To infer the functional interactions within and between spatio-temporal gridded data sets of climatological parameters, we adopt the δ-MAPS algorithm proposed by . This algorithm is rooted in network sciences/graphical modelling, in which graphs are used to express the dependence structure between random variables. A graph or network consists of a set of nodes connected by a set of edges, which describe the interactions between the nodes. Networks can be classified depending on their topology: simple networks like lattices and fully connected networks or complex networks like scale-free and small-world networks. Small-world networks are often observed in climate and other earth sciences, in the human brain, and in social networks. Their nodes are strongly clustered into semi-autonomous components, and the average shortest path length between any two nodes is small.

In contrast to structural networks or flow networks, where the edges are physically observable (like wired connections or trajectories of particles, respectively), functional networks are inferred from the behaviour of the nodes. We consider the grid cells of a selected climatological field as the nodes of the graph. The spatial embedding is naturally given by the locations of the grid cells. In the edges of a fully connected grid-cell-level network are defined using the unpruned Pearson correlation ϱ of the time series as an association measure between any pair of nodes. Based on this weighted network, the δ-MAPS algorithm identifies semi-autonomous components D1…DK, called domains. A domain is a spatially contiguous set of grid cells with highly correlated temporal activity. propose an iterative algorithm that alternately expands and merges a preliminary set of domain seeds S (neighbourhoods with locally maximal correlation, 3×3 grid cells in our case) so as to find the maximum possible sets of grid cells that satisfy the homogeneity constraint δ: let D be a spatially contiguous set of grid cells with cardinality |D| 1δ≤ϱD:=1|D|(|D|-1)∑i≠j∈Dϱij, where ϱij is the correlation between the time series at grid cells i and j, and δ is a chosen parameter to regulate the number and size of the domains. The domains are expanded to neighbouring grid cells (one at a time) as long as ϱD≤δ. Two domains Di and Dj are merged if they contain at least one pair of adjacent grid cells, and their union still satisfies the threshold δ. The algorithm stops when no more domains can be merged or expanded.

The number of domains K generated by this algorithm is not predefined. Overlapping domains are allowed in δ-MAPS because grid cells might be influenced by more than one physical process. If a grid cell does not satisfy the homogeneity constraint with any of its neighbours, it remains unassigned. Deviating from , we use Spearman's rank correlation to determine the similarity between grid cells to allow for monotone, yet non-linear association. Furthermore, we set the threshold δ for minimal average correlation within a domain to equal a selected high quantile of all pairwise correlations (our δ is not based on a significance test; therefore there is no need to correct for auto-correlation). Lower thresholds allow the domains to expand and merge, further resulting in a smaller number of spatially larger domains, which means lower parcellation, and vice versa. In Sect. 4, we choose δ so as to produce “intuitive” domains evocative of known teleconnection patterns.

In , the identification of domains was further refined: grid cells are assigned to a common domain if their time-varying complexity (quantified by recurrence entropy) evolves coherently. Coherent evolution of complexity reflects coherent dynamical evolution and is thus an even stronger indicator of semi-autonomous component organisation than correlation between the original climatological time series. But for complexity time series to be constructed, the proposed recurrence measure has to be evaluated on moving time windows (100-year windows over 6000 years of monthly values in ). Unfortunately, our time series are not long enough to detect complexity changes by means of recurrence entropy (nor to actually occur in the real climatological fields), so we have to stick to the original definition of δ-MAPS in .

The first stage of δ-MAPS is a local community detection algorithm, where the criterion to maximise is the number of grid cells assigned to a minimum number of communities under the conditions (i) ϱD≥δ, (ii) D being spatially contiguous, and (iii) D containing a seed s∈S . As this problem is NP-hard (solvable in polynomial time; ), the greedy algorithm of only approximates one possible solution. Despite this, it is able to detect meaningful communities of any size (no preferred scale) and independently from the network structure in other spatial regions.

Subsequently, the domains identified above serve as super-nodes in the second stage of δ-MAPS. A functional weighted network is inferred between the domains on the basis of a dependence measure (in , the lagged Pearson correlation is used; we use distance correlation; see Sect. 3.3). The time series of a domain is defined as 2xD=(xD1…xDT),xDt=1∑i∈Dcos⁡φi∑i∈Dxitcos⁡φi, where φi is the latitude of grid cell i. In contrast to , we use the means instead of the sums of the grid cells for domain time series. We do so because otherwise the variances of the domains would grow with their size, something that would hinder interpretation. On the other hand, the spatial correlation within the domains, the precondition for grid cells to form a domain, impedes the decrease in the variance of the domain mean following the central limit theorem at the rate of |D|. Instead, the variances of the domain means are of comparable magnitude regardless of the domain size.

Every possible link with every possible lag -L≤l≤L is tested for significance, which constitutes a multiple-testing problem such that the cumulative probability of type I errors increases. One way to control the false discovery rate FDR to be smaller than a predefined level α was proposed by : the p levels of the individual tests are in ascending order, p(1)≤…≤p(12K(K-1)(2L+1)), and the hypothesis (H0; link is insignificant) is rejected only for those tests where p(k)<2kαK(K-1)(2L+1).

The network consists of two maps, D (D: set of nodes (grid cells) ⟶ power set of domains PD1…DK, which assigns one/several/no domains to every grid cell) and W (W: set of pairs of domains  {D1…DK}×{D1…DK}⟶ maximal (lagged) dependence ∈R, which assigns every pair of domains a link that equals the maximal (lagged) dependency between them; we allow lags up to 10 seasons).

The distinction between grid cells that are dependent within the same domain and grid cells that are dependent across two different domains allows δ-MAPS to differentiate between local diffusion phenomena and remote interactions as for instance an atmospheric bridge or an oceanic tunnel .

Since the techniques to construct the δ-MAPS network are statistical, long time series are convenient in order to obtain robust estimates of the dependence measures. In the case of non-stationarity, such estimates would be biased and reflect only a temporal average connectivity between the components of the network. The time dependence can be addressed using evolving networks, which are constructed over sliding time windows (see for instance , and ). The present study considers a time-constant network for the period 1901–2010 and a shorter-period network for 1951–2010, where more observations are available for assimilation into the reanalyses. To investigate the temporal evolution, a third network is constructed for 1901–1955.

The complex network framework offers a lot more approaches in order to exploit the richness of the data, as for instance multi-scale, causal, and multi-layer networks. Wavelet multi-scale networks were proposed for investigating interactions in the climate system simultaneously at different temporal scales, revealing features which usually remain hidden when looking at one particular timescale only . Interactions between processes evolving on different timescales are investigated by . Moreover, as the number of identified domains within a climatological field is drastically smaller than the number of original grid cells, this also opens up the possibility of investigating the causal relationships between them , although the basic assumption of causal network inference that the dependence structure can be represented by a directed acyclic graph is questionable in the climate context. The construction of both dependence-based and causal networks can naturally be extended to cross-networks, which include multiple fields .

Our algorithm, presented in Sect. 3.2.1, combines grid cells with highly rank-correlated time evolution into domains. Domains have to be contiguous but may overlap; grid cells may remain unassigned. Average mutual rank correlation within a domain has to be higher than a selected threshold δ; we examined the quantiles q0.9(ϱij|i≠j=1…n)≤δ≤q0.99(ϱij|i≠j=1…n) of all pairwise rank correlations. The plots included in this paper refer to thresholds q0.95 for SST and δ=q0.93 for Z500, chosen for their intuitive parcellation of the fields evocative of known teleconnection patterns. As varying the threshold affects the networks for different data sets in a similar way, the choice of δ changes the results only marginally.

The domains constructed this way from the detrended, deseasonalised SST anomalies of the CERA-20C ensemble mean include all important SST teleconnection patterns with interannual to decadal timescales (see for example ). The map of the CERA-20C SST domains (Fig. a) resembles the corresponding maps for COBEv2 and HadISST in reasonably well, taking into account the differing data sets and time periods. Their main domains are clearly identifiable: El Niño–Southern Oscillation (ENSO; o11, for its broad extension also reminiscent of region 2 of the Interdecadal Pacific Oscillation (IPO) tripole in ), the horseshoe pattern (o7), the South Pacific (o9), the Indian Ocean (o3), the North Tropical Atlantic (o15, with extension to the extratropics), the South Tropical Atlantic (o1). Furthermore there are domains in the extra-tropical southern (o2) and eastern Indian Ocean (o4), the extra-tropical southern (o12) and north-eastern (o14) Atlantic and the Norwegian Sea (o16), the Gulf Stream (o13), the North Pacific Current (o8, region 1 of the IPO tripole), a domain corresponding to region 3 of the IPO tripole (o10), the Kuroshio Extension (o5), and a domain south of Australia including the Great Australian Bight (o6). Areas where sea ice occurs are omitted because of the confounding effect on SST.

In the CERA-20C Z500 map of domains (Fig. b), the seasonally migrating Tropical Belt (TB; a15) formed by the Hadley circulation and the two polar cells (Arctic a1 and a13 largely overlapping and Antarctic a3) stand out, stretching around the whole globe. The mid-latitudes are populated by numerous domains with more (over ocean) or less (over land) pronounced zonal extension (cyclone tracks). The missing segmentation of the tropical belt into several domains probably results from the seasonal time resolution.

The domains of SST and Z500 are now ready for network construction (see Sect. 3.2.2). Figure c illustrates the maximum lagged (-10≤L≤10) distance correlations (Sect. 3.3) for all pairs of SST domains in the CERA-20C ensemble mean, omitting the geographic information for enhanced clarity. Only significant links to the FDR level α=0.05 (Sect. 3.2.2) are shown. However, even weak links are assessed as significant because the time series are long enough (4 seasons × 110 years) to allow the distance correlation to be estimated accurately. The darkest shades (except of the self links) correspond to the links between ENSO (o11), the horseshoe (o7), and IPO3 (o10): o11↔o7, o11↔o10, o7↔o10. We see enhanced connectivity of o7, o10, and o11 to the northern and southern Pacific Ocean (o8, o9) and from the Pacific to the Indian Ocean (o7/o10/o11↔o3), corresponding to known ENSO teleconnections, but not to the Kuroshio Extension (o5). The southern Indian Ocean domain is furthermore linked to the South Atlantic (o2↔o12).

The intra-Atlantic links are much weaker: o14, o15, and o16 are largely overlapping domains and together conceivably form the Atlantic Multidecadal Oscillation (AMO); the Gulf Stream is linked to the north-eastern Atlantic (o13↔o14) as well as the tropical to the extra-tropical South Atlantic (o1↔o12). The South Atlantic is also weakly connected to all North Atlantic domains (o12↔o13/o14/o15/o16), but there is no link between the North Atlantic and the South Tropical Atlantic (o1). It might be hypothesised that this bypass is related to the thermohaline circulation that tunnels the shallow subtropical cell . According to the network, the Atlantic is connected to the other oceans only via the Southern Ocean, with links o12/o13/o14/o16↔o9, o14/o15/o16↔o6, o1/o12↔o2, and o16↔o2 that appear rather weak, although visible against their virtually zero background. A link between the South Tropical Atlantic (o1) and ENSO (o11) as proposed in and is not apparent in our network. This absence is likely caused by the non-stationarity of this link, which was not observed before 1970. Nevertheless, it does appear when a network is constructed for the period 1971–2010 (not shown).

Note that allowing for lagged dependence changes the network only marginally compared to a network with only instantaneous links. Few connections are increased in strength of distance correlation by more than 0.05 and none by more than 0.1. All links already exist in the instantaneous network, and the structure of the network remains unchanged.

The network between CERA-20C Z500 domains (Fig. e) is considerably weaker than the SST network, possibly a consequence of the stronger high-frequency variability in the Z500 time series in response to seasonally varying solar forcing combined with weaker low-frequency variability caused by stronger mixing of the freely flowing air masses. Moreover, many of the known atmospheric teleconnections vary considerably throughout the year, which weakens the all-season dependence between the involved domains. Apart from the overlapping domains a1/a13, a9/a10, and a7/a12, the Tropical Belt (a15) is the most strongly connected domain with links to the mid-latitudinal Ferrel cell domains, enveloping the cyclone tracks, over all oceans (a15↔a4/a7/a9/a10/a12/a14), to which the undisturbed Hadley circulation releases a substantial amount of energy. Domains over land have fewer and weaker links. Known atmospheric teleconnections are clearly identifiable: the Pacific North America Pattern (PNA) with links a10↔a11, a10↔a14, but interestingly not a11↔a14, and the North Atlantic Oscillation (NAO) with a link a13↔a14 (and much weaker a1↔a14). Other complex teleconnections also seem to involve the Arctic domains: a1↔a2/a6/a8/a16 and a13↔a8. In contrast, the Antarctic domain (a3) is largely autonomous, as discussed in . Lagged dependence is irrelevant in the Z500 network.

We notice that many known atmospheric teleconnections are defined as higher-order modes of some EOF decomposition. As such they exist only as additive modulations of their corresponding leading modes. We would therefore not expect to find many of them in our networks.

Network methods allow the investigation of interactions between different climatological fields in a straightforward way, constructing cross-networks between (in our case) SST and Z500 domains that describe the coupled ocean–atmosphere variability . We notice that the inference of links between the domains of two unipartite networks is different from the construction of bipartite communities in multi-layer networks as in . Here, we just calculate the distance correlations between pairs of one SST and one Z500 domain. The inferred CERA-20C SST–Z500 cross-links are shown in Fig. d. The connectivity is mostly quite weak, except for the cross-links from the Tropical Belt (a15) and the northern and southern Pacific Z500 domains (a9, a10, a12) to the ENSO-related SST domains (o7, o8, o9, o10, o11) and the tropical Indian Ocean (o3), but also the Great Australian Bight (o6). This feature was also observed by , who related it to the Walker circulation. Z500 domains a4 and a7 participate in this pattern, but to a lesser extent. Z500 domains over oceans are usually connected to their underlying SST counterparts (a4↔o2, a7↔o6, a9/a10↔o8, a14↔o13 (SST modulating the NAO), a15↔o3/o11), although in the Atlantic this dependence is exceptionally weak (a15↔o1/o15, a16↔o14, a3↔o12). But teleconnections to more distant SST domains are, in some instances, as strong as or even stronger than those proximate cross-links (a4↔o3/o10/o11/o12, a7↔o12, a14↔o15). Interestingly, the Arctic Z500 domain (a13) is weakly linked to the AMO domain (o15), but not to the North Pacific. Except for slightly increased overall connectivity levels, supposedly mediated by the SST, allowing for lagged dependence does not change the network.

The analogous plot for 20CRv3 can be found in Fig. S3.

As before, this subsection presents only results for CERA-20C over the time period 1901–2010. The overall high level of connectivity between SST domains motivated us to take a deeper look into the dependence structure of the climate system. In a modest first attempt, we search for interacting triples in the sense of Lancaster, in graph theory termed as 2-hyperedges, taking all combinations of three SST domains and calculating their third-order distance multicorrelation as introduced in Eq. () in Sect. 3.3.1. As discussed there, we choose a large FDR level α=0.2 in order to not suppress too many distance multicorrelations. To avoid cumbersome evaluations with different lag combinations, we stick to instantaneous networks.

Only a small number (13) of significant third-order dependencies are detected (we list them in Table instead of plotting them), all somehow related to the ENSO phenomenon, one of them the IPO tripole itself. The hyperedges also include the tropical Indian Ocean (o3) and the Great Australian Bight (o6). As the nature of Lancaster interaction is inherently non-linear, this concentration on ENSO corresponds to the findings in , who detect substantial non-linear contributions to mutual information in SST (apart from trends and seasonal variance) mainly in the central tropical Pacific. Likewise, find synchronised extremes of uncorrelated PCs of SST in the Pacific that cannot be explained by linear interaction. Despite this, one distance multicorrelation is also detected in the North Atlantic: the SST triple (o14, o15, o16), which corresponds to the AMO.

Note that not every triple with strong pairwise dependencies also has a significant third-order dependence. Table  shows the hyperedges along with their distance multicorrelation and the sum of their pairwise distance correlations. As distance multicorrelation is symmetric, every significant hyperedge is listed only once in the table. Note also that the sum of pairwise distance correlations is not bounded by 1 because the pairwise dependencies are not mutually exclusive. Although the detected distance multicorrelations are significant, they are at most 20 % of the sum of the respective pairwise distance correlations. That means third-order interactions complement but not outweigh pairwise dependence in the three-dimensional joint dependence.

The same comments essentially apply to cross-hyperedges consisting of two SST domains and one Z500 domain or one SST domain and two Z500 domains. We detected 15 and 5 significant cross-hyperedges, respectively, in the Pacific, which all resemble some ENSO interaction. The Z500 domains a13 and a14 (NAO) have no notable distance multicorrelation with North Atlantic SST domains, indicating that the North Atlantic is linked to the NAO domains on a pairwise basis (o15↔a13/a14), but no higher-order interaction is taking place. There is no hyperedge of three Z500 domains with significant multicorrelation. Known atmospheric tripoles like the Arctic Oscillation (a9, a13, a14) and the Pacific North America Pattern (a10, a11, a14) apparently lack significant third-order dependence.

We believe that the construction of higher-order networks including hyperedges by means of distance multicorrelation might well be one step towards understanding the synergies emerging from multivariate coupling of large-scale oceanic/atmospheric teleconnections.

We turn to the comparison of reference networks in terms of the NQS and MNQS criteria (see Sect. 3.4), calculated from the adjacency matrices M containing the regionally distributed distance correlation links between all pairs of domains (Eq. ). As CERA-20C was produced as a 10-member ensemble representing the inevitable sampling and modelling uncertainty inherent in the production process, we take this opportunity to construct the δ-MAPS networks individually for each member. The results are matched to the networks derived for the CERA-20C ensemble mean.

The CERA-20C individual networks for the complete time period 1901–2010 are very similar to each other, with average NQSs close to 1 for all three parameters (average NQS_SST = 0.98, average NQS_Z500 = 0.94, average NQS_SST–Z500 = 0.96), such that the MNQSs have a mean of 0.96 with only a small spread. The average MNQS between the individual CERA-20C runs and the CERA-20C ensemble mean is 0.96. The small differences are brought about by the pattern correlation factor in netSSIM, the mean and variance factor being virtually equal to 1.

The networks for the shorter periods 1901–1955 and 1951–2010 are equally similar with average MNQS = 0.95 and 0.95 between runs and 0.95 and 0.96 for the ensemble mean, respectively. Because the networks for individual CERA-20C runs and the CERA-20C ensemble mean are nearly indistinguishable, we only take the CERA-20C ensemble mean networks for reference in the following comparisons.

When analysing the temporal evolution of the connectivity in the CERA-20C ensemble mean, we find good agreement between the first and second half of the century (MNQS = 0.87; Table ), resulting from comparable differences in the SST and SST–Z500 networks and higher similarity at Z500 (NQS_SST = 0.84, NQS_Z500 = 0.93, NQS_SST–Z500 = 0.84). In contrast, the full period is more similar to the first half in all networks (MNQS = 0.96, NQS_SST = 0.95, NQS_Z500 = 0.96, NQS_SST–Z500 = 0.95) than to the second half because especially the SST–Z500 networks bear more differences (MNQS = 0.92, NQS_SST = 0.92, NQS_Z500 = 0.94, NQS_SST–Z500 = 0.89). We emphasise that the networks contain only information about the strength of the dependencies between the domains and not about their functional form.

Because of deviating domain extension and numbering, comparing the networks by means of the rectangular network plots (like in Fig. c–e) is cumbersome. In Fig.  we have plotted two-modal slices of the spatially distributed adjacency hypermatrices M with respect to grid cells in the ENSO domain, in the AMO domain and in the Tropical Belt, respectively. The comparison of these slices is evidently not exhaustive but may give a hint regarding the nature of the differences between the networks.

The domains in the three CERA-20C SST network slices for the ENSO domain (Fig. a–c) are very similar in shape and size, but the links between the domains are differently distributed. The networks most obviously disagree in link strength from ENSO to the tropical Indian Ocean, but also from ENSO to the North Tropical Atlantic, to the North Pacific, and to the Southern Ocean. The same is visible in the network slices for the AMO domain (Fig. d–f).

In contrast, the CERA-20C Z500 network slices for the Tropical Belt (Fig. g–i) bear more apparent similarity than the SST network slices, which was already apparent in the network quality scores above. Although the shape of the tropical belt differs slightly more than the shape of the ENSO domain, the links to the rest of the globe resemble each other more strongly. However, the domains over the North Pacific and the southern Indian Ocean seem somewhat ambiguous.

The cross-links from ENSO to the Z500 domains (Fig. j–l) and from the Tropical Belt to the SST domains (not shown) show differences similar to the unipartite networks. Yet, the stabilising effect of the self-links (large patches with distance correlation 1) does not apply to the SST–Z500 cross-networks, such that the network scores may turn out a little lower.

As regards the second reanalysis 20CRv3, we observe strong similarity to the CERA-20C ensemble mean in the two shorter time periods 1951–2010 and 1901–1955 (MNQS = 0.89 and MNQS = 0.88; Table  and Fig. S4), where disagreement within the same time period is mainly restricted to higher southern latitudes (remember that the SSIM includes an area weighting). But dissimilarities between the first and the second half of the century are stronger in 20CRv3 than in CERA-20C (MNQS = 0.81 and MNQS = 0.87; Table ). Notably, in 1901–1955 20CRv3 shows the same strong connection between SST domains around the whole tropics as CERA-20C, which is lost in 1951–2010 in both reanalyses. In contrast, the similarity between 20CRv3 and CERA-20C is slightly reduced in 1901–2010 (MNQS = 0.82; Table  and Fig. S4) mainly due to differing atmospheric interactions and the weaker cross-links in 20CRv3 compared to CERA-20C (NQS_SST = 0.94, NQS_Z500 = 0.81, NQS_SST–Z500 = 0.72). In all three networks (SST, Z500, SST–Z500) we observe that regional unsimilarity increases with latitude. Table S1 shows pairs of most similar domains between CERA-20C and 20CRv3 along with their domain-wise network quality score.

Besides, the similarity between the different time periods in 20CRv3 is not the same as in CERA-20C, with 1901–2010 more similar to 1951–2010 than to 1901–1955 (MNQS = 0.84 and MNQS = 0.90; Table  and Fig. S4). For example, in contrast to CERA-20C, the link between ENSO and the South Pacific vanishes after 1950 in 20CRv3. This might be a consequence of sparse observations in the first half of the century and thus a stronger dynamical heritage from the models used to produce the reanalyses. On the other hand, there might have been changes in connectivity driven by increasing GHG levels, which are not equally reflected in CERA-20C and 20CRv3 (they are model results after all). Caution leads us therefore to restrict the comparison of CMIP6 data sets to reanalyses in the period 1951–2010.

The networks belonging to the CMIP6 historical projections (listed in Table ) are compared in Fig.  to the CERA-20C ensemble mean (bold black cross marks) and to the 20CRv3 best estimate (bold red cross marks) in the time period 1951–2010 in terms of individual network NQSs (for SST networks (a), for Z500 networks (b), and for the cross-networks (c)) and in terms of MNQSs for each reference, respectively (d). Finally we take the average of both MNQSs to account for the uncertainty inherent in the reanalyses: 1/2(MNQS(CERA-20C)+MNQS(20CRv3)) ((e), bold cross marks). As expected, the similarity between models and references is generally weaker than between references, although in the Z500 networks some models reach a comparable level. Network quality scores are highest for Z500, followed by SST and SST–Z500. SST–Z500 cross-networks show the greatest deviations across models as well as across references. The seemingly contradictory scores for Z500 with respect to CERA-20C and 20CRv3 have to be put into perspective with their very high values and can be traced back to the differences between the reanalyses.

When applying the alternative, pointwise SSIM calculation (Fig. , thin black and red cross marks), the final average MNQS values are somewhat lower in their overall level, but similar in spread, and the model ranking suffers only minor changes.

The differences between the reanalyses are also reflected in the MNQSs of the models, where the reanalyses agree very well upon some models (HadGEM3-GC31-LL, IPSL-CM6A-LR, MPI-ESM11-2-HR, MIROC-ES2L, MIROC6) but less upon others (MRI-ESM2-0, TaiESM1, CNRM-CM6-1, CNRM-ESM2-1). But altogether, a tendency to differentiate between more/less similar models with respect to reanalyses is clearly visible. We conclude that, when combining several references from independent sources, the average MNQS over these references is a valid evaluation instrument for assessing whether the teleconnections between large climate components in a general circulation model are realistically represented. Still, as our evaluation is restricted to a single run per model, we are not able to differentiate between good runs and good models as such.

Using the example of four of the highest-ranking GCM runs with respect to MNQS, we illustrate in short the opportunities offered by the δ-MAPS approach to detect model deficiencies. We examine some of the pointwise adjacency maps of EC-Earth3, UKESM1-0-LL, MPI-ESM1-2-HR, and IPSL-CM6A-LR in comparison to CERA-20C and 20CRv3 over 1951–2010 (Figs. S5–S9). In the SST networks, we notice that differences are not restricted to higher latitudes, as was the case for the two reanalyses. Even in the main feature of interannual variability, ENSO, spatial connectivity deviates significantly. In all models the tropical Indian Ocean depends much more strongly, although to varying degrees, on ENSO than in both reanalyses (Fig. S5). EC-Earth3 and IPSL-CM6A-LR do not at all reproduce the northern extension of the ENSO domain seen in both reanalyses (Fig. S5a, d, e, f), which reflects the widely recognised low-frequency interdependency between ENSO and the Pacific Decadal Oscillation (PDO) . The links to the southern Indian Ocean and the South Atlantic differ considerably across models, but no model shows better performance in all domains. In MPI-ESM1-2-HR the dependence between AMO and ENSO is exaggerated, whereas in IPSL-CM6A-LR the Norwegian Sea is nearly disconnected from the Tropical North Atlantic, which is not consistent with AMO (Fig. S6c and d). As regards Z500, UKESM1-0-LL shows an unrealistic link between the Tropical Belt and the Antarctic domain (Fig. S7b). At the same time, the dependence of the Arctic domain is matched well only in UKESM1-0-LL (Fig. S8b). In contrast, the cross-links from ENSO to Z500 are well represented in all four models (Fig. S9).

Continuing the analysis of all pointwise adjacency maps, it would be possible to identify regions/climate phenomena of higher and lower confidence in any model, an exercise that might be instructive for both modelling groups and downstream users of climate projections.