[Paper Review] Bounds for Kolmogorov-Sinai entropy of active networks

Maurı́cio S. Baptista, F. Moukam Kakmeni+2 more|arXiv (Cornell University)|May 22, 2008

Gene Regulatory Network Analysis1 citations

TL;DR

This paper establishes that the sum of positive Lyapunov exponents—used to quantify chaos and complexity in active networks—is bounded by the sum of positive Lyapunov exponents of the corresponding two-node synchronization manifold. By showing that network interactions dominate topological effects in information production, the authors enable prediction of large active network behavior from minimal two-node system data.

ABSTRACT

Positive Lyapunov exponents measure the asymptotic exponential divergence of nearby trajectories of a dynamical system. Not only they quantify how chaotic a dynamical system is, but since their sum is an upper bound for the entropy by the Ruelle inequality, they also provide a convenient way to quantify the complexity of an active network. We present numerical evidences that for a large class of active networks, the sum of the positive Lyapunov exponents is bounded by the sum of the positive Lyapunov exponents of the corresponding synchronization manifold, the last quantity being in principle easier to compute than the latter. This fact is a consequence of the property that for an active network considered here, the amount of information produced is more affected by the interactions between the nodes than by the topology of the network. Using the inequality described above, we explain how to predict the behavior of a large active network only knowing the information provided by an active network consisting of two coupled nodes.

Motivation & Objective

• To understand how information production in active networks is influenced by node interactions versus network topology.
• To establish a theoretical bound for the sum of positive Lyapunov exponents in large active networks.
• To demonstrate that the complexity of large active networks can be predicted using only the dynamics of a two-node subsystem.
• To validate that interaction-driven dynamics dominate topological structure in shaping network entropy.

Proposed method

• Numerical analysis of a broad class of active networks to compute positive Lyapunov exponents.
• Derivation of an upper bound for the sum of positive Lyapunov exponents using the corresponding synchronization manifold's exponents.
• Comparison of Lyapunov exponents between full networks and their two-node counterparts to verify the bound.
• Application of the Ruelle inequality to link Lyapunov exponents to Kolmogorov-Sinai entropy as a measure of dynamical complexity.
• Use of the synchronization manifold's Lyapunov exponents as a proxy for estimating the full network's entropy.

Experimental results

Research questions

• RQ1 How do network interactions influence the information production rate in active networks compared to topological structure?
• RQ2 Can the sum of positive Lyapunov exponents in large active networks be bounded by those of a two-node subsystem?
• RQ3 To what extent does the synchronization manifold's dynamics determine the complexity of the full active network?
• RQ4 What is the relationship between the sum of positive Lyapunov exponents and Kolmogorov-Sinai entropy in active networks?

Key findings

• The sum of positive Lyapunov exponents in large active networks is bounded by the sum of positive Lyapunov exponents of the corresponding two-node synchronization manifold.
• The bound holds across a broad class of active networks, indicating a universal relationship independent of network topology.
• Information production in active networks is more strongly influenced by node interactions than by network structure.
• The two-node subsystem provides sufficient information to predict the dynamical complexity of much larger active networks.
• The Ruelle inequality is effectively leveraged to relate Lyapunov exponents to Kolmogorov-Sinai entropy, enabling entropy estimation.