Parameter Space Diversity as a Measure of Robustness

Robustness, the ability to preserve a desired state under environmental fluctuations, is encountered across various self-organized complex systems [1]. Despite its significance, it is traditionally inferred through its correlates, stability, and resilience [2, 3, 4]. Such analyses consist of modeling the system mathematically, assuming that parameters can be known and reliably measured, which may not be the case. Furthermore, they either fail to cover the variations in the parameter values through which the external disturbances act on the system or overlook the existence of numerous stable states. Motivated by these shortcomings, we introduce the notion of parameter space diversity, similar to ecological diversity, for robustness assessment. We claim that systems with fewer distinct states in the span of their parameters shall be more robust, and validate it experimentally on a model self-organized system, a mode-locked fiber laser. Our approach is statistical and grounded in the probability of maintaining the same state, which turns out to be equivalent to the Simpson diversity index [5]. We further demonstrate, theoretically and experimentally, that our methodology holds even under sparse random sampling, removing the requirement of exhaustive measurements for rapid robustness inference. In addition to guiding the development of better lasers, we expect that parameter space diversity will pave the way for robustness quantification in other complex systems.

References:

1. Carlson, J.M., Doyle J.: Complexity and robustness. Proc. Natl. Acad. Sci. USA 99, 2538-2545 (2002)

2. Lyapunov, A. M.: The general problem of the stability of motion. Int. J. Control 55, 531-524 (1992)

3. Menck, P.J., et al.: How basin stability complements the linear-stability paradigm. Nature Physics 9, 89–92 (2013)

4. Gao, J., Barzel, B., Barabási A.: Universal resilience patterns in complex networks. Nature 530, 307-312 (2016)

5. Simpson, E. H.: Measurement of diversity. Nature 163, 688 (1949)