Why is the equilibrium network size can be less than the socially optimal size in networks w/ positive externalities for single system?
The equilibrium network is smaller than that chosen by a social planner because agents do not internalize network externalities.

Vaccination against a contagious disease is a standard such example.
Someone who is not vaccinated exposes others to potential infection and prevents others from, say, using public facilities. This is a negative externality for which he pays no cost. A social planner would therefore impose vaccination when an individual agent might choose not to.