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A simple model of the two dimensional collective motion of a group of mobileagents have been studied. Like birds, these agents travel in open free spacewhere each of them interacts with the first $n$ neighbors determined by thetopological distance with a free boundary condition. Using the sameprescription for interactions used in the Vicsek model with scalar noise it hasbeen observed that the flock, in absence of the noise, arrives at a number ofinteresting stationary states. In the `single sink state' the entire flockmaintains perfect cohesion and coherence. In the `cyclic state' every agentexecutes a uniform circular motion, and the entire flock executes a pulsatingdynamics i.e., expands and contracts periodically between a minimum and amaximum size of the flock. When refreshing rate of the interaction zone is thefastest, the entire flock gets fragmented into smaller clusters of differentsizes. On introduction of scalar noise a crossover is observed when the agentscross over from a ballistic motion to a diffusive motion. Expectedly thecrossover time is dependent on the strength of the noise $\eta$ and diverges as$\eta \to 0$. In simpler version the translational degrees of freedom of theagents are suppressed but their angular motion are retained. Here agents arethe spins, placed at the sites of a square lattice with periodic boundarycondition. Every spin interacts with its $n$ = 2, 3 or 4 nearest neighbors. Inthe stationary state the entire spin pattern moves as a whole when interactionsare anisotropic with $n$ = 2 and 3; but it is completely frozen when theinteraction is isotropic with $n=4$. These spin configurations havevortex-antivortex pairs whose density increases as the noise $\eta$ increasesand follows an excellent finite-size scaling analysis.

Cyclic and Coherent States in Flocks with Topological Distance