Fragile balances: understanding global stability

Predicting the fate of world order using a toy model

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Introduction

In a world shaped by interconnected political and economic systems, international stability cannot be easily given for granted. Today’s global order relies on a dense network of treaties and institutions — many created after World War II — to coordinate cooperation, prevent conflict and address cross-border threats like pandemics and climate change. From FAO to the United Nations and World Health Organization, from the World Trade Organization to the European Union.

Nevertheless, as we enter an era marked by multipolar power dynamics, shifting alliances and institutional stress, the possibility of sudden breakdowns in global governance becomes real. Understanding which parts of this complex architecture are most vulnerable — and, overall, why — is crucial for anticipating future disruptions and strengthening resilience.

A couple of years ago we have used publicly available data about political and economic alliances to build a multilayer network representation of socio-economic and political interests (see the paper here). Nodes represent countries and edges represent existing interactions between different relationships, such as political and economic ones, which are mapped to different layers.

Then we have proposed hypothetical scenarios where either single countries could leave an existing alliance, or entire alliances/organizations could be wiped out. Started as an academic inquiry, it turned out that real perturbations like the Brexit or the U.S. leaving the WHO would be perfectly covered by the model. This illustrates that even simple models can offer meaningful insights into the structural vulnerabilities of the international system when real-world shocks occur.

The overall analysis reveals a counterintuitive insight: it is not the major powers but rather medium, small and micro-states that carry the highest potential for disrupting global stability. Politically, these include former colonial territories and tax havens; economically, mid-sized countries in Europe and Africa play pivotal roles.

The most fragile alliance? The World Trade Organization. Importantly, we have shown that structural factors — such as colonial legacies and offshore financial networks — matter more than traditional fragility indices in determining systemic risk (within the modeling assumptions and limitations, of course).

Cartography of world order resilience. The central panel shows the distorted maps of the fragile states index (FSI, top) vs. the political damage index (PDI, middle) and economic damage index (EDI, bottom), as defined in the text. Cartograms are built according to Gastner and Newman (33). In the left and right panels, we show the values of the damage index with respect to economic and political relationships, separately, in response to country-based disruptions (left) and treat-based disruptions (right): values range from 0 to 1, and we refer to the text for further details. [Source]

At that time we did not use other potential layers, such as the one encoding trade of goods, and some additional dimensions of interest to better characterize world order. But the goal of that paper was to show how multilayer network science (see this small technical book, if interested) can be used for the purpose. And it was mostly about the structure and its response to hypothetical stressors and disruptive scenarios, by simulating cascade failures to study the robustness of the global order.

A dynamical systems perspective on stability

It might be interesting to understand why countries evolved to their current sizes and network of alliances. To get an idea about the “dynamics”, see this nice video with a visualization based on historical data:

Dynamical systems theory offers a powerful lens to describe international behavior. In this context, a country’s “size” (naively used as a proxy for population, capacity, influence or economic weight) helps to shape at least three key variables: its internal instability (I), exposure to external risk (E) and overall systemic health (S). By modeling these interactions over time by means of equations, we can simulate how small shocks — like the defection from a treaty — can propagate and destabilize broader networks, as in the study described above, and we can try to gain some basic insights about the stability of the system in the long term, even in absence of shocks. This approach helps identify equilibrium conditions, tipping points and nonlinear effects that would be missed by static analyses.

It’d be impossible to describe even the basics of (linear) stability analysis in dynamical systems theory, therefore I am not going to do it, referring to technical sources. What it might be useful to understand at this stage, to follow the subsequent part of the post, is to know that:

• we can map our assumptions into mathematical equations, expressing how some variables of interest (such as the size or the health of a country) change over time as a function of the intervening factors — that we can name X — that we hypothesize, depending on some control parameters that we can indicate by Θ:

  \(\frac{dX}{dt} = f(X;\Theta)\)

• Using this prescription, we model each country as a dynamic system (like the above one) defined by three variables:

  • S: “size” (political/economic/systemic health)

  • I: internal instability (e.g. unrest, institutional fragility, internal conflicts)

  • E: external risk (geopolitical or economic threats)

  These variables evolve over time based on internal dynamics and external interactions, where instead of a generic parameter Θ we would like to consider key parameters such as:

  • α (alpha): intrinsic growth rate of “health”

  • K: carrying capacity, i.e. the maximum attainable “health”

  • β (beta): sensitivity of “health” to internal instability

  • γ (gamma): sensitivity of “health” to external risk

  • μ (mu): rate at which “health” feeds internal instability

  • δ (delta): decay rate of internal instability

  • σ (sigma): strength of destabilization due to conflicts with other countries

  • η (eta): baseline external risk, higher in small or isolated countries

  • ζ (zeta): natural decay rate of external risk

  • ρ (rho): strength of external stabilization due to alliances

  • A⁻(i,j) and A⁺(i,j): conflictive and cooperative interactions, denoting a network, between populations i and j belonging to distinct geographic areas

  • z⁻, z⁺: average number of conflictual and cooperative ties in the international network

Yes, it’s a bit complex and we should be really get rid of most parameters (spoiler: we’ll do). But for the moment let’s see where this approach will lead us. The equations for the model can be written as follows:

\(\begin{eqnarray} \frac{dS_i}{dt} &=& \alpha_i\, S_i \left(1 - \frac{S_i}{K_i}\right) - \beta_i\, I_i\, S_i - \gamma_i\, E_i\, S_i \\ \frac{dI_i}{dt} &=& \mu_i\, S_i - \delta_i\, I_i + \sigma \sum_{j=1}^{N} A_{ij}^{-} \frac{S_j}{S_i}\\ \frac{dE_i}{dt} &=& \frac{\eta_i}{S_i} - \zeta_i\, E_i - \rho \sum_{j=1}^{N} A_{ij}^{+} S_j \end{eqnarray} \)

It looks more complicated than what it really is: the model describes how a country's internal dynamics and its international environment can jointly shape its stability over time. The index i allows us to identify the behavior of the i-th country in terms of the aforementioned variables.

Health: dSᵢ/dt

A country tends to grow toward a stable state, much like populations grow in ecosystems. This growth is naturally limited by a carrying capacity (e.g. resources, geographic space, etc.) and enhanced by the country’s ability to self-organize. This explains the first term, the one proportional to αᵢ. However, stability can be eroded due to:

• Internal instability (Iᵢ): unrest or dysfunction reduces capacity, and we can assume that the larger the size (someone would name it “complexity”, using an umbrella term) the higher the rate at which internal instability can have an impact. This assumption results in the negative term -βᵢ Iᵢ Sᵢ

• External risk (Eᵢ): global pressures — like sanctions, wars or economic shocks —also weaken the health of a country, therefore we can assume that the larger the size the larger the rate at which external risks can have an impact. This assumption results in the negative term -βᵢ Eᵢ Sᵢ

The more unstable or externally threatened a country is, the harder it becomes to maintain or grow its health.

Internal instability: dIᵢ/dt

Instability might arise from two main sources:

• Domestically, when a system is very healthy (high Sᵢ), paradoxically it may generate more tension (e.g., reform pressures, inequality), captured by a production rate and the term +µᵢ Sᵢ.

• Internationally, instability can spread from other nations through conflict networks. If a country is exposed to unstable or aggressive neighbors, this raises its own internal pressures. This behavior is captured by the terms with σ

• Meanwhile, instability tends to decline over time as societies adapt or crises subside, as captured by the term -δᵢ Iᵢ

External risk: dEᵢ/dt

This captures the risk a country faces from the broader international system. It includes:

• A baseline risk, especially relevant for small or geopolitically exposed nations, that we can model as +ηᵢ / Sᵢ. We will likely keep this term equal to zero to simplify a bit the model.

• Natural decay, as external threats often fade unless reinforced, captured by the term -ζᵢ Eᵢ

• Stabilizing alliances, which reduce external risk: stronger partners and treaties help buffer shocks. This behavior is captured by the terms with ρ

At this point we have all the ingredients to start our analysis. First of all, let’s simplify the model, by considering that all actors (nations/countries) have an average number of positive (due to alliances) and negative (due to enemies) interactions, and that the other parameters are the same for any country. This is not very realistic, but we can still learn a lot from the following:

\(\begin{equation} \begin{aligned} \frac{dS}{dt} &= \alpha\, S\left(1 - \frac{S}{K}\right) - \beta\, I\, S - \gamma\, E\, S\\ \frac{dI}{dt} &= \mu\, S - \delta\, I + \sigma\, z^-\\ \frac{dE}{dt} &= - \zeta\, E - \rho\, z^+\, S \end{aligned} \end{equation}\)

We are interesting in the equilibrium condition, that is when the situation is stable and we observe no variations in our variables. In this case, the derivative with respect to time in the left-side of the three equations is just zero. If we solve the equations accordingly and discard the trivial solution with S = 0 , we obtain

\(S^{\star} = \frac{\frac{\beta \sigma}{\delta}z^- - \alpha}{\frac{\gamma \rho}{\zeta} z^+ - \frac{\alpha}{K} - \frac{\beta \mu}{\delta}}\)

Since we are assuming the non-trivial case, to have a meaningful solution we must require that both numerator and denominator have the same sign and that the ratio is positive, leading to the following conditions:

\( z^+ > z^+_{\mathrm{crit}} \quad \text{with} \quad z^+_{\mathrm{crit}} = \frac{\zeta}{\gamma \rho}\left(\frac{\alpha}{K} + \frac{\beta \mu}{\delta}\right),\)

and

\( z^- > z^-_{\mathrm{crit}} \quad \text{with} \quad z^-_{\mathrm{crit}} = \frac{\alpha \delta}{\beta \sigma}\)

or, equivalently, both parameters have to be smaller than their critical values. Note that another way to better appreciate the result is to consider an adequate rescaling of the variable S* and rewrite the equilibrium state as

\(s^{\star} = \frac{S^\star}{K}\left(1+\frac{\mu K}{\sigma z^-_{\text{crit}}}\right) = \frac{\frac{z^-}{z^-_{\text{crit}}} - 1}{\frac{z^+}{z^+_{\text{crit}}} - 1} = \frac{\xi^- -1}{\xi^+ - 1}\)

A quick and dirty R code to plot this transition gives the following heatmap, which is handy since we have removed the dependence on the specific parameters by rescaling our variables, leading to the true essence of the transition:

Interestingly, the model shows that “rescaled systemic health” doesn’t require high cooperation and high conflict simultaneously. What matters is balance: the (rescaled) average number of cooperative and conflictual interactions must either both exceed or both fall below their respective critical thresholds (which is 1, after rescaling). This leads to two possible “regimes” where stability is possible:

• In a high-engagement world, intense cooperation and rivalry coexist to generate a dynamic but stable equilibrium.

• In a low-engagement world, countries are relatively isolated, facing little conflict but also few external stabilizing alliances.

However, if a country is caught between these extremes (e.g., it faces significant conflict but has too few alliances) then no stable solution exists (the black areas in the above plot). This asymmetry highlights how mismatched exposure to threats and support can lead to instability.

Two words about unbounded growth

Of course, this model is just a toy: it is useful for exploring dynamics, not for predicting real-world outcomes. Still, it’s worth saying a few words about a theoretically interesting case: unbounded growth. This is a common (though often unrealistic) assumption in many simplified models.

To achieve unbounded growth in our framework, the carrying capacity must be infinite, while both internal instability and external risk must vanish in the long run. For this to happen, several stringent conditions must hold:

• The parameter μ, which controls how systemic "health" feeds instability, must be zero.

• The coupling σ, which captures destabilization through conflict with other countries, must also be zero.

• The decay rate δ for internal instability must be very large, implying that the system must be intrinsically able to rapidly resolving its own problems.

In short: conflicts must have no impact, domestic complexity must not generate any internal tension and the country must be inherently self-stabilizing.

On the external side, large values of ρz⁺ and ζ are needed to suppress external risk : through strong alliances and rapid risk dissipation.

Honestly, I struggle to imagine how such conditions could ever be sustainable or realistic. Our simplistic model allows for unbounded growth, but only in a highly idealized (and arguably unfeasible) world.

I have discussed with Pierluigi Sacco — Professor of Economics at the Department of Neuroscience, Imaging and Clinical Studies, University of Chieti-Pescara; Senior Adviser to the OECD; Research Affiliate at the metaLAB (at) Harvard — about this.

What are the minimal insights that we can get from a model like this?

The most compelling insight of the model is that stability doesn't require minimizing all conflicts or maximizing all cooperation, as a straightforward linear thinking would easily conclude. Rather, it's about achieving balance. The model shows two stable regimes:

• A high-engagement world where cooperation and rivalry coexist

• A low-engagement world with minimal conflicts but also fewer alliances

This reflects some real-world patterns we have observed. For example, the post-WWII order features significant cooperation (UN, WTO, etc.) alongside persistent rivalries (cold war, USA-China, etcetera) which would often relfect into the global cooperation institutions themselves. Similarly, historically isolated regions sometimes maintain stability through limited external engagement (this is the case of many dictatorships such as North Korea, Turkmenistan, Myanmar, etcetera).

What are the missing ingredients to make it more realistic?

The model would add some interesting feature by contemplating the possibility of strategic interaction between nations in a game-theoretic framework, which would better reflect the complexity of international relations. Another important aspect is that interactions among countries are often mediated by factors of cultural distance/proximity.

A more sophisticated model would recognize multiple non-trivial equilibria and should reproduce the move from unipolarity toward multipolarity that we are currently witnessing.

Can we use a simplified model like this to “read” some aspects of the current geopolitical situation?

The "black areas" in the model's stability map — where countries face significant conflict but have too few alliances — might describe nations caught in geopolitical crossfires like Ukraine or Taiwan. These nations experience heightened external risk without sufficient stabilizing alliances, making their positions inherently precarious.

The notion that structural factors like colonial legacies and offshore financial networks matter more than traditional fragility indices also has contemporary relevance. It helps explain why seemingly stable systems can experience sudden disruptions when these underlying structural connections are stressed.

Conclusions

The non-trivial equilibrium solution, while mathematically consistent, results in a positive level of internal instability (which is fine, while not desirable) and a negative external risk (which is not desirable, or even unphysical). This outcome reflects a limitation in the model’s construction: we haven’t precisely defined what external risk is, nor how to bound or interpret it. As a result, negative values were allowed by design, even if they lack some real-world meaning.

Another limitation is that the model is not really considering a game-theoretical framework where nations have payoffs for their decisions: our model is deterministic and we are assuming precise — yet stringent — dynamical rules, leaving no room for stochastic fluctuations.

A more realistic model would require variables with appropriate domains and interpretation. It would also need to relax the assumption that all actors are identical, recognizing, instead, that states differ in capacity, exposure and behavior. In that richer setting, we’d likely observe multiple non-trivial equilibria, similar to what happens in ecological models, leading to a rugged stability landscape with basins, thresholds and even tipping points.

That would be a fun direction to explore: but at that point, we’re no longer writing a blog post… we’re writing a paper.