  Criticality has been conjectured as an integral part of neuronal network
dynamics. Operating at a critical threshold requires precise parameter tuning
and a corresponding mechanism remains an open question. Recent studies have
suggested that topological features observed in brain networks give rise to a
Griffiths phase, leading to power-laws in brain activity dynamics and the
operational benefits of criticality in an extended parameter region. Motivated
by growing evidence of neural correlates of different states of consciousness,
we investigate how topological changes affect the expression of a Griffiths
phase. We analyze the activity decay in modular networks using a
Susceptible-Infected-Susceptible propagation model and find that we can control
the extension of the Griffiths phase by altering intra- and intermodular
connectivity. We find that by adjusting system parameters, we can counteract
changes in critical behavior and maintain a stable critical region despite
changes in network topology. Our results give insight into how structural
network properties affect the emergence of a Griffiths phase and how its
features are linked to established topological network metrics. We discuss how
those findings can contribute to understand the observed changes in functional
brain networks. Finally, we indicate how our results could be useful in the
study of disease spreading.
