What are some examples of undirected weighted networks in ecology?
Epidemiology, the spreading of diseases, is probably one of the most famous applications of network theory in biology, going all the way back to John Snow.

A more relevant example would maybe be something like graph models of habitat mosaics. Populations are often both spread out over landscapes and clumped together in smaller pockets of suitable habitats. While fragmented, these populations still have a certain amount of contact, resulting in gene flow and the creation of what is often called a meta-population. These things can happen on a vast range of scales, from your back yard to whole continents, but by modeling them like graphs they can all be studied comparatively and tested for resilience agains disturbances, habitat loss etc.

You mention social networks, but there is no reason why you can't study the same in animal populations. Say one individual learns something useful, maybe the location and route to a rich feeding ground. That information can then be shared with other individuals, either by show-and-tell, or maybe by some kind of abstract communication. The information spreads in familiar exponential fashion until everyone knows, then any one of those individuals can relay or be "eaves dropped" on by a neighbouring population, it doesn't even have to be the same species in some cases, and the information flows again. As with the habitat mosaics the benefit of these models is that very different systems can be compared and tested on an equal footing, anything from fish, to birds, mammals, even some social insects, lend themselves well to this.