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

Abstract

In both natural and engineered systems, communication often occurs dynamically over networks ranging from highly structured grids to largely disordered graphs. To use, or comprehend the use of, networks as efficient communication media requires understanding of how they propagate and transform information in the face of noise. Here, we develop a framework that enables us to examine how network structure, noise, and interference between consecutive packets jointly determine transmission performance in complex networks governed by linear dynamics. Mathematically, normal networks, which can be decomposed into separate low-dimensional information channels, suffer greatly from readout noise. Most details of their wiring have no impact on transmission quality. Non-normal networks, however, can largely cancel the effect of noise by transiently amplifying select input dimensions while ignoring others, resulting in higher net information throughput. Our theory could inform the design of new communication networks, as well as the optimal use of existing ones.

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

“…Another aspect that is unavoidably associated with the high asymmetry of real networks is their non-normality [ 13 ], namely their adjacency matrix satisfies the condition [ 12 ]. The non-normality can be critical for the dynamics of networked systems [ 13 , 14 , 15 , 16 , 17 , 18 ]. In fact, in the non-normal dynamics regime, a finite perturbation regarding a stable state can undergo a transient instability [ 12 ], which, because of the nonlinearities, could never be reabsorbed [ 13 , 14 ].…”

“…The effect of non-normality in dynamical systems has been studied in several contexts, such as hydrodynamics [ 19 ], ecosystems stability [ 20 ], pattern formation [ 21 ], chemical reactions [ 22 ], etc. However, it is only recently that the ubiquity of non-normal networks and the related dynamics have been put to the fore [ 13 , 14 , 15 , 16 , 17 , 18 ]. In this paper, we will elaborate on these lines showing the impact of non-normality on the stability of a synchronous state.…”

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.…”

Spreading Control in Two-Layer Multiplex Networks

“…Spreading processes in complex networks have attracted recent attention for the purpose of analyzing the intertwined dynamics of epidemics [ 8 , 9 , 10 , 11 , 12 , 13 ] or information transmission in [ 14 , 15 , 16 , 17 , 18 ]. The control of such problems has to address fundamental questions as (i) which parameters of the system are amenable to manipulation and (ii) which nodes must be actively controlled.…”

“…Spreading processes in complex networks have attracted recent attention for the purpose of analyzing the intertwined dynamics of epidemics [8][9][10][11][12][13] or information transmission in [14][15][16][17][18].…”