In general, a 2-state Markov chain (MC) is characterised by its two possible states S0 and S1 with two transition probabilities p and q. In particular, p is the probability to to go from state S0 to S1 in one time-step while similarly q is the probability from S1 to S0. This is summarised by the transition matrix T=(TSS′:S,S′∈{S0,S1}) given in Eq. (). 1T=1-ppq1-q

The update equation for the probability to be in one of the states, say state S1, at time t+1, Pt+1(S1), can then be found in terms of p,q and the probability at the previous time-step t, Pt(S1), given by Eq. (). 2Pt+1(S1)=p(1-Pt(S1))+(1-q)Pt(S1)

The dynamics of the whole chain are fully captured by the dynamics of one state since we have P(S0)+P(S1)≡1. Given an initial distribution of P0(S1) at time t=0, the explicit time evolution at time t is given by Eq. (). 3Pt(S1)=pp+q(1-(1-p-q)t-1)+(1-p-q)tP0(S1)

At equilibrium, Pt+1(S1)=Pt(S1), giving the stationary distribution, P∗(S1), in Eq. () which, assuming fixed p,q, is stable if p+q>0. 4P∗(S1)=pp+q

To find the stability of the stationary distribution in Eq. () corresponds to finding the eigenvalues of the Jacobian matrix at the stationary distribution, which, in this case, is simply the derivative of Eq. (). Formally, for x=Pt(S1) and a map f:[0,1]→[0,1] such that Pt+1(S1)=f(x), the stationary distribution x∗=P∗(S1) is stable if |f′(x∗)|<1. From Eq. (), f′(x)=1-(p+q) such that the stability condition simply becomes 0<p+q<2.

For a set of Markov chains, V, let G={V,E} be a graph of pairwise interactions between individual Markov chains. A directed edge (i,j)∈E exists if there is a causal link or effect due to MC i∈V on another MC j∈V, disallowing self-loops. An element aij of the adjacency matrix A=(aij:∀i,j∈V) corresponds to the probability factor PF(i→j) due to one tipping element i being triggered on another element j. In general the edge from i to j is represented by aij which may be weighted – indicating some positively valued coupling strength – and signed – indicating the polarity of the effect (e.g. stabilising or destabilising). For signed edges, we denote the adjacency matrix of destabilising edges with A+=(aij+:∀i,j∈V) with with aij+=PF(i→j) when PF(i→j)>1. Similarly, we denote A-=(aij-:∀i,j∈V) all the stabilising edges, i.e. those with PF(i→j)<1 has a corresponding adjacency element aij-=PF(i→j)-1.

The state of each MC i∈V is denoted by Si which can take one of two environmental states – either Prosperous P or Degraded D – and has an associated internal collapse probability pc,i∈(0,1) and recovery probability pr,i∈(0,1), which are the Markov transition probabilities between the possible environmental states due to the internal processes alone. Moreover, let the probability of node i being in the Degraded state be denoted Di≡P(Si=D). As probability factors are conditional upon one element having already been triggered, then, in general, degraded destabilising in-neighbours increase the collapse probability, while stabilising ones decrease the probability to collapse. We therefore modify the individual collapse probability by the interactions due to all in-neighbours into an effective probability to collapse p̃c,i. 5p̃c,i=pc,i1+∑j∈Vaji+Dj1+∑j∈Vaji-Dj

Each Markov chain evolves according to the transition matrix in Eq. () and the corresponding update equation in Eq. () with p=p̃c,i and q=pr,i, however there is now a dependence on the states of its in-neighbours. As such an explicit expression for the time evolution of each node requires solving a set of non-linear coupled maps, such that a general solution is cumbersome to write down as a closed form expression. Instead, as in Eq. (), we derive the update equation Di[t+1] in Eq. (). Note for notational brevity, where relevant, all terms in the right hand side of the equation are evaluated at time t. 6Di[t+1]=1+∑j∈Vaji+Dj1+∑j∈Vaji-Djpc,i(1-Di)+(1-pr,i)Di

Moving away from the abstract, in this subsection, we address the question of how long a time-step is and make explicit the one-to-one correspondence between probabilities and tipping timescales. We then introduce how global warming temperatures, as a proxy of human impact, change the transition probabilities.

First, consider that the update equations describing the dynamics of a Markov chain generally, e.g. in Eq. (). There they describe how the probability to be in one state S1 at the next model time-step t+1 depends on the probability at the current time-step t. In other words, the Markov chain evolves over discrete and dimensionless time-steps. In order to translate such model time-steps into physical units of time, we set the duration of a time-step to be τm years. In principle τm could be any value and represents the temporal precision of the model. However note, as we will see in Sect. , that the discretisation of time can influence ergodicity and therefore stability of the stationary distribution.

We now make the correspondence between tipping timescales and probabilities. Consider the transition from one state to another, which on average is expected to take τ amounts of real time, say τ years to be concrete. Then, given some duration of time τm, the effective probability for the state transition to occur within that time-frame τm, conditional upon having been triggered, is given by p=(τ/τm+1)-1. Here, the plus one is to ensure that if τ=0 then certainly, i.e. with p=1, the transition has occurred within τm. With regards to tipping elements, once an element i has crossed its threshold, it does not instantly transition to its new state, rather it collapses over some timescale τc,i. Similarly, we can construct a recovery transition probability pr,i given a subsystem's recovery timescale.

The dynamics of nature are not purely driven by natural/physical processes but also by the actions of individuals, stakeholders and institutions. Although there are many anthropogenic drivers of environmental/ecological collapse, some of which are not directly correlated with global warming (e.g. deforestation for agricultural land), we consider all anthropogenic effects as being mediated by – or at the very least proxied/dominated by – global warming. That is, as finds, when the global mean surface temperature (GMST) exceeds the critical threshold of an individual tipping element, that element is far more likely to be triggered and thus begin its timely collapse. To that end, for each tipping element or ENSO, i, we do a similar decomposition into the likelihood to be triggered (which is a function of global warming) multiplied by the conditional probability of tipping discussed previously (τc,i/τm+1)-1. In particular, given a GMST anomaly relative to pre-industrial times of ΔT, for element i with timescales τc,i and τr,i and an estimated critical temperature threshold of Tlim,i we have that the internal probabilities to recover and collapse are given by, 7pr,i=1(τr,i/τm)+1,pc,i=1(τc,i/τm)+1⋅11+exp⁡(-βi(ΔT-Tlim,i)) where τm is the duration of a model time-step in units of years – unless otherwise stated we henceforth set τm=1 year – and βi is the sensitivity of the logistic function.

One way to interpret this parameter is physically: if a tipping point with low βi has crossed its threshold by a small amount then its tipping probability is only marginally increased; whereas one with a high βi would experience a large increase in its tipping probability. Another way, as we do in this work, is to quantify the deep uncertainty amongst the expert assessments: that is when the range of plausible tipping thresholds is large, indicating the wide range in expert opinion, the sensitivity βi is low; whereas a small range in the threshold suggests a high degree of confidence and/or agreement and therefore a high βi. For details on how we infer the value of βi see Appendix .

comprehensively assessed 16 tipping, through their extensive multi-disciplinary literature review, providing estimates on, amongst others, the tipping threshold and timescales. In total the authors had reviewed 33 possible tipping elements, however in our work, due to the limited amount of TE-TE interaction data available , we focus only on nine of them – 6 global core tipping elements, two regional elements (low-latitude coral reefs and the Sahel and West African monsoon) and the El Niño-Southern Oscillation (ENSO) whose status as a tipping element has been disputed . As our model is agnostic to the precise definition of a tipping element – for example as one containing hysteresis or irreversibility – and regardless of its classification, we include ENSO in our case study since it has direct, strongly destabilising impacts on other subsystems, notably on the coral reefs and on the Amazon rainforest . ENSO's “collapse” transition can be thought of as the shift towards a state of increased El Niño amplitude and/or increased rainfall amplitude variability . Table  summarises for each tipping element and ENSO the set of parameters inferred from the review, namely the timescales for collapse and recovery, the tipping threshold and the sensitivity to temperature. For specific timescale values and inference methods, especially for the sensitivity, see Appendix .

For the interactions between tipping elements we rely on a similarly extensive literature review by , in which the authors had assessed four properties of 19 pairwise interactions: response type, response strength, level of agreement within the literature and level of evidence. Although the review described response strengths in qualitative terms, a core foundation of the review comes from the quantitative probability factors of the expert elicitation . We use a similar methodology from an earlier work to, first, convert the qualitative terms back into probability factors, thus allowing us to include the new interactions considered. For specifics, see Appendix .

A major challenge here is the deep uncertainty within the literature where interactions are concerned . That is, in some cases, there is not even general consensus as to whether a particular interaction is destabilising or stabilising. For example, there are multiple opposing mechanisms which allow the West Antarctic ice sheet to either strengthen or weaken AMOC, while other works suggest the disintegration of WAIS can avoid or at least delay the subsequent collapse of AMOC. Therefore, in order to reflect the uncertainty both from the original expert elicitation and the more recent review , we associate with each pairwise interaction a uniform random distribution whose range is the interval of plausible PF values (see Fig.  for illustration or Table  for details). Note, that since a uniform random distribution maximizes entropy or uncertainty given a constrained interval, no further assumptions other than the range of plausible PF values are needed. Moreover, note that, in this work, we discard all interactions with both unclear response type and unclear response strength (in total, 2 of the original 19 interactions from ), as we found no clear, reasonable way to quantify such interactions. Given the 17 dimensional space of parameters, we use a Monte Carlo ensemble of simulations using the Latin hypercube sampling method , which covers the parameter space more efficiently than i.i.d sampling (for more computational detail see Appendix ).

The stable stationary distribution of Eq. () can give us valuable insights into the effects of interactions – following the logic of probability factors  – on the nine climate subsystems. In particular, as the distribution is always stable, this distribution is precisely the long-term risk, after millennial times, of each tipping element being Degraded. In this subsection, we explore this equilibrium or long-term behaviour for a range of fixed GMST anomalies in terms of the expected number of degraded tipping elements (Fig. ) and the individual risk (Fig. ). In the former, we see that the equilibrium response of tipping elements to the global warming level is highly non-linear, due in large part to the sigmoidal temperature-dependence of each element's internal collapse probability. If we were to permanently (i.e. at least over millennial timescales) commit to the Paris Agreement global warming levels of 1.5–2 °C, then 3 tipping elements – both polar ice sheets and the low-latitude coral reefs – are very likely (> 90 %) to be in their respective degraded states as their individual critical thresholds have been surpassed.

Interactions between tipping elements on the whole increase the risk of being degraded, as indicated by the no-interactions case (dashed lines) generally lying below the ensemble mean for the with-interactions case. This is largely due to the majority of TE-TE interactions being destabilising ; for some tipping elements, in fact, all of their known incoming interactions are destabilising (see Table ) such that interactions can only ever increase stationary risks. AMOC, although mostly dominated by its destabilising neighbours, has a moderately strong though of unclear type interaction due to WAIS, meaning that for some interaction matrices, the interaction-free scenario is riskier. Similarly, as AMAZ only has a single incident edge, also of unclear response, the interactions can decrease the risk; mostly, however, since its only neighbour is AMOC, the interactions typically increase the risk to be degraded.

In Fig.  we display the stationary risk-response for each individual tipping element. For all elements, when global warming is near pre-industrial levels (ΔT≤ 1 °C) the risk to be Degraded is zero or negligible. At very high temperatures (ΔT = 10 °C), the degraded risk for all tipping elements have more or less converged to the respective D∗,ihot discussed previously in Sect. . Most tipping elements under such catastrophically hot temperatures are very likely (D∗,ihot > 90 %) to be degraded. Note that, as can be seen in Eq. (), D∗,ihot can never be precisely 100 % unless pr,i=0 (or equivalently τr,i=∞) in other words unless the tipping element is precisely irreversible. In other words even at high equilibrium temperatures – e.g. at 10 °C global warming – a tipping element's equilibrium risk will not necessarily be 100 % due to this property of the Markov chain model. Moreover for an individual tipping element i, having more destabilising neighbours increases D∗,ihot while if there are enough stabilising neighbours D∗,ihot will be reduced. We see this in the case of both the Greenland ice sheet and Arctic sea ice; the risk of both at very high temperatures is lower, though still likely (D∗,ihot≈ 80 %), due to being partially stabilised by AMOC. We stress that, assuming all thresholds have been crossed, increasing temperature monotonically increase the equilibrium risk.

As temperature increases the long-term (i.e. equilibrium) risks for all but one element increase monotonically, from which we see a threshold effect emerging. That is, below some global warming level – within the estimates of the critical threshold from , which is denoted by the dotted vertical lines in Fig.  –, an individual tipping element's risk nears 0 while above this threshold the risk nears D∗,ihot. The effects of interactions, on the other hand, are highly non-linear in temperature, being most strongly felt over intermediate temperatures, especially around the critical thresholds of one or both elements involved. For example, as seen in Fig. S1 in the Supplement, the equilibrium risk for AMOC increases by around 30 % at around 3 °C, while the risk for the Arctic sea ice is reduced by around 20 % at 6 °C.

The slight differences between the emergent thresholds and the estimates of is a consequence of the probabilistic model and, to various degrees, the interactions between tipping elements. For instance the destabilising interactions felt by the AMOC reduces its effective critical threshold by about 0.4 °C. On the other hand, as the Arctic sea ice has a single incoming interaction from the AMOC, which is moreover stabilising, its effective critical threshold is in fact increased, relative to no interactions, by around 0.2 °C.

The Greenland ice sheet stands out most amongst its counterparts as it shows a non-monotonic response to fixed global warming levels. Equilibrium risk increases with temperature until around 1.7 °C, after which the risk in fact decreases with temperature in the range 1.7–4 °C. This corresponds to the range of temperatures over which the risk of AMOC quickly increases, thus helping to somewhat stabilise GIS. Between 4–5 °C the risk plateaus at around 77 % as no other thresholds are being crossed, while between 5–6 °C the Arctic sea ice crosses its critical threshold and destabilises GIS. Another consequence of ASI crossing its threshold can then be seen in the corresponding rise in the permafrost risk to be degraded up to D∗,ihot≈ 97.5 %.

While the stable stationary distribution highlights the long term risks for each tipping element, given that global warming level is fixed in time, the short term evolution (up to multi-centennial times) is of most importance to policy and human life. For instance, it took the Earth less than a century to reach an annual GMST of +1.6 °C in 2024 from around +0.25 °C in 1950 . In this section, therefore, using both historical levels of global warming (1850–2014) and future projections (2015–2350) under the five extended shared-socioeconomic pathways (SSPs) outlined in the IPCC's Sixth Assessment Report , we look at how the risk of each tipping element evolves over the course of 500 years (from 1850 to 2350). As different SSPs represent different policy and commitment scenarios for the 21st century onwards, we can see in Fig. , in particular, how different tipping elements respond, in essence, to human activity.

As the dynamics are dominated by the faster timescales of GMST evolution rather than the slower timescales of an individual element's recovery or collapse, which SSP scenario the tipping elements are subjected to has a huge impact on the risk of being Degraded certainly by 2350 and even in 2100 for some. For instance, under the fossil-fuelled scenario of SSP5-8.5, in which exploitation of global resources are intensified and lifestyles become increasingly energy-intensive, the Sahel and West African monsoon (WAM) has a 58 % likelihood on average to be Degraded in 2100 and is very likely (94 %) to be so in 2350. On the other hand, under the sustainable and “green” development pathway of SSP1-1.9, where ecologies are preserved, planetary boundaries respected and income inequality being actively reduced, on average WAM is exceptionally unlikely (< 0.003 %) to be degraded. Suffice to say, in all cases, following such a green and sustainable scenario minimises the risk of degrading tipping elements, as compared to other scenarios.

That being said, under the most optimistic scenario SSP1-1.9, the coral reefs are at least likely to be degraded (> 74 %) in 2100 (see Fig.  due to the initial overshoot in temperature and their decadal collapse times. Fortunately, as the overshoot is temporary, the coral reefs are able to recover sufficiently quickly that by 2350 (see Fig. S2 in the Supplement) they are very unlikely (< 10 %) to remain degraded. Under the middle-of-the-road scenario, SSP2-4.5, its risk does begin to fall in 2100–2150 but remains likely to be degraded by 2350, unlike AMOC and the Amazon whose risks increase an intermediate amount under the same scenario from 2100 onwards. Although the reefs' singular interaction from ENSO does destabilise it over time, the increase from the interaction-free case is less than 1 % for the first three scenarios and almost 7 % for SSP3-7.0 and SSP5-8.5 (see Fig. ); the vast majority of its risk is due to its fast timescales (a few decades) and to its low tipping threshold (+1.5 °C). On the other hand for very slow tipping elements their dynamics are dominated instead by the evolution of SSP warming and their faster neighbours. For instance, GIS, which evolves over millennia, is stabilised quickly enough by AMOC, which evolves over decades to centuries, such that its 2100 risk is actually reduced by around 1 % for the hotter scenarios. Intriguingly, under both SSP1 scenarios, the ensemble mean risk is instead slightly increased, though by a very small amount, although some interaction matrices still result in a reduction.

We find that, on the one hand, AMOC is at worst about as likely as not (40 %) to be degraded by 2100 under SSP5-8.5, despite the inclusion of other, mostly destabilising, interactions – depending on the SSP scenario this risk may be even less than 5 %. On the other hand, by 2350, we can see how dependent the risk of tipping is on the specific SSP scenario, with the two hottest scenarios (SSP3-7.0 and SSP5-8.5) leading to AMOC being very likely to tip, while under both SSP1 scenarios the risk is negligible. We find that interactions on the whole play a destabilising role in AMOC though these effects are most prominent only after 2100, increasing the risk by 10 %–15 % on average and by up to 18 % (see Fig. ). Although, under the other 3 scenarios, some interaction matrices can reduce the tipping risk by up to 5 % in 2350 – and thus somewhat stabilise AMOC as for instance claimed – we find that the destabilising interactions not considered by dominate the stabilising effects of WAIS producing a net increase in risk by 2350 across all scenarios.

Although this subsystem cannot sufficiently stabilise itself nor the entire network (see Fig. S4 in the Supplement), individual tipping elements are less at risk, in large part due to the stabilising effects from AMOC. For instance, for equilibrium temperatures hotter than 2 °C, GIS is on average less likely to tip than without interactions. The effect can be as large as a reduction of 10 %–20 % in absolute value at equilibrium (see Fig. S1). In the short term too, especially after 2100, interactions lower the risk by up to 2 %, depending on the scenario (see Fig. ). Moreover due to its millennial timescales and despite its lower threshold of 1.5 °C, GIS is very unlikely to tip (< 3.5 %) across all scenarios by 2350, thus allowing for some safe overshoots in temperature as has been previously found . Similarly, the Arctic sea ice too is stabilised greatly by the AMOC, both at equilibrium (by up to 20 %) and in the short term (for the SSPs 3-7.0 and 5-8.5 the reduction is most prominent in the period 2150–2200 at around 20 %–30 %).

In fact for all elements barring the coral reefs and polar ice sheets, we see that the effects of interactions under SSP3-7.0 and SSP5-8.5 are most strongly felt between 2150–2200 in Fig.  and gradually weaken past 2200. Meanwhile, the effects of interactions plateau for the coral reefs, since they are already maximally at risk under SSP3-7.0 and SSP5-8.5. Given their millennial timescales, the strength of the effects of interactions increase slowly over time across all SSPs for both polar ice sheets. In general, interactions play a much smaller role for all climate subsystems under SSP1-1.9, SSP1-2.6 and even the middle of the road scenario of SSP2-4.5 for most, since temperatures are, first, too low to impact the important nodes in the network (i.e. to begin network-wide cascades). Second, the tipping points that are crossed under these 3 scenarios are either sink nodes in the network (the coral reefs) so does not impact others, or are too slow to respond to begin a cascade.

In this work, we have presented our novel risk assessment framework, modelling 8 interacting tipping elements and ENSO under various global warming scenarios and at equilibrium. In summary, we find that interactions, on the whole, increase the risk of individual climate subsystems being degraded, similarly to . Moreover, we quantify the effects of interactions by comparing against an interaction-free baseline, unlike , finding that, at the level of total risk, the net effect due to interactions is actually somewhat weak, for example as seen in Fig. . Despite the fact that the majority of pairwise interactions are destabilising , we find that the few strongly stabilising interactions (namely those due to the collapse of AMOC) offset much of this total risk, but not entirely. For the Greenland ice sheet and the Arctic sea ice the decrease in their respective risks can be as much 30 % at equilibrium (Fig. S1) while in the short term the latter can feel up to a 40 % decrease in risk under SSP3-7.0 and SSP5-8.5.

With respect to AMOC, conceptual models and Earth System models , linking the polar ice sheets to the AMOC via meltwater flux, have found that AMOC can effectively be stabilised depending on the collapse rates of and time delay between tipping of the two polar ice sheets, even without the cooling effects a weakening AMOC has on the Greenland ice sheet . However, at the inclusion of more interactions and other tipping elements, we find, in contrast, that AMOC is on the whole destabilised by its overall interactions, especially after the year 2100 in the short term (Fig. ) and above 1.5 °C of global warming at equilibrium (Fig. S1).

Considering a robust representation of interactions, global warming temperature is still a critical factor in the risk of tipping element degradation. We find that keeping long term temperatures below 1.5 °C – less than the lower limits of the Paris Agreement – risks fewer than 3 degraded tipping elements (the coral reefs and both polar ice sheets), while in the short term “taking the green road” in global development and economic paradigms can completely avoid any tipping risks. In particular, as exemplified by the SSP1-1.9 scenario  – one of the most optimistic overshoot scenarios – temporary overshoots in temperature mean that the coral reefs will be very likely (94 %) to be in their prosperous state by 2350, despite being, first, likely (74 %) to be degraded in 2100.

Although large tipping risks can be avoided by limiting the convergence temperature – for long-term GMST under 1 °C, we find that all climate subsystems considered are very unlikely to be degraded, similarly to  – we are still well above such temperatures, having reached 1.6 °C in 2024 and likely entering a 20 year period with an average of at least 1.5 °C . Our stylised model study suggests that we are already close to having up to 3 tipped elements – the coral reefs (> 90 %), the Sahel and West African monsoon (> 48 %) and the AMOC (up to 36.7 %) – by 2100. By 2350 under the SSP3-7.0 and SSP5-8.5 scenarios, this total risk drastically increases to an equivalent of almost 7 tipped elements (all except the Greenland ice sheet and perhaps the West Antarctic ice sheet). Moreover, we find that even if global warming temperatures are permanently maintained within the Paris Agreement limits of 1.5–2 °C, on average 3 elements (the coral reefs and the polar ice sheets) will still be degraded at equilibrium, suggesting that such limits be treated only as short-term overshoot limits which cannot be sustained in the long term.

Our framework differs from other stylised models on interacting tipping elements as we model the interactions more coherently to the limited data, originally quantified from expert elicitation as probability factors, while greatly expanding the set of elements to those found by . In following this conceptual logic, the probability to collapse is inversely-proportional to (and therefore non-linear in) stabilising interactions while remaining a linear polynomial in the destabilising interactions. However, since state variables of the dynamical systems models and the degraded state in our model are not necessarily linear, then linear interactions in probability space may not correspond to linear interactions in state variable space. Moreover, compared to the simple Boolean model of , a big advancement of our approach is an explicit inclusion of timescales and temperature-dependence in the tipping process. Stylised models, including ours, are useful to complement physics-based models while physically based ESMs should, reciprocally, be used to complement and inform conceptual models.

We acknowledge that the models of are more suitable to explain other phenomena such as critical slowing down, while explicitly including bifurcations. However, these models conceptualise interactions deterministically as opposed to the explicitly probabilistic interactions discussed in . Furthermore, they are more computationally demanding. We use a 2-state Markov chain as a higher level abstraction of the double-fold bifurcations to gain a more computationally efficient and ultimately more interpretable model fit for a risk assessment framework. The outcomes of our model qualitatively reflect their results while being computationally cheaper and remaining coherent to the original expert elicitation. For our ensemble of 500 interaction matrices, numerically solving for the long-term risk of all tipping elements simultaneously across 104 values of temperature took 15 min on a personal computer. For the short-term risk over all SSP scenarios all simulations took less than 5 s to compute.

Although we have an explicit temperature dependence for collapse, we do not do so for recovery, due to the deep uncertainty within the literature. In some climate subsystems, such as the polar ice sheets , strong hysteresis lead to threshold effects for recovery that may not occur at the same temperature value as the corresponding collapse threshold. In others, even the question of reversibility is still under debate, let alone having a fully quantified temperature dependence. For example, while note a growing amount of evidence to suggest Arctic sea ice loss is irreversible, they find it is not yet sufficient to assess it as such. Therefore, we try to remain as agnostic as possible with respect to the temperature dependence of recovery, minimising the amount of additional assumptions, by assuming temperature independence, in this paper. We suspect in the short-term (multi-centennial scales) that temperature-dependent recovery will not change our results greatly, given the extra order of magnitude in the recovery timescales, however the effects in the long term and at equilibrium may be large – dominating any effects due to interactions – and warrants further detailed investigation.

Another major limitation was the (lack of) data regarding interactions themselves, in particular in their quantification. Although is a substantial review, the strength of each interaction is qualitatively assessed. While some of these interaction strengths were quantified in the earlier expert elicitation of , others were reassessed or were entirely new in the 2024 review, and thus lacked any quantification. In order to synthetically fill the data gap, we had converted qualitative strengths into numerical values (see Appendix ), potentially introducing additional errors and uncertainty.

More fundamentally, interactions between tipping elements simply have received little attention in the broader literature. As seen in Fig. , some interactions can affect a subsystem's risk to be degraded by 10 % (ENSO's effects on the coral reefs) and even by up to 30 % (AMOC's effects on the Greenland ice sheet). To that extent, a missing interaction could change the results, particularly for individual subsystems, by a large amount. Expert elicitations can help to coalesce large scant parts of the literature over multiple models and might even provide insights into events not captured well by Earth system models , however the lack of explicit inclusion of general climate model results and/or historical observations speak to their limited reliability more generally.

Fortunately, more research is being devoted to tipping points in the Earth system, of which our work is one. One major set of works are the Global Tipping Points (GTP) Reports that extensively focus on all aspects and variety of tipping points. The set of interactions evaluated in the latest 2025 report is bigger than (as they include interactions with the North Atlantic subpolar gyre and the Southern Ocean), however there was no longer any mention of the strength of the interactions, let alone a qualitative evaluation, again highlighting the need for more research regarding inter climate subsystem interactions.

Another international collaboration is the the Tipping Points Modelling Intercomparison Project (TIPMIP) which seeks to systematically asses tipping risks using state-of-the-art coupled Earth system models, following the success of the Coupled Model Intercomparison Projects (CMIPs) used for climate projections. As these large projects continue to output detailed and complex models and simulations – some of the preliminary Tier 1 results and experimental protocols of TIPMIP are being released – it will be important to compare the outputs of those models to our simplified framework, given the significant difference in computational and time costs. At the conceptual and mathematical level, it will also be important to draw a more rigorous equivalence between the stylised bifurcation-driven models for individual/interacting elements (e.g. the , model) and our probabilistic approach, in order to derive more realistic and process-based transition rates. Moreover, given the importance of long transients, for example as can be found in AMOC , more research is needed to quantify the timescales associated with transitions to and from intermediate states in order to, say, model each climate subsystem as some N-state Markov chain with more complex transitions and for N>2.

While the coral reefs are at most risk both at equilibrium and in the short term, they form a sink node in the interaction network, with only incident edges, that affects no other nodes in the network, and thus any intervention or policy directed at them would be limited in effect only to the reefs. A long-term and sustainable policy must instead take into account and consider the entire network. For example, we find that AMOC has the highest betweenness centrality , regardless of the network instantiation, further supporting the findings of and that AMOC is a dominant mediator of tipping cascades. Given limited resources, AMOC might therefore be an effective node to intervene on first, in our stylised network model. On the other hand, we stress, once again, the critical importance of limiting global warming and global emissions of CO2 and other short-lived climate pollutants. Future work is needed, therefore, to rigorously test how interventions on single nodes affect the short- and long-term dynamics of the entire Earth system network.

Finally, in this work, we have used the SSP warming scenarios as proxies for the potential effects of humans on the environment and ecology. However, given how complex social-environmental relationships are, it is necessary to integrate agency more directly. To conclude, we note that eco-evolutionary games , multi-agent reinforcement learning dynamics in stochastic games , and networked and spatial games are promising frameworks to do so. Our networked model lays the groundwork for future works that consider the social dilemma of interacting Earth tipping elements and climate subsystems.