Introducing cluster tomography for complex systems

In nature, many phenomena lead to cluster formation, including magnetic domains, bacteria swarming, cell migration and the collective motion of animal groups, as well as distinct regions of the brain or cities. Here we propose a quantitative approach to assess the geometric complexity of such clustered systems, called cluster tomography. Cluster tomography asks how many clusters are crossed by a probe that is shot through a complex system of clusters. To answer this question, we consider lines that cross the entirety of finite 2d and 3d percolation systems. The number of clusters N crossed by these lines scales linearly with the size of the system L as aL where a is non-universal and therefore depends on microscopic details of the system. However, at criticality we find an additional singularity of the form b_γ log(L) where the universal prefactor b_γ depends only on the angle γ at which the line intersects the system’s surface. Our numerical results are verified by analytical arguments found by extending known results for line segments that cross only a part of the system (i.e., partial tomography). With the universal singularities only being observed at criticality, cluster tomography can be used in a wide variety of applications including detecting phase transitions and identifying universality classes in complex systems.