Systemic Metrics

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PrinceRank is a measure of a particular agent's satisfaction within a given power structure. There are also metrics that are not specific to individual agents but that characterize power structures as a whole.


In the academic literature on international relations, the polarity of the international system is often an attribute of concern. This metric is typically determined by counting the number of great powers which dominate a system. For instance, a system dominated by one large state is a unipolar system; a system with two relatively strong powers is bipolar; and a system with more than two is considered multipolar.

Polarity is usually not defined in a rigorous way. It typically entails an informal counting of the great powers in the system at any given time, based on states' "market share" of total power — or equivalently, MarketShare[s_] := s/Total[s] when size vectors are used. It may very well be that this procedure is sufficient and that no additional sophistication is needed; in fact, there are rarely debates about what the polarity is at any given point in time. On the other hand, if quantification is possible, we should like to know how.

Singer Concentration

To this end, a definition of power concentration was proposed by Singer (1972). Here Singer concentration is expressed using normalized size vectors:

SingerConcentration[s_] := Sqrt[(Total[MarketShare[s]^2] - 1/Length[s])/(1 - 1/Length[s])]

In other words:

[math]\displaystyle{ \text{SingerConcentration} = \sqrt{\frac{\sum_{}^{} \frac{s^2}{(\sum_{}^{}s)^2} - \frac{1}{n}}{1 - \frac{1}{n}}} }[/math]

This formula returns a number between 0 (least concentrated) and 1 (most concentrated). Unipolar systems are thought to have values from 0.4-0.5, and bipolar and multipolar systems have values from 0.2-0.4. Though conceptually coherent, this formula is never used in the literature.

Herfindahl-Hirschman Index

Another missed opportunity is that international relations theorists have not repurposed the Herfindahl-Hirschman Index (HHI) of market concentration as a way to quantify polarity. The formula for HHI, Total[MarketShare[s]^2], shares the same beating heart as Singer's formula, but it is simpler and well-understood from its use in the antitrust context. Though international relations scholars could have easily adapted the HHI for this purpose, no such reference in the literature has yet been found.

However, we can do better than HHI. HHI always returns a value between 0 and 1, and it is interpreted to have three ranges of market concentration, essentially low, medium, and high. But with a slight hack, and by applying it to international power as opposed to market dominance, we can make it approximate the number of relatively large agents in the system. We do this by dividing each agent's market share ms of power by the market share of the largest agent, before squaring:

[math]\displaystyle{ \text{ModifiedHHI} = \sum_{i=1}^{n} \frac{ms_{i}^2}{\operatorname{Max}(ms)^2} = \sum_{i=1}^{n} s_{i}^2 }[/math]

or in code:

ModifiedHHI[s_] := Total[s^2]

In other words, if we square a normalized size vector and then total it, the result is a summary of the number of large agents in a given power structure. The table below shows the modified HHI for a few simple size vectors. As indicated, this is a simple way to get an objective count of the number of great powers.

s ModifiedHHI
{1} 1
{1, 1} 2
{1, 0.2} 1.04
{1, 1, 1} 3
{1, 0.5, 0.5} 1.5
{1, 0.1, 0.1} 1.02
{1, 0.75, 0.2} 1.6
{1, 1, 0.05, 0.05} 2


But we can do better still. The problem with Singer's formula, the HHI, and the modified HHI is not so much that they aren't used, but that they only take into account the sizes of the relevant entities and not the relationships among them. The figure below illustrates why this is an issue. In the two power structures shown, the agent sizes are exactly the same. However, it is clear to the eye that the system on the left represents a unipolar reality while the one on the right represents a bipolar system.

Polarity - need.png

We can solve for this by building upon PrinceRank to create a measure of polarity that takes into account the entire power structure — both the sizes and the agents' relationships:

Polarity[s_, T_] := Total[NormalizeList[PrinceRank[s, T]]^2]

This function yields a fairly intuitive, albeit non-integer, value of how many poles or hubs there are in the network. Applying this definition to the two examples above, we get:

Polarity - example 1.png

This new polarity function provides a more holistic classification of power structures. Of course, the specific results are dependent upon six parameters: β, μ, λ, ρ, δ, and α. Let's look at some examples using the parameters β=2, μ=3, λ=1, ρ=0.9, δ=0.9, and α=2.25:

Polarity - example 2.png

Roughly speaking, polarity is the number of agents that are tied for first in PrinceRank. In a way, it is more a statement about the potential future of the system rather than its immediate present, because PrinceRank looks ahead by simulating the flow of power through the network. So Polarity recognizes that a state conventionally believed to have unipolar dominance can have that superiority eroded due to the realignment of the smaller agents in the system. To illustrate, in the example of rebellion below, the polarity value anticipates a future composed of five equally dominating agents:

Polarity - example 3.png

Because the polarity function is forward-looking, the relationships among the agents play a large role in how it characterizes them, as shown in these other examples:

Polarity - example 4.png

One final feature to note about the polarity function is the correct way that it handles equal-sized agents with no interrelationships:

Polarity - example 5.png


An important feature of the international system is how stable it is at any given time and understanding this is one of whose primary objectives of international relations scholars, who would like to know when war is likely. Researchers often take the polarity of the system as an independent variable, and the question is how stable the system is, given its polarity. There are a variety of opinions as to whether unipolar, bipolar, or multipolar systems are the most stable, or indeed whether there is any correlation at all.

Here we establish quantitative measures related to stability that can be used to investigate these questions. These metrics are based on two different concepts. First, we quantify stability, or the expected resistance to change of any kind, by measuring how different the size vector and tactic matrix are likely to be in the near term. Second, we use the flow of power to quantify the systems's trajectory towards positive (cooperative) versus negative (violent) change, and its volatility, or tendency toward violence. All of these metrics are based on simulations in which every agent simultaneously makes its "best move" in response to a given power structure.

Size Instability

If we want to know how different the sizes of the agents will be after they make their best moves, size instability uses the Euclidean distance to quantify that change:

[math]\displaystyle{ \operatorname{SizeInstability} = ||\mathbf{s}_{1}-\mathbf{s}_{0}||_{2} }[/math]

where [math]\displaystyle{ \mathbf{s}_{0} }[/math] is the original size vector and [math]\displaystyle{ \mathbf{s}_{1} }[/math] is the size vector after the agents have made their best moves. In code:

SizeInstability[s_, T_] := Norm[MonteCarloBestMove[s,T]["s"]-s]

When there is no change in the size vector, this metric returns a 0, indicating maximum stability. It is not necessarily a problem that a particular power structure has size instability, as it could be that all agents are likely to grow. Stability should be used with the other metrics below to understand the degree and nature of expected change. Of particular concern are situations that are highly unstable with a negative trajectory.

Relationship Instability

If one is concerned with the change in the tactic matrix (and not the size vector) after agents play their best moves, relationship instability can be used to determine how different the two matrices are:

[math]\displaystyle{ \operatorname{RelationshipInstability} = ||\mathbf{T}_{1}-\mathbf{T}_{0}||_{\operatorname{F}} }[/math]

where [math]\displaystyle{ \mathbf{T}_{0} }[/math] is the original tactic matrix and [math]\displaystyle{ \mathbf{T}_{1} }[/math] is the tactic matrix after the agents have made their best moves. In code:

RelationshipInstability[s_, T_] := Norm[MonteCarloBestMove[s,T]["T"]-T, "Frobenius"]

Like size instability, this value is 0 when there is no change in the tactic matrix after the agents make their best moves. When relationship instability is 0, none of the agents have incentives to change their tactics. Accordingly, we say that such a power structure is in equilibrium.

EquilibriumQ[s_, T_] := RelationshipInstability[s,T] == 0

Note, however, that even if none of the agents alters their foreign policy, the law of motion will continue to alter the agents' sizes, thus perturbing the incentives that yielded the equilibrium. For this reason, equilibria are rarely permanent.


The trajectory of a power structure is its tendency as a whole to become more positive or more negative. It is defined as the growth percentage of a power structure, given the agents' best simultaneous moves. Expressed mathematically:

[math]\displaystyle{ \operatorname{Trajectory} = \frac{\sum (\mathbf{s}_{1}-\mathbf{s}_{0})}{\sum \mathbf{s}_{0}} }[/math]

where the variables have the same definitions as above. Trajectory can be thought of as the percent increase (or decrease) in size of the "total economy" of a power structure. Trajectory can be coded as:

Trajectory[s_, T_] := Total[MonteCarloBestMove[s,T]["s"] - s] / Total[s]

The resulting number can be positive, reflecting net growth, or negative, reflecting net decay. A result of 0 means that there is zero net growth, possibly masking the fact that some agents grow and other agents shrink. With this metric, we can get a sense of whether a given power structure is likely to get better or worse in the near term.


Though the stability and trajectory of a power structure help us understand its latent tendencies, these definitions do not speak to what truly concerns the international community. What is critical to know is: How prone to violence is a given power structure? This metric, which we call volatility, is defined as the amount of negative flow emitted after agents have made their best moves, divided by the total flow that could be emitted.

[math]\displaystyle{ \operatorname{Volatility} = \frac{\sum (\mathbf{s}_{1} \circ \sum \operatorname{Ramp}(-\mathbf{T}_{1}))}{(1-\rho) \sum \mathbf{s}_{1}} }[/math]

where [math]\displaystyle{ \mathbf{s}_{1} }[/math] and [math]\displaystyle{ \mathbf{T}_{1} }[/math] are parts of the power structure after all agents have made their best moves and [math]\displaystyle{ \circ }[/math] is elementwise vector multiplication. In code:

Volatility[s0_, T0_] := Block[{m = MonteCarloBestMove[s0,T0]},
  Total[m["s"]*Total[Ramp[-m["T"]]]] / Total[(1-ρ)*m["s"]]

A power structure can be described as volatile if the latent tensions within it create incentives for violence. Of the metrics just discussed, this is probably the one that most concerns observers of the international system, as it quantifies the degree of violent conflict that can be expected to emerge in the near term as a result of structural tensions in the system.

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