  We consider inhomogeneous spatial random graphs on the real line. Each vertex
carries an i.i.d. weight and edges are drawn such that short edges and edges to
vertices with large weights occur with higher probability. This allows the
study of models with long-range effects and heavy-tailed degree distributions.
We introduce a new coefficient $\delta_\text{eff}$ which quantifies the
influence of heavy-tailed degrees on long-range connections. We show that
$\delta_\text{eff}<2$ is sufficient for the existence of a supercritical
percolation phase in the model and that $\delta_\text{eff}>2$ always implies
the absence of percolation. In particular, our results complement those in
Gracar et al. (Adv. Appl. Prob., 2021), where sufficient conditions were given
for the soft Boolean model and the age-dependent random connection model for
both the existence and the absence of a subcritical percolation phase. Our
results further provide a criterion for the existence or non-existence of a
giant component in large finite graphs.
