  Some of the most important information on a radio pulsar is derived from its
average pulse profile. Many early pulsar studies were necessarily based on only
few such profiles. There, discrete profile components were linked to emission
mechanism models for individual stars through human interpretation. For the
population as a whole, profiles morphology must reflect the geometry and
overall evolution of the radio emitting regions. The problem, however, is that
this population is becoming too large for intensive studies of all sources
individually. Moreover, connecting profiles from a large collection of pulsars
rapidly becomes cumbersome. In this article, we present ToPP, the first-ever
unsupervised method to sort pulsars by profile-shape similarity, using graph
topology. We apply ToPP to the publicly available European Pulsar Network
profile database, providing the first organised visual overview of
multi-frequency profiles representing 90 individual pulsars. We find discrete
evolutionary tracks, varying from simple, single component profiles at all
frequencies, towards diverse mixtures of more complex profiles with frequency
evolution. The profile evolution is continuous, extending out to millisecond
pulsars, and does not fall in sharp classes. We interpret the profiles as a
mixture of pulsar core/cone emission type, spin-down energetics, and the
line-of-sight impact angle towards the magnetic axis. We show how ToPP can
systematically classify sources into the Rankin empirical profile scheme. ToPP
forms one of the key unsupervised methods that will be essential to explore
upcoming pulsar census data such as expected by the Square Kilometer Array.

High Energy Astrophysical Phenomena;Instrumentation and Methods for Astrophysics