1.6.1.4 Network methods

Another way to monitor resilience loss in systems is to conceive of them as a network. In a spatial system, this would involve edges connecting neighbouring points. This framework can be applied to other, non-spatial, systems which are not necessarily linked in space but through other variables. 

Multivariate systems (i.e. systems with multiple measurable variables) can pose problems for early warning signals. For instance, two different variables may give conflicting information, or obscure a clear signal (Boerlijst et al., 2013; Weinans et al., 2021). Multivariate systems relevant for climate science include examples such as interaction networks with different plant or animal species, or spatial systems where every grid cell can be represented as a variable in the system or a node in the network (Tsonis and Roebber, 2004; Donges et al., 2009).

Changes in network structure can show an approaching tipping point and have been observed in some systems, including climate (Lu et al., 2021) and lake systems (Wang et al., 2019). More generally, monitoring structural changes properties (e.g. connectivity, node centrality) in network systems (i.e. a network of interacting components, such as spatially connected sites, interacting actors, or species in a community (Mayfield et al., 2020; Cavaliere et al., 2016; Yin et al., 2016) can be used for EWS. Alternatively, correlations in time between components in multivariate systems has been used to construct an interaction network and analyse its structural properties (Tirabassi et al., 2014).

Once the nodes – or variables – are chosen, there are a number of ways the analysis can proceed. One such method evaluates network statistics. To create a network, the method calculates if the correlation between each set of two nodes is above a predetermined threshold and, if it is, connects the two nodes with an edge (a network connection). If this analysis is repeated on a moving window (measuring the correlation between two variables on a moving window like the temporal EWS), changes in the network topology (i.e. the arrangement of node connections) over time can be used as EWS. For instance, as the system moves towards a tipping point, the network will display a higher number of connections between nodes and an increase in variance in connections (Kuehn et al., 2013). 

Unlike spatial methods, which examine a ‘snapshot’ of the system at a given time, these methods require the use of a time window to measure the changing structure on, and thus reasonably complete time series are needed. Another possible disadvantage is that, in some networks, the edges do not necessarily have a physical foundation (Ebert-Uphoff and Deng, 2012). Recent research explores a complementary approach where causal links are calculated instead of correlation links and the strength of the causal link works as the indicator of resilience (Nowack et al., 2020; Setty et al., 2023).

Alternatively, ‘dimension reduction’ techniques can capture overall network dynamics into a representative statistic. For instance, Principal Component Analysis (often referred to as ‘Empirical Orthogonal Functions’ (EOF) in climate science) can be used on a time series to get directions of change (Held and Kleinen, 2004; Weinans et al., 2019), although these linear projections may eliminate existing tipping points so care must be taken. It can often be used in spatial systems to detect the leading mode of variability over a region, such as a climate index like the Pacific Decadal Oscillation (Mantua and Hare, 2002). Next, data can be projected onto the leading principal component, effectively yielding a univariate (i.e. single variable) time series on which time-based univariate EWS can be calculated (Held and Kleinen, 2004; Bathiany et al., 2013; Boulton and Lenton, 2015). 

From a network point of view, this analysis does not make any a priori assumptions about the interactions between the different network nodes, and is therefore quite flexible in its use. However, it requires large amounts of high-quality data to yield accurate results. The underlying assumption is that, as the system approaches the tipping point, the dynamics become more correlated, leading to a high explained variance of a PCA and clear directionality in the dynamics (Lever et al., 2020).