Early warning signs for saddle-escape transitions in complex networks

Abstract

Many real world systems are at risk of undergoing critical transitions, leading to sudden qualitative and sometimes irreversible regime shifts. The development of early warning signals is recognized as a major challenge. Recent progress builds on a mathematical framework in which a real-world system is described by a low-dimensional equation system with a small number of key variables, where the critical transition often corresponds to a bifurcation. Here we show that in high-dimensional systems, containing many variables, we frequently encounter an additional non-bifurcative saddletype mechanism leading to critical transitions. This generic class of transitions has been missed in the search for early-warnings up to now. In fact, the saddle-type mechanism also applies to lowdimensional systems with saddle-dynamics. Near a saddle a system moves slowly and the state may be perceived as stable over substantial time periods. We develop an early warning sign for the saddle-type transition. We illustrate our results in two network models and epidemiological data. This work thus establishes a connection from critical transitions to networks and an early warning sign for a new type of critical transition. In complex models and big data we anticipate that saddletransitions will be encountered frequently in the future.The low-dimensional systems that are commonly investigated in the context of critical transitions (or tipping points) 1,2 typically settle to stable, but not necessarily stationary, states 3 . The defining features of such states is that the system returns exponentially to the state after sufficiently small perturbations 4 . When environmental parameters change, a critical transition may occur when thresholds are crossed, where the system becomes unstable to certain perturbations. Low dimensional systems then typically depart quickly from the original state before approaching another, possibly distant, state. The thresholds at which such transitions occur are called bifurcation points 5 and the corresponding transitions are local bifurcations 3,6 . Warning signs for such bifurcation-induced transitions detect the loss of exponential restoring dynamics either through its impact on the statistics of noise-induced fluctuations 1,7,8 or by direct measurement of recovery rates 9,10

Anticipating epidemic transitions with imperfect data

“…For mathematical reasons, proposed EWS for disease emergence have assumed access to regular recordings ("snapshots") of the entire infectious population [8][9][10][11][12][13]. However, epidemiological data are typically aggregated into periodic case reports subject to reporting error.…”

“…Combined, they can make each pathogen's emergence seem idiosyncratic. In spite of this apparent particularity, there is a recent literature on the possibility of anticipating epidemic transitions using model-independent metrics [6][7][8][9][10][11][12][13][14]. Referred to as early-warning signals (EWS), these metrics are summary statistics (e.g.…”

“…A major obstacle to deploying early-warning systems is the type of data available to calculate the EWS. Theoretical predictions assume the data will be sequential recordings (or "snapshots") of the true number of infected in the population through time [8][9][10][11][12][13]. In this paper we refer to this as snapshots data.…”

Linear stability theory as an early warning sign for transitions in high dimensional complex systems

“…More recently in [14] the authors describe transitions in a different way. In this new interpretation transitions are named saddle-escape transitions and metastable states are interpreted as high dimensional saddle points.…”

“…the dot-com bubble in 1997-2000). Contrary to [14], to build our indicator we do not make use of a rigid macroscopic variable to monitor the system; instead we analyse the interactions between the single microscopic components.…”