  The concept of network immunity, i.e., the robustness of the network
connectivity after a random deletion of edges or vertices, has been
investigated in biological or communication networks. We apply this concept to
a self-assembling, physical network of microemulsion droplets connected by
telechelic polymers, where more than one polymer can connect a pair of
droplets. The gel phase of this system has higher immunity if it is more likely
to survive (i.e., maintain a macroscopic, connected component) when some of the
polymers are randomly degraded. We consider the distribution $p(\sigma)$ of the
number of polymers between a pair of droplets, and show that gel immunity
decreases as the variance of $p(\sigma)$ increases. Repulsive interactions
between the polymers decrease the variance, while attractive interactions
increase the variance, and may result in a bimodal $p(\sigma)$.
