  We analyse the topology of the state space of two systems: i) N Ising spins
+/-1 with the antiferromagnetic interactions on a triangular lattice, with the
condition of minimum of energy, ii) a roundabout of three access roads and
three exit roads, with up to 2 cars on each road. The state space is
represented by a network, and states - as nodes; two nodes are linked if an
elementary process (spin flip or car shift) transforms the respective states
one into another. Information is collected on the number of neighbours of
states, what allows to distinguish classes and subclasses of states, and on the
cluster structure of the state space. In the Ising systems, the clusters are
characterized by anisotropy of the spin-spin correlation functions. In the case
of a roundabout, the clusters differ by the number of empty or full roads. The
method is general and it provides a basis for applications of the random walk
theory.
