  We introduce a Cucker-Smale-type model for flocking, where the strength of
interaction between two agents depends on their relative separation (called
"topological distance" in previous works), which is the number of intermediate
individuals separating them. This makes the model scale-free and is motivated
by recent extensive observations of starling flocks, suggesting that
interaction ruling animal collective behavior depends on topological rather
than metric distance. We study the conditions leading to asymptotic flocking in
the topological model, defined as the convergence of the agents' velocities to
a common vector. The shift from metric to topological interactions requires
development of new analytical methods, taking into account the
graph-theoretical nature of the problem. Moreover, we provide a rigorous
derivation of the mean-field limit of large populations, recovering kinetic and
hydrodynamic descriptions. In particular, we introduce the novel concept of
relative separation in continuum descriptions, which is applicable to a broad
variety of models of collective behavior. As an example, we shortly discuss a
topological modification of the attraction-repulsion model and illustrate with
numerical simulations that the modified model produces interesting new pattern
dynamics.
