Hirsch's theorem

Hirsch's theorem states that for non-negative integer matrices that describe systems where things flow or move between states (like in certain networks or Markov processes), the number of distinct types (orbits) of states that are interconnected is finite. In simple terms, it guarantees that such systems can't have infinitely many separate, mutually reachable groups of states—they are limited to a finite number of interconnected clusters, which helps in analyzing and understanding the structure and long-term behavior of these systems.