  States of self stress (SSS) are assignments of forces on the edges of a
network that satisfy mechanical equilibrium in the absence of external forces.
In this work we show that a particular class of quasilocalized SSS in
packing-derived networks, first introduced in [D. M. Sussman, C. P. Goodrich,
and A. J. Liu, Soft Matter 12, 3982 (2016)], are characterized by a lengthscale
$\ell_c$ that scales as $1/\sqrt{z_c-z}$ where $z$ is the mean connectivity of
the network, and $z_c\!\equiv\!4$ is the Maxwell threshold in two dimensions,
at odds with previous claims. Our results verify the previously proposed
analogy between quasilocalized SSS and the mechanical response to a local
dipolar force in random networks of relaxed Hookean springs. We show that the
normalization factor that distinguishes between quasilocalized SSS and the
response to a local dipole constitutes a measure of the mechanical coupling of
the forced spring to the elastic network in which it is embedded. We further
demonstrate that the lengthscale that characterizes quasilocalized SSS does not
depend on its associated degree of mechanical coupling, but instead only on the
network connectivity.
