  Consider a navigation rule defined on a graph that maps every vertex of the
graph to a vertex in such a way that the navigation rule commutes with every
automorphism of the graph. It is to say that the navigation rule applied to the
vertices remains the same after taking any automorphism of the graph. Such a
navigation rule is said to have the covariance property. This study delves into
a collection of such covariant navigation rules, indexed by locally finite
graphs, and subject to a measurability condition. This ensemble of rules is
termed a vertex-shift. More generally, one can consider vertex-shifts on
networks, graphs that have labels on edges and on vertices. A vertex-shift
induces a dynamic on the space of locally finite rooted networks. The central
focus of this work lies in investigating the dynamic associated to a specific
navigation rule called the record vertex-shift. It is defined on the
trajectories of any one dimensional discrete random walk whose increments have
finite mean and the random walk can jump only one step to the left.
Additionally, the work includes several notable results concerning record
vertex-shifts applied to processes with stationary increments. The thesis also
contains results on more general vertex-shifts on unimodular networks.
