A handful of matches falls on the table. Each lies however it lands, but in our minds we turn each into an arrow — from one end to the other. Now ask: can you get from one point to another by moving only in the direction of the arrows? A group of points where you can reach any one from any other is called a strongly connected cluster. The authors calculated how such clusters emerge in lattices of different dimensions — from a flat grid to a seven-dimensional cube.
It turns out that as long as the dimension is less than six, the shape of the clusters is intricate. Their internal "roughness" (fractal dimension) differs from the dimension of the space itself. It's as if a ball of yarn in a three-dimensional room had its own measure of tangling, not equal to three. But starting at six dimensions, everything simplifies: the clusters become "smooth," following the same laws as in a network where every point is connected to every other. However, in such a fully connected network, a surprise emerged. Large clusters grow according to one scenario, while small ones follow another, as if in a crowd, micro-groups first huddle together, and only then merge into a giant procession.
These results show that even in a multidimensional world, where intuition fails, strongly connected clusters remain sharp fractal objects. And the transition to simple behavior at high dimensions is accompanied by unexpected rearrangements in the statistics of sizes. As if increasing the size of the room changes not only the shape of the yarn ball but also the very manner in which the threads tangle.
In a fully connected network (where everyone is connected to everyone else), small strongly connected clusters behave as if they live in their own world with different rules: their size distribution follows a power law with exponent 1, not 4, as for large clusters.
In Stanisław Lem's novel "His Master's Voice," mathematicians study a random signal, trying to extract meaningful structures from it — much like here, where one looks for islands of directed connectivity in a randomly oriented network.