.8 | - |

25 K | 1 | 1 | 1 | 1 | 1 | 0.5 | 1 | 1 | |

100 K | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

**Table 2.**${F}_{1}$-scores for four-node ($M=4$) networks. We present the classification summary for the three arbitrary topologies of coupled Lorenz systems represented by Figure 4f–h (network ${G}^{5}$ has no edges and thus an undefined ${F}_{1}$-score). The p-value of the TEE score is given in the top row of each table, with ∞ signifying using no significance testing, i.e., score (27).

$\mathit{p}=\mathit{\infty}$ | $\mathit{p}=0.01$ | $\mathit{p}=0.001$ | $\mathit{p}=0.0001$ | ||||||
---|---|---|---|---|---|---|---|---|---|

Graph | N | ${\mathit{\sigma}}_{\mathit{\psi}}=1$ | ${\mathit{\sigma}}_{\mathit{\psi}}=10$ | ${\mathit{\sigma}}_{\mathit{\psi}}=1$ | ${\mathit{\sigma}}_{\mathit{\psi}}=10$ | ${\mathit{\sigma}}_{\mathit{\psi}}=1$ | ${\mathit{\sigma}}_{\mathit{\psi}}=10$ | ${\mathit{\sigma}}_{\mathit{\psi}}=1$ | ${\mathit{\sigma}}_{\mathit{\psi}}=10$ |

${G}^{6}$ | 5 K | 0.57 | 0.5 | 0.57 | 0.29 | 0.57 | 0.29 | 0.57 | - |

25 K | 0.75 | 0.33 | 0.75 | 0.33 | 0.75 | 0.29 | 0.75 | 0.33 | |

100 K | 1 | 0.33 | 1 | 0.57 | 1 | 0.4 | 1 | 0.33 | |

${G}^{7}$ | 5 K | 1 | 0.25 | 1 | 0.29 | 0.75 | 0.25 | 0.75 | 0.57 |

25 K | 1 | 0.5 | 1 | 0.86 | 1 | 0.86 | 1 | 0.5 | |

100 K | 1 | 0.86 | 1 | 0.86 | 1 | 0.86 | 1 | 0.86 | |

${G}^{8}$ | 5 K | 1 | 0.25 | 1 | 0.57 | 1 | 0.75 | 1 | 0.25 |

25 K | 1 | 0.86 | 1 | 0.86 | 1 | 0.86 | 1 | 0.86 | |

100 K | 1 | 0.86 | 1 | 0.86 | 1 | 0.57 | 1 | 0.86 |

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**MDPI and ACS Style**

Cliff, O.M.; Prokopenko, M.; Fitch, R. Minimising the Kullback–Leibler Divergence for Model Selection in Distributed Nonlinear Systems. *Entropy* **2018**, *20*, 51.
https://doi.org/10.3390/e20020051

**AMA Style**

Cliff OM, Prokopenko M, Fitch R. Minimising the Kullback–Leibler Divergence for Model Selection in Distributed Nonlinear Systems. *Entropy*. 2018; 20(2):51.
https://doi.org/10.3390/e20020051

**Chicago/Turabian Style**

Cliff, Oliver M., Mikhail Prokopenko, and Robert Fitch. 2018. "Minimising the Kullback–Leibler Divergence for Model Selection in Distributed Nonlinear Systems" *Entropy* 20, no. 2: 51.
https://doi.org/10.3390/e20020051