  Consider a particle whose position evolves along the edges of a network. One
definition for the displacement of a particle is the length of the shortest
path on the network between the current and initial positions of the particle.
Such a definition fails to incorporate information of the actual path the
particle traversed. In this work we consider another definition for the
displacement of a particle on networked topologies. Using this definition,
which we term the winding distance, we demonstrate that for Brownian particles,
confinement to a network can induce a transition in the mean squared
displacement from diffusive to ballistic behaviour, $\langle x^2(t) \rangle
\propto t^2$ for long times. A multiple scales approach is used to derive a
macroscopic evolution equation for the displacement of a particle and uncover a
topological condition for whether this transition in the mean squared
displacement will occur. Furthermore, for networks satisfying this topological
condition, we identify a prediction of the timescale upon which the
displacement transitions to long-time behaviour. Finally, we extend the
investigation of displacement on networks to a class of anomalously diffusive
transport processes, where we find that the mean squared displacement at long
times is affected by both network topology and the character of the transport
process.
