A network is commonly represented by a graph G(V, E), which is specified by its node set V = {1, …, N} and edge set E, with N the number of nodes. Thus, the number of possible edges in an undirected graph (meaning the links are non-directional) is N(N - 1)/2. In the current context, each node corresponds to a grid cell at which a valid monthly time series is available and N is the total number of such cells in a gridded data set. Construction of a network generally proceeds in two steps, network growth and pruning. In the network growth step, similarity between all potential node pairs (i.e., edges) in graph G is quantified. Common measures of similarity are statistical correlation (either Pearson or Spearman), mutual information, and synchronization (Boers et al., 2013; Donges et al., 2009a). In the pruning step, an appropriate similarity threshold (τ) is imposed to the edge set to retain only those connections that exceed the threshold. The main purpose of network pruning is to improve network analysis efficiency. If the correlation between two time series is used as a measure of statistical similarity, then τ represents the minimum correlation coefficient (R) above which a pair of nodes is considered connected. The absolute value of correlation is used such that both strongly positive and negative correlations are counted.

Several methods have been used in the CCN literature to determine τ. In the significance testing method (Tsonis et al., 2006), τ is based on the two-sided Student's t test. The critical t value, tc, for a given sample size ns and user-defined significance level α are determined using the Student's t cumulative distribution function (CDF), from which the value of τ can be solved: tc=τns-21-τ2. A similar method uses the probability value (i.e., p value) of test statistics directly: a pair of nodes is considered connected if the p value is less than a critical value; for instance, Steinhaeuser et al. (2011) set the critical value to 10-10. Yet another method defines τ from an edge-density function ρ(τ) defined as ρ(τ)=nc(τ)N(N-1)/2, where nc is the number of active edges retained in a network when the threshold is set to τ. Thus, edge density is closely related to the CDF of R.

Obviously, all methods involve a certain degree of subjectivity. The selection of τ thus incurs a tradeoff between network maneuverability and preservation of network features: if too many edges are included, the main network features will be obscured, not to mention a significant increase in computational effort required to characterize a large network. In this work, the edge-density method is used because it allows for a direct comparison of network properties computed from different data sets (Donges et al., 2009a). Additional statistical analyses (see Sect. 4) are performed to ensure that all statistically significant features are retained in the constructed networks.

The outcome of the network construction process is a Boolean-valued, symmetric N × N matrix, referred to as the adjacency matrix and denoted by A. Elements of A, aij, are set according to the following rule: aij=1,if|Rij|>τ0,otherwise, in which |Rij| is the absolute value of correlation between time series at nodes i and j. A number of network measures can then be applied to A to quantify network topology. The main metrics adopted in this work include the degree of centrality and connection length.

The degree of centrality of a node, ki, is defined as the number of first neighbors of node i and reflects the importance of node i in a network. Regions having high ki values are referred to as “supernodes” in network theory because these nodes tend to have not only local connections but also long-range connections or teleconnections. However, ki itself does not reveal the actual type of connections. Because of nonuniformity of cell areas at different latitudes, the degree of centrality ki is usually weighted by cell areas, leading to the area-weighted connectivity, ACi (Tsonis et al., 2006; Heitzig et al., 2012): ACi=∑j∈nicos⁡λj∑j=1Ncos⁡λj,i=1,…,N, where ni is the set of all first neighbors of the node i, and λj is the latitude of its jth first neighbor. Thus, ACi is a normalized value representing the fraction of the Earth's surface area that a node is connected to.

A classic measure of network integration is the average distance between node i and all other nodes, Di, and is defined as (Rubinov and Sporns, 2010) Di=1N-1∑j∈V,j≠idij,i=1,…,N, where dij is the number of edges traversed along the shortest path between the node pair (i, j). If (i, j) is not connected, dij is defined as infinity. The characteristic path length of the network is obtained by taking an average of all Di and it represents the average number of edges to be traversed along the distance between two randomly selected nodes in a network. Calculation of pairwise shortest path lengths becomes computationally expensive when the number of node pairs is large. In this work, the average distance between node i and all other nodes, Li, is approximated according to Li=1ki∑j∈nilij, where only the first neighbors of node i are included in the calculation, and lij is the physical distance between node pair (i, j) measured by using the respective cell-center latitudes and longitudes, (λi, ϕi) and (λj, ϕj). The physical-based characteristic path length of the network, L‾, is simply the average of all Li (i = 1, …, N). The probability distribution of Li provides a sense of the average edge lengths in a network and L‾ provides a measure of network integration.

The number of possible edges represented by the TWS data sets is more than 100 million for N = 14 540. After removing seasonal trends from GRACE and GLDAS and calculating the correlation coefficient R for all node pairs, the edge-density method is applied to determine a similarity threshold τ. Note in the discussion below, R is calculated at zero lag unless otherwise specified.

Figure 1a shows edge-density functions constructed using GRACE and GLDAS TWS data, both are monotonically decreasing (i.e., fewer connected edges at higher τ values) and are similar in shape. As mentioned in Sect. 2, edge density provides an indicator of the fraction of connected edges at different threshold values. Figure 1b plots the maximum correlation coefficient as a function of edge length, which is defined as the shortest physical distance between a pair of nodes in this work. To arrive at Fig. 1b, all R values are first sorted according to nodal separation distances, a bin width of 250 km is applied to the resulting distribution, and the maximum R value within each bin is recorded. Figure 1b suggests that the maximum correlation stays relatively high (> 0.7) for most distances. Recall that the main purpose of network pruning is to improve the computational efficiency of network characterization while preserving the most significant network features. In this study, we set the threshold τ to 0.57 because (a) the corresponding fraction of connected edges is relatively small (0.036); i.e., at this level more than 96 % of edges is removed; (b) the edge densities of GRACE and GLDAS happen to be the same at that level; and importantly (c) the cutoff τ threshold is still below the maximum correlation exhibited at all separation distances, as suggested by Fig. 1b. Thus, the selected τ value ensures that all statistically significant network features are retained in the constructed networks.

A basin analysis is useful for helping visualize the TWSA connection patterns at the basin level. As some examples, Figure 2 shows the results for six river basins around the world. To generate a plot in Fig. 2, a cell is first fixed, and all its edges are colored according to the actual R (not the absolute values). For our purpose, the centroid of each basin is used. While the basin centroid may not be representative of the connection patterns of an entire basin (especially when the basin spans several climatic regions), it serves as a basis for comparing multiple basins at a qualitative level. Figure 3 applies the cutoff threshold τ defined in Fig. 1 to all plots in Fig. 2. Results suggest that interannual TWSA connections in the Amazon and Congo basins are dominated by local connections. The mid-latitude basins (Ganges, Mississippi, and Tigris) generally show more teleconnections, although Yangtze is an exception. In the case of the Tigris Basin, a large number of strongly positive and negative correlations are observed and the local connections extend far beyond the basin boundary. A detailed interpretation of this observation will be given in the next section.

Extensive teleconnection is an advantage from a forecasting perspective because climate indices, such as El Niño–Southern Oscillation (ENSO) and North Atlantic Oscillation (NAO), can be used as possible indicators of future changes. For those basins without strong teleconnection, water resources planning must rely mainly on regional data. Such distinction sheds light on the significance of GRACE data to long-term basin planning and natural hazard mitigation strategies, as we will elaborate on in the following sections.

As a sensitivity study, Fig. 4 (left column panels) shows the results of basin analysis for the Mississippi Basin, the largest basin in North America, using different thresholds corresponding to τ values of 0.41, 0.57 (the base case), and 0.76. The corresponding edge density is labeled in the figure. Because the cutoff threshold increases as ρ decreases, a significant reduction in the number of edges can be observed. For comparison, the modeled TWS connections obtained from GLDAS are provided in the second column of Fig. 4. In general, the connections modeled by GLDAS are much weaker (i.e., smaller in spatial extent) than those obtained from GRACE. In some cases, the locations of connections are also different. For example, the negative correlation obtained by GLDAS in northern Africa for ρ = 0.1 is not seen by GRACE. The complex networks thus provide a useful tool for examining the agreement, or the lack of it, between GLDAS and GRACE.

Using the selected cutoff τ, a network adjacency matrix A is formed and various network measures described under Sect. 2 are applied to quantify network topology. Figure 5a shows the area-weighted connectivity map constructed using GRACE data. On the map, red colors highlight regions of high connectivity. Recall that a high degree of connectivity indicates that a node interacts strongly with the rest of the nodes in a network (i.e., a supernode); however, the connectivity map itself does not explain the type of connections per se, and needs to be analyzed jointly with the connection length map to be shown in the next section. The largest cluster of supernodes appears in the Middle East region, where the connected neighbors account for more than 0.16 of the global area. To a lesser extent, the Pacific northwest and east coast of the USA, southern Africa, southern South America, and eastern Australia all show smaller supernode regions. In contrast, most of Asia, central USA, and Europe exhibit little or no connectivity (blue color). These observations are consistent with patterns observed during basin analyses (see Figs. 3 and 4).

The supernode regions shown in Fig. 5a reflect the superposed effects of climate variations and anthropogenic activities. These can be explained in terms of global precipitation and atmospheric circulation patterns. In general, the poorly connected regions have stronger precipitation variations over shorter spatial scales, leading to the emergence of high-precipitation gradients which, in turn, are responsible for regional extreme events that are more localized in time and space (Scarsoglio et al., 2013). Those with high connectivity tend to be directly influenced by ocean and climatic oscillations (e.g., ENSO and NAO). Kahya and Dracup (1993) studied streamflow variations in the contiguous USA and identified northeast, northcentral, Pacific northwest, and Gulf of Mexico states as regions with potentially significant streamflow responses to ENSO forcing. These four regions can be easily identified on Fig. 5a, among which the Gulf of Mexico region shows the weakest connection. Similarly, Chiew et al. (1998) reported that the ENSO can be used to help forecast spring runoff in southeast Australia and summer runoff in the northeast and east coasts of Australia. This teleconnection pattern is also indicated clearly by Fig. 5a.

At the global scale, Dai et al. (2009) studied the monthly streamflow records of the world's 925 largest ocean-reaching rivers from 1948 to 2004. They concluded that (a) the interannual variations of streamflows are correlated with the ENSO events for discharge into the Atlantic, Pacific, Indian, and the global oceans as a whole and (b) the effects of anthropogenic activities on annual streamflow are likely to be small compared to those of climate variations; however, anthropogenic activities can create more disturbances in arid and semi-arid regions, where the discharge magnitudes are low (e.g., Indus, Yellow, and Tigris–Euphrates river basins). To elaborate the latter point further, Fig. A1 in Appendix A plots the proportion of total renewable water resources withdrawn by country for human uses in the agricultural, municipal, and industrial sectors, using long-term data compiled by the Food Agricultural Organization of United Nations. Figure A1 indicates that the Middle East and northern African countries show the highest withdraw proportions. In a recent GRACE study focusing on northcentral Middle East, Voss et al. (2013) reported that GRACE data show an “alarming rate” of decrease in TWS of approximately 143.6 km3 during 2003–2009. Thus, the resemblance between Figs. 5a and A1 in those regions is not coincidental and can be corroborated using multiple sources. Because interannual TWS anomalies are well connected in clustered supernode regions, these regions tend to exhibit more vulnerability to both climate and human-induced disturbances.

Having elaborated the close relationship between GRACE TWSA and climate patterns, it is important to point out that the TWS also includes effects of soil moisture and groundwater storage (mostly unconfined aquifers) changes that may not synchronize with climate patterns.

Figure 5b shows the same area-weighted connectivity map, but constructed using the GLDAS-NOAH outputs. Although GLDAS-NOAH shows many of the similar patterns detected by GRACE, it also indicates stronger connectivity in the Arabian Peninsula, northern Africa, and in middle South America, and much weaker connectivity in southern Africa. These discrepancies may be caused by GLDAS-NOAH's parameterization and other errors. The other main reason is the lack of representation of the deeper groundwater storage in GLDAS. The discrepancies highlighted here provide additional spatial calibration constraints for land surface models. In areas dominated by shallow TWS components, GLDAS needs to show similar patterns as those derived from GRACE, whereas discrepancies are only expected in areas dominated by deep TWS components and/or impacted by significant anthropogenic activities. We emphasize here the connectivity maps shown in Fig. 5 are for TWSA. Thus, the high-precipitation areas (e.g., Amazon Basin) do not necessarily exhibit high anomaly connectivity after removing the intra-annual variability.

So far, all results have been based on zero-lag correlations. The effect of temporal lag on connectivity is examined in Fig. 6, in which the connectivity map is built using the maximum (absolute) correlation found between -18 and +18 monthly lags of each node pair. The figure suggests that incorporation of lagged correlation further strengthens connectivity. The supernode regions are more expanded in space, notably in eastern Australia and in the Colorado River basin and Gulf Coast states in the USA. Further, Appendix B shows the maximum correlation and phase lags for the six basins studied in Fig. 2, which suggest that each river basin is in phase with most cells in itself and the immediate surroundings. However, significant phase lags exist between each river basin and other river basins.

Figure 7a shows maps of the physical-based average nodal connection length Li (i = 1, …, N). Nodes that exhibit the longest connection lengths are mostly located in the southern part of South America (∼ 12 000 km). Other regions with relatively long connections are found in Pacific northwest, northcentral, Colorado River, and northeastern regions of the USA, southern Africa, and eastern Australia. Interestingly, the Middle East region is mostly characterized by connection lengths less than 5000 km; thus, the supernodes in that region are dominated by local connections. The connection length patterns observed here support the previous discussions in the context of teleconnection and forecasting potential. Importantly, the connection length map can help evaluate the influence of teleconnection on TWS for a particular region.

The average nodal connection length map constructed using GLDAS data suggests much wider connections, although most are local. Again this can be attributed to model parameterization schemes, forcing resolution, and spatial correlation constraints, as discussed before.

The probability distribution of the average connection length, Li, is shown in Fig. 8. Most nodes in the GRACE network are dominated by short-range edges with lengths less than 2000 km, although several other smaller modes appear in the 4000–6000, 6000–8000, and 8000–10 000 km ranges. In contrast, the GLDAS network shows a weaker local connection mode in < 2000 km range, but a wider and more persistent second mode in 4000–6000 km. Interestingly, the two modes of GLDAS coincide with those of GRACE. The characteristic path length (L‾) is 2300 km for GRACE and 4000 km for GLDAS, respectively.