Emergent stability in complex network dynamics

Abstract

Will a large complex system be stable? This question, first posed by May in 1972, captures a long standing challenge, fueled by a seeming contradiction between theory and practice. While empirical reality answers with an astounding yes, the mathematical analysis, based on linear stability theory, seems to suggest the contraryhence, the diversity-stability paradox. Here we settle this dichotomy, by considering the interplay between topology and dynamics. We show that this interplay leads to the emergence of non-random patterns in the system's stability matrix, leading us to relinquish the prevailing random matrix-based paradigm. Instead, we offer a new matrix ensemble, which captures the dynamic stability of real-world systems. This ensemble helps us analytically identify the relevant control parameters that predict a system's stability, exposing three broad dynamic classes: In the asymptotically unstable class, diversity, indeed, leads to instability à la May's paradox. However, we also expose an asymptotically stable class, the class in which most real systems reside, in which diversity not only does not prohibit, but, in fact, enhances dynamic stability. Finally, in the sensitively stable class diversity plays no role, and hence stability is driven by the system's microscopic parameters. Together, our theory uncovers the naturally emerging rules of complex system stability, helping us reconcile the paradox that has eluded us for decades.-all naturally emerging features of real-world networks -indeed, impact the system's dynamics [25][26][27][28][29][30] . However, on their own, these topological features are insufficient to ensure stability 31,32 , leaving May's mathematical challenge unresolved.Here, we show that the desired devious strategies arise not just from the network topology, but mainly from its interplay with the system's intrinsic, often nonlinear, interaction dynamics. Constrained by the physical mechanisms that drive the system's interacting components, these dynamics provide the desired, naturally emerging, design principles that determine the system's stability. Our analysis shows that, under real-world dynamics, stability can be fully predicted from a small set of relevant parameters that capture the combined contribution of both topology and dynamics. Most crucially, we identify a broad class of frequently encountered mechanisms, for which a large complex system can, and, at times, even must be stable, therefore settling the long overdue diversity-stability debate 33,34 .

Information transfer pathways: signal propagation in complex global topologies

“…Furthermore, the perturbation Δx(t) for states x(t) satisfies the following dynamic equation, obtained by subtracting equations (1) from (2). The detailed proof has been provided in [18][19][20].…”

“…Here, we apply near linear method proposed in Hu et al [19] and propagation time for path proposed in equation (20), and   ( ) m i itp is denoted as information transfer paths.…”

“…The outbreak and spread of infectious diseases from one location to all over the world [1][2][3][4][5][6][7], the control of proteins or genes over the biochemical processes [8][9][10][11][12], and many other dynamic processes involve the propagation of perturbations from some nodes to the overall network [13], which indicates that the patterns of global propagation for small perturbations have a wide range of applications in the real world [14]. Specifically, studying the propagation time of dynamical systems can effectively monitor the speed of signal propagation, which is of great significance for research on network control and network stability [15][16][17][18][19][20][21]. However, due to the complexity and heterogeneity of network interactions, conventional analytical tools are often challenging to decouple global topology in real complex networks, and are difficult to apply to the field of signal propagation.…”

The biological role of local and global fMRI BOLD signal variability in human brain organization

“…It is a crucial feature of complex ecological and biological systems 1 : from day-to-day climate variability driving climate change and climate mitigation efforts 2,3 , to heart rate variability serving as a clinical tool to predict overall cardiovascular health and mortality 4 . Acting as a catalyst for system adaptability, variability determines the organization of complex systems, defines their spatiotemporal properties and guides their behavior 5 . Gaining insights into the variability of a system may thus unlock a deeper understanding of the system from which variability emerges.…”

Network Information Dynamics Renormalization Group

“…Furthermore, network science has been used to study the flow of information within networks, from the electrochemical signals traveling between brain areas and pathogens spreading among social systems to the flights in airline networks and the messages being shared on social media. To this aim, often, a dynamical process is exploited that governs the short-to long-range flow between the components [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41] . Yet, the largeness of many real-world systems poses a challenge, due to computational complexity and components' redundancies in roles and features, hampering a clear understanding of how complex systems operate.…”