[Paper Review] On Monotonicity and Propagation of Order Properties

Aivar Sootla|arXiv (Cornell University)|Mar 9, 2015

Gene Regulatory Network Analysis10 references4 citations

TL;DR

This paper establishes a link between deterministic monotonicity and stochastic order propagation in Markov processes, showing that systems with convex vector fields propagate the increasing convex order (F_icx), while those with standard monotonicity propagate the usual order (F_d). The key result is that Linear Noise Approximations (LNAs) of unimolecular reactions preserve the F_icx order through their mean and covariance, enabling robust comparison of biological processes via moment-based analysis.

ABSTRACT

In this paper, a link between monotonicity of deterministic dynamical systems and propagation of order by Markov processes is established. The order propagation has received considerable attention in the literature, however, this notion is still not fully understood. The main contribution of this paper is a study of the order propagation in the deterministic setting, which potentially can provide new techniques for analysis in the stochastic one. We take a close look at the propagation of the so-called increasing and increasing convex orders. Infinitesimal characterisations of these orders are derived, which resemble the well-known Kamke conditions for monotonicity. It is shown that increasing order is equivalent to the standard monotonicity, while the class of systems propagating the increasing convex order is equivalent to the class of monotone systems with convex vector fields. The paper is concluded by deriving a novel result on order propagating diffusion processes and an application of this result to biological processes.

Motivation & Objective

• To clarify the connection between deterministic monotonicity and stochastic order propagation in Markov processes.
• To provide infinitesimal characterizations of increasing and increasing convex orders in deterministic systems.
• To identify conditions under which Markov processes propagate stochastic orders, especially F_icx and F_d.
• To apply these results to biological systems modeled via Linear Noise Approximation (LNA).
• To demonstrate that unimolecular reaction networks preserve the F_icx order through their mean and covariance matrices.

Proposed method

• Derives infinitesimal conditions for F_d and F_icx order propagation using gradient and Hessian analysis of vector fields.
• Shows that F_d order propagation corresponds to standard monotonicity, while F_icx propagation requires convex vector fields.
• Applies techniques from stochastic calculus and conditional expectation to analyze discrete-time LNA processes.
• Uses Jensen’s inequality and positive semidefiniteness of covariance differences to prove order preservation.
• Employs the conditional expectation framework to compare distributions via moment-based stochastic orders.
• Analyzes the transition operator of the LNA process to verify propagation of the F_icx order under specific network structures.

Experimental results

Research questions

• RQ1 What deterministic conditions ensure propagation of the increasing convex order (F_icx) in dynamical systems?
• RQ2 How does the structure of the vector field relate to the propagation of stochastic orders in Markov processes?
• RQ3 Which classes of biochemical reaction networks preserve the F_icx order under Linear Noise Approximation?
• RQ4 Why is the F_icx order more suitable than F_d for comparing unimolecular reaction processes?
• RQ5 Can moment-based analysis (mean and covariance) be used to infer stochastic order relations in LNA processes?

Key findings

• The class of systems propagating the F_icx order is equivalent to monotone systems with convex vector fields.
• The F_d order is equivalent to standard monotonicity, corresponding to almost-sure sample path comparison.
• Only decoupled birth-death processes in LNA propagate the F_d order, limiting its applicability to biological systems.
• All unimolecular reaction networks in LNA propagate the F_icx order, which is preserved through the mean and covariance matrix of the process.
• The F_icx order enables comparison of system distributions via moment-based stochastic dominance, even when sample path comparison fails.
• The proof relies on conditional expectation and Jensen’s inequality, showing that E[h(TZ₂)|Z₁] ≥ E[h(TZ₁)] for all convex functions h.