  We study push-pull rumour spreading in small-world models for social networks
where the degrees follow a power-law. In a non-geometric setting Fountoulakis,
Panagiotou and Sauerwald have shown that rumours always spread fast (SODA
2012). On the other hand, Janssen and Mehrabian have found that rumours spread
slowly in a spatial preferential attachment model (SIDMA 2017). We study the
question systematically for the model of geometric inhomogeneous random graphs
(GIRGs), which has been found to be a good theoretical and empirical fit for
social networks. Our result is two-fold: with classical Euclidean geometry both
slow and fast rumour spreading may occur, depending on the exponent of the
power law and the prevalence of weak ties in the networks, and we fully
characterise the phase boundaries between those two regimes. Depending on the
parameters, fast spreading may either mean polylogarithmic time or even doubly
logarithmic time. Secondly, we show that rumour spreading is always fast in a
non-metric geometry. The considered non-metric geometry allows to model social
connections where resemblance of vertices in a single attribute, such as
familial kinship, already strongly indicates the presence of an edge. Classical
Euclidean Geometry fails to capture such ties.
  For some regimes in the Euclidean setting, the efficient pathways for
spreading rumours differ from previously identified paths. A vertex of degree
$d$ can transmit the rumour efficiently to a vertex of larger degree by a chain
of length $3$, where one of the two intermediaries has constant degree, and the
other has degree $d^{c}$ for some constant $c<1$.
