  Localization phenomena permeate many branches of physics playing a
fundamental role on dynamical processes evolving on heterogeneous networks.
These localization analyses are frequently grounded, for example, on
eigenvectors of adjacency or non-backtracking matrices which emerge in theories
of dynamic processes near to an active to inactive transition. We advance in
this problem gauging nodal activity to quantify the localization in dynamical
processes on networks whether they are near to a transition or not. The method
is generic and applicable to theory, stochastic simulations, and real data. We
investigate spreading processes on a wide spectrum of networks, both
analytically and numerically, showing that nodal activity can present complex
patterns depending on the network structure. Using annealed networks we show
that a localized state at the transition and an endemic phase just above it are
not incompatible features of a spreading process. We also report that epidemic
prevalence near to the transition is determined by the delocalized component of
the network even when the analysis of the inverse participation ratio indicates
a localized activity. Also, dynamical processes with distinct critical
exponents can be described by the same localization pattern. Turning to
quenched networks, a more complex picture, depending on the type of activation
and on the range of degree exponent, is observed and discussed. Our work paves
an important path for investigation of localized activity in spreading and
other processes on networks.
