Graphs show up all over the place. Social media: who is connected, which people interact. Cybersecurity: which computers/programs/users interact with which other computers/programs/users. Retail analytics: which products are bought with which other products; which products are more important in a graph than others.

Basically, any problem where you can establish relations between elements can be treated as a graph. I've used graphs for image analysis before too: pixels are vertices, edges represent neighborhood relations - especially useful when you make nonlocal connections (e.g., nonlocal means; graph-cut methods for segmentation; etc...)

I've worked with them in three of the above contexts: cybersecurity (my current projects), retail analytics, and image analysis. I've avoided social network stuff - never cared for that area much.