Resilience for stochastic systems interacting via a quasi-degenerate network

Abstract

A stochastic reaction-diffusion model is studied on a networked support. In each patch of the network two species are assumed to interact following a non-normal reaction scheme. When the interaction unit is replicated on a directed linear lattice, noise gets amplified via a self-consistent process which we trace back to the degenerate spectrum of the embedding support. The same phenomenon holds when the system is bound to explore a quasi degenerate network. In this case, the eigenvalues of the Laplacian operator, which governs species diffusion, accumulate over a limited portion of the complex plane. The larger the network, the more pronounced the amplification. Beyond a critical network size, a system deemed deterministically stable, hence resilient, may turn unstable, yielding seemingly regular patterns in the concentration amount. Non-normality and quasi-degenerate networks may therefore amplify the inherent stochasticity, and so contribute to altering the perception of resilience, as quantified via conventional deterministic methods.Models of interacting populations are of paramount importance in a broad range of applications of interdisciplinary breath. Beyond the simplified arena of deterministic approaches, stochastic effects play a role of paramount importance and might yield a large of plethora of non trivial behaviors. Furthermore, to account for the inherent complexity of the existing interactions, the inspected model are embedded on a network architecture. In this paper, we show how a non-normal reaction model coupled to a directed, quasi-degenerate, network can drive a resonant amplification of the noisy component of the dynamics. This observation, that we here substantiate analytically, calls for a revised concept of resilience, the ability of a system to oppose external disturbances.

Synchronization Dynamics in Non-Normal Networks: The Trade-Off for Optimality

“…Recently, a set of systematic statistical studies has brought to light the fact that most real networks, including biological, ecological, social, transport, economic, etc. [5][6][7][8][9], possess a high level of non-normality with peculiar structural features. In this regard, it has been recently shown for autonomous systems in [7] that despite a negative Master Stability Function (MSF) [10] indicating a strictly stable fixed point, the orbits starting near such a state could escape from it, stabilizing in other equilibrium.…”

Non-normality and non-monotonic dynamics in complex reaction networks

“…Even when a fixed point is linearly stable in a nonlinear system described by ordinary differential equations, if the corresponding Jacobian matrix is non-normal, a small but finite perturbation can transiently grow beyond the validity of the linear approximation and enter into the nonlinear regime, preventing the perturbation from decaying to zero. The discovery of this phenomenon has led to the thorough study of the spectral properties of non-normal matrices in the context of transient dynamics [2]; it has also inspired recent works on implications of non-normality for network and spatiotemporal dynamics [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Given the common perception that linear dynamics are fully understood, the possibility of such transient growth offers interesting alternative interpretations for behavior usually attributed to nonlinearity, such as ignition dynamics in combustion and temporary activation of biochemical signals.…”

Efficient communication over complex dynamical networks: The role of matrix non-normality

“…Non-normal networks appear ubiquitous, with strong non-normality having been found in food webs, transport, and biological, social, communication, and citation networks (22). In addition, we mention that, besides information transfer, non-normality turns out to be the key to explaining and understanding a variety of other equally important phenomena, for instance, the process of pattern formation in natural and biological systems (21,35), the selective amplification of cortical activity patterns in the brain (32), and the emergence of giant oscillations in noise-driven dynamical systems (36)(37)(38).…”

“…Also, noise could play an active role in the information transfer process as the input source of the communication channel. This change of perspective could lead to an information-theoretic interpretation of the findings in (36)(37)(38), wherein non-normality has been linked with the emergence of amplified oscillations in noise-driven interconnected nonlinear systems.…”