Small Countries, Good Expanders

by radimentary

Small Countries

Political scientists have known for a very long time a simple scaling law: smaller nations are better run. To my knowledge, Montesquieu was the first to publicize this matter:

It is natural for a republic to have only a small territory; otherwise it cannot long subsist. In an extensive republic there are men of large fortunes, and consequently of less moderation; there are trusts too considerable to be placed in any single subject; he has interests of his own; he soon begins to think that he may be happy and glorious, by oppressing his fellow-citizens; and that he may raise himself to grandeur on the ruins of his country.

In an extensive republic the public good is sacrificed to a thousand private views; it is subordinate to exceptions, and depends on accidents. In a small one, the interest of the public is more obvious, better understood, and more within the reach of every citizen; abuses have less extent, and, of course, are less protected.

The long duration of the republic of Sparta was owing to her having continued in the same extent of territory after all her wars. The sole aim of Sparta was liberty; and the sole advantage of her liberty, glory.

One can see positive evidence for this law today, most of the highest functioning – at least in the sense of bureaucratic efficiency – are barely visible on the map: here is an arbitrary list I found: Switzerland, Iceland, Sweden, Norway, Denmark, Ireland, Netherlands, Austria, New Zealand, Finland, Germany, Luxembourg, Canada. The next set of highest: Belgium, Australia, Czech Republic, Singapore, United Kingdom, Slovenia, France, Spain, Japan, Slovakia, Taiwan, Italy, Mauritius, Portugal, Uruguay, Lithuania, South Korea, Latvia, Costa Rica, Romania.

There are a handful of exceptions, but is truly remarkable how many tiny countries there are on this list. I would mention that many of the dysfunctional countries at the bottom of the list are also tiny, but muh narrative…

In any case, it is reasonable to speculate that most of the highest performing nations are tiny, even if the converse is false.

Scaling Laws

I want to suggest that all scaling laws are the same: surface area grows quadratically, but volume grows cubically. In nature, this means that cells cannot be bigger than a certain size, that a elephant-sized ant would need elephant-sized legs to stand up, that there is a natural range of sizes where different types of biological organization exist, and pushing past these size constraints need real feats of ingenuity.

Whereas a single-celled organism can simply wait to absorb oxygen from its membrane, the human body has an entire organ system dedicated to expending energy to transport oxygen throughout its interior, essentially artificially increasing the “surface area” of the human body. A body with a thousand times the mass has only a hundred times the surface area naturally.

What kind of scaling law controls political efficiency? Montesquieu suggests that republics too large will altogether fail due to splintering, factionalism, and self-interest. History has already borne witness to the falsehood of this original idea, but it contains at least a grain of truth.

Good Expanders

I think bureaucratic efficiency is directly tied to the connectivity of the social network. In some sense, the relevant scaling law is: smaller countries are better connected. What does this mean? Is it just a corollary of the one true scaling law I described already?

You might define connectivity as average degree – then I think the scaling law must be false. People generally have some bounded number of friends and acquaintances, the number and strength of which doesn’t vary depending on the size of the nation they live in. There is no nation so small that this constrains significantly the size of a person’s contact list.

The sophisticated notion of connectivity is expansion, which is closely related to the notion of conductance. A graph is a good expander if every subset has a lot of neighbors outside itself – that is, there are no large, relatively isolated subcultures. Formally, we might define the conductance of a set of vertices as the ratio of edges leaving the set to edges staying inside the set. The conductance of the graph then is the worst conductance of all its subsets (other than the very big ones). A good expander is a graph with high conductance, so that every subset has lots of edges leaving it.

Why is this a useful notion of connectivity and how might it relate to politics or efficiency? The basic and important property of expanders is that they act like sparse random graphs – in particular, random walks on expanders converge rapidly to uniform. One corollary: information (or any other resource we’re trying to spread: values, amenities, culture) spreads quickly and uniformly across an expander. In contrast, with bad expanders information will get stuck in “bottlenecks” of low conductance for long periods of time.

Then I propose that the law of scaling for conductance is: conductance grows like perimeter/area, which will generally depend inversely on the size of the nation grows.

Double the area of a nation. The number of people multiplies roughly by four, and so too the number of edges in the social network multiplies by four. However, the length of the border of the nation only doubles, so in net the larger nation is relatively better isolated from its neighbors due to geography. The same argument applies to any geographically contiguous region inside the nation – as these get larger, they also get paradoxically more isolated from the other. As long as social networks are bounded degree and tied to geography in a serious way, this is a simple artifact of our original scaling law one dimension down: perimeter grows linearly and area grows quadratically. It is no wonder that bigger nations are more divided.