In this study, we analyze the statistical and topological measures of abrupt precipitation and PET changes at multiple scales, ranging from regional to global scales. For global-scale analysis of early-warning signals, we retrieved the long-term (1901–2018) gridded global monthly precipitation and PET data with a spatial resolution of 0.5∘×0.5∘ from the University of East Anglia Climatic Research Unit Time-Series (CRU TS) version 4.03 dataset (Centre for Environmental Data Analysis, 2022). This gridded dataset covers all land domains of the world except Antarctica (Harris et al., 2020). CRU TS was produced using the angular distance weighting (ADW) method to interpolate monthly climate anomalies based on extensive weather station observations. Here, we calculated the annual global means of precipitation and PET over land as the weighted averages of the Northern Hemisphere and Southern Hemisphere, following Osborn and Jones (2014). Note that means for each hemisphere are the areal-weighted averages of all non-missing values.

For regional-scale analysis of early-warning signals, we obtained annual precipitation time series for selected cities in the CONUS from the National Oceanic and Atmospheric Administration (NOAA) National Centers for Environmental Information (NCEI) Climate at a Glance database (NOAA, 2022). This city-level database contains monthly temperature and precipitation data for 215 US cities: 27 cities have data recorded by Automated Surface Observing System (ASOS) stations, and the remaining 188 cities use Global Summary of the Month (GSOM) data. In particular, the GSOM dataset is based on the Summary of the Day observations of the Global Historical Climatology Network–Daily (GHCN-Daily) dataset (Menne et al., 2012), in which the total monthly precipitation is based on daily or multiday (if daily is missing) precipitation reports. GHCN-Daily data have been quality controlled using a suite of automated algorithms designed to detect as many errors as possible while maintaining a low false-positive rate (valid observations erroneously identified as invalid) (Durre et al., 2010). Additional quality control has also been performed with a validation process that involves independent calculations and cross-comparisons to ensure computational accuracy.

For the subsequent network analysis, we focus on precipitation over all cities in the CONUS. We retrieved the 1 km × 1 km monthly precipitation data over the region (1980–2018) from the Daymet Version 3 dataset (Thornton et al., 2018). This dataset uses spatial convolution of a truncated Gaussian weighting filter applied to a number of stations (on average 15 for precipitation) (Thornton et al., 1997). Daymet uses daily precipitation measured by ground-based meteorological stations from the GHCN-Daily dataset (Menne et al., 2012). In contrast to the regional-scale analysis of early-warning signals (single station for each city), here we derived the time series of monthly precipitation spatially averaged over each urban area in the CONUS based on Daymet dataset. Note that urban areas (or cities) in this study are defined as areas with densely developed land and a population over 50 000, and the city boundaries are retrieved from the Topologically Integrated Geographic Encoding and Referencing (TIGER) system, US Census Bureau (2022).

The hydrological system dynamics often evolve on top of complex topological structures such as climate networks (Konapala and Mishra, 2017); their interactions modulate the potential tipping of the system. In this section, we extend the analysis of early-warning signals for precipitation transitions beyond the changes in conventional means (AR1 and variance) and investigate the system structure represented by the precipitation network of all CONUS cities.

In its simplest form, a network (or graph) can be mathematically represented as a group of nodes or vertices that are connected together. The connection between a pair of nodes is called a link (edge), representing the similarity (or inverse distance) of attributes of the two nodes. Note that we focus here on undirected and unweighted networks only, although the proposed procedure of analysis in this study remains applicable to more complicated (e.g., directed and/or weighted) networks with slight modification.

More specifically, in this study, we treat 481 CONUS cities (see Sect. 2.1) as nodes and construct the network based on monthly precipitation retrieved from the Daymet dataset (Thornton et al., 2018) for the period from 1980 to 2018 (see Sect. 2.1). The connectivity of precipitation networks is constructed using the Pearson correlation coefficient (a commonly adopted distance function) between monthly precipitation time series of two cities i and j (ρij), which is defined as follows: 7ρij=EPi-μPiPj-μPjσPiσPj, where Pi is the precipitation time series, with the subscript i denoting the node (city) number; and μ and σ are the mean and standard deviation of the precipitation time series, respectively. Note that the correlation coefficient is only one kind of measure to describe connectivity, and other similarity (or dissimilarity) functions (e.g., Minkowski distance) (Wang et al., 2020a) are also applicable in network construction. The connectivity between a pair of nodes forms the adjacency matrix A (or Aij, i, j=1, 2, …, N), which is defined as follows (Tsonis and Roebber, 2004): 8Aij=Θρij-ρthreshold, where Θ is the Heaviside step function, ρthreshold is the threshold value, and N is the number of cities. Here, we choose a threshold of 0.5, which is a statistically significant example, as suggested by previous studies (Tsonis and Roebber, 2004; Wang and Wang, 2020).

To investigate the topological feature of the precipitation network and how it evolves over time, four representative metrics are then analyzed: the mean distance (or equivalently the network-average Pearson correlation coefficient), the density, the network modularity, and the clustering coefficient. The mean distance describes the overall connectivity of the network and is given by (Tsonis et al., 2007) 9d(t)=2N(N-1)∑dijt∈Dtdijt, where d(t) denotes the mean network distance at current time t at the center of a sliding window, dijt represents the distance between a pair nodes i and j at time t, and Dt is the distance matrix for each sliding window. Equivalently, the mean network distance can be measured using the Pearson correlation coefficient: 10dijt=21-|ρijt|. The distance can be thought of as the average correlation between all possible pairs of nodes and is interpreted as a measure of synchronization among different components of a network: a distance of zero corresponds to a complete synchronization (Tsonis et al., 2007). The mean (nodal) Pearson correlation coefficient is 11ρ(t)=2N(N-1)∑ρijt∈Dtρijt, and it is clear that the mean distance and the mean Pearson correlation coefficient are inversely correlated when the latter is positive.

The density of a network is the fraction of edges that are present and can have a value between zero and one. It is calculated as follows: 12ρ′=2mN(N-1), where m is the total number of edges. The density represents the probability that a pair of nodes picked at random from the whole network are connected by an edge. It plays an important role in the random graph model (Newman, 2018). The larger the density value, the denser the network.

The modularity is the number of edges falling within groups minus the expected number in an equivalent network with edges placed at random. It describes how strong the community structure is (Newman et al., 2006) and is defined as follows: 13Q=12m∑ijAij-kikj2mδgigj, where gi is the group to which node i belongs, δij is the Kronecker delta function, and ki is the degree (number of links) of node i. A network with positive and large modularity values is preferable for research and real-world applications (Newman and Girvan, 2004).

The clustering coefficient of a network is defined as follows: 14C=1N∑i=1Nnikiki-1/2, where ni is the number of edges among the nearest neighbors of the ith node. Topologically, the clustering coefficient is the fraction of paths with the length equal to two in the network that are closed, and it quantifies the extent to which a pair of nodes with common neighbor are also neighbors of each other (Newman, 2018).

We first identify the conventional early-warning signals at the global scale, the AR1 and SD, for the potential evapotranspiration and precipitation anomalies using the algorithm detailed in Sect. 2.2. Figure 2 shows PET and precipitation anomalies and the early-warning signals prior to critical transitions. The critical transition of global PET occurred in the year 1994 (denoted by the red dashed line in Fig. 2a), which is generally consistent with the transition of the solar radiation trend at the Earth's surface in ∼1990 (Pinker et al., 2005; Wild et al., 2005). This transition from decreasing to increasing global radiation (also known as the transition from global dimming to brightening) has been found in many observational records, primarily due to changes in cloudiness, aerosol loadings, and atmospheric transparency (Pinker et al., 2005). It is clear from Fig. 2a that both the AR1 and SD of global PET increased gradually over time for more than 1 decade (one sliding window) prior to the critical transition. Per theoretical analysis in Sect. 2, the increase in the AR1 and SD apparently presages the critical slowing down of the rate of recovery from perturbation, as the system evolved approaching the transition in 1994.

We then proceed to identify early-warning signals for precipitation at the city scale, following the same procedure as that utilized for the global-scale analysis above. Figure 3 shows the results for four CONUS cities (New York City, NY; Seattle, WA; Fresno, CA; and Miami, FL). These cities were preselected as being representative of the distinct regional geographic conditions and background climates of their locations. Note that the monthly precipitation data are from 1895 to 2019 (125 years) for New York City and Fresno, whereas the data are from 1948 to 2019 (72 years) for Seattle and Miami. For individual cities, the times (years) at which the critical transition occurred in the precipitation records differ: 1967 for New York City, 1993 for Seattle, 1971 for Fresno, and 1993 for Miami. The results are generally in good agreement with studies on climate phase changes. Those phase changes have been associated with significant changes in climate variability, such as global temperature, ENSO, and the IPO. For example, the switch of the IPO from a cold phase to a warm phase around 1977 induced a clear upward trend in precipitation over much of the western and central regions (e.g., Oklahoma, Kansas, and Missouri) of the CONUS, whereas a latter shift in around 1999 (back to a cold phase) resulted in decreased precipitation (Dai, 2013). The results of early-warning signals are also shown in Fig. 3. Here, again, we find that the statistical measures of the AR1 and SD generally increase with time in all four cities prior to the emergence of transitions.

In this section, we extend the concept of early-warning signals for critical transitions in hydrological systems from conventional time series analysis to the topological analysis of their network structure. We first construct the precipitation networks based on the monthly precipitation data for all CONUS 481 cities in the period from 1980 to 2018 retrieved from the Daymet dataset (Thornton et al., 2018), with a threshold similarity of 0.5 (detailed in Sect. 2.3). The CONUS precipitation network generated using the entire time series (39 years × 12 months) is shown in Fig. 4a. All CONUS cities are further subdivided into nine geographic regions following Wang and Wang (2020), as listed in Table 1. The connectivity, represented by the adjacency matrix, of the precipitation network is shown in Fig. 4b, with the nine regions marked.

From Fig. 4, it is clear that the 39-year aggregated CONUS urban precipitation network is largely occupied by intra-regional connection, manifest as regional clusters on the geographic map (Fig. 4a) and dark (connected) diagonal blocks in the adjacency matrix (Fig. 4b). This is consistent with a clustering analysis based on monthly precipitation (1981–2010) in CONUS cities (Wang et al., 2020a). This is physical, as the patterns of precipitation and its anomalies in a city are predominated by the geographic controls and background climate conditions in the region where the city is located. Thus, the dense intra-regional connections (diagonal blocks in Fig. 4b) reveal the similarity of precipitation patterns among cities with a similar climate environment, or, in other words, “like is connected to like” (Newman and Girvan, 2004). This topological feature also shows that the CONUS precipitation network is highly assortative with large modularity (Q=0.589), as the modularity reveals network community structures.

In addition to the modular structure reflected as dense intra-regional connectivity, the precipitation network also includes some inter-regional (Fig. 4a) or “off-diagonal” (Fig. 4b) connections. The presence of long-range, out-of-region connectivity is usually formed by complex interactions via atmospheric gateways (teleconnection) (Boers et al., 2019). For example, it is found that precipitation patterns are similar in the Ohio Valley (Region 1) and Upper Midwest (Region 2) (same for the West and Northwest), which is shown as the off-diagonal connection in Fig. 4b. Similar hydrological teleconnections have been observed in Konapala and Mishra (2017) and Wang et al. (2020a). For example, Konapala and Mishra (2017) found that the Ohio Valley region plays an important role in propagating droughts (with higher values of outward strength) toward adjacent regions such as the Upper Midwest (Region 2) and Northeast (Region 3), consistent with the off-diagonal connections shown in Fig. 4b.

To find how the network structure responds to (or presages, as early-warning signals of AR1 and SD) critical transitions in the CONUS precipitation system, we then proceed to construct the time-varying networks (cf. the aggregated 39-year network in Fig. 4) using a sliding window of 7 years (following Sect. 2.2) from the same 39-year (1980–2019) set of Daymet precipitation data. In addition, we follow the same procedure for analyzing the time series of cumulative precipitation in the CONUS and determine the year of critical transition at the emergence of abrupt change in the slope of cumulative precipitation. In this case, the critical transition for precipitation in the CONUS urban areas was around the year 1998, which is consistent with the IPO phase changes in the 1990s (Dai, 2013; Deser et al., 2004). This can also be seen from Fig. 5a, as the early-warning signal of SD had a clear increasing trend, whereas the trend of AR1 was slightly obscure with a 7-year window size. Nevertheless, a clear increase immediately prior to the transition is still observed for the AR1 in Fig. 5a.

In addition, we also find that the network topology has a significant response to the critical transition, as shown in Fig. 5. The mean nodal correlation coefficient ρ, the network density ρ′, and the clustering coefficient C all exhibited a clear trend of increase prior to the transition. Note that ρ is the structural parameter prior to the construction of networks, whereas ρ′ and C are posterior parameters determined from the topology of constructed networks. Furthermore, the mean distance d is (nearly) inversely correlated with the mean nodal correlation coefficient, so it is expected that d decreased prior to the transition. The increase in ρ (or, equivalently, the decrease in d) suggests that, on average, the connectivity among all cities in the CONUS precipitation network was enhanced prior to the transition. This is likely attributable to the strengthening of synchronization of precipitation events (Boers et al., 2016; Rheinwalt et al., 2016) via, e.g., propagation of precipitation fronts entraining larger urban areas than before, as a result of critical slowing down in recovery from system perturbation (e.g., extreme rainfalls). Similar patterns of increases in structural correlation ρ and decreases in mean distance have been observed in other climate networks by Tsonis et al. (2007).

The enhanced network connectivity prior to critical transitions, manifested as an increase in nodal correlation ρ (or a decrease in mean distance d), led to the increase in the total number of links and the network density ρ′ (the fraction of edges in the network) (Fig. 5b). The clustering coefficient C also increased as a result of enhanced community structure predominated by the intra-regional connections (Fig. 4). In contrast, the modular structure of the precipitation network remained insusceptible to the emergence of critical transition and enhanced synchronization of precipitation events, with the value of modularity Q fluctuating around 0.55–0.57 in the 7-year window size. This is seemingly because the critical transition tends to enhance both the local community structure (intra-regional connection) and the teleconnection (inter-regional connection), thereby leaving the modular organization of the entire network unchanged. Qualitatively, this means that the topological structure of the CONUS precipitation network, despite its temporal evolution and changes in the overlying dynamics (the occurrence of critical transition or slowing down), remains “like is connected to like”.