  Dependency links in single-layer networks offer a convenient way of modeling
nonlocal percolation effects in networked systems where certain pairs of nodes
are only able to function together. We study the percolation properties of the
weak variant of this model: nodes with dependency neighbours may continue to
function if at least one of their dependency neighbours is active. We show that
this relaxation of the dependency rule allows for more robust structures and a
rich variety of critical phenomena, as percolation is not determined strictly
by finite dependency clusters. We study Erd\"os-R\'enyi and random scale-free
networks with an underlying Erd\"os-R\'enyi network of dependency links. We
identify a special "cusp" point above which the system is always stable,
irrespective of the density of dependency links. We find continuous and
discontinuous hybrid percolation transitions, separated by a tricritical point
for Erd\"os-R\'enyi networks. For scale-free networks with a finite degree
cutoff we observe the appearance of a critical point and corresponding double
transitions in a certain range of the degree distribution exponent. We show
that at a special point in the parameter space, where the critical point
emerges, the giant viable cluster has the unusual critical singularity $S-S_c
\propto (p-p_c)^{1/4}$. We study the robustness of networks where connectivity
degrees and dependency degrees are correlated and find that scale-free networks
are able to retain their high resilience for strong enough positive
correlation, i.e., when hubs are protected by greater redundancy.
