At the turn of this century it was realized that social and communication networks were best modeled by graphs that were "small worlds" and/or had power law degree distributions. I will discuss two examples. The first is a situation where physicist's mean field arguments give the wrong answer about the spread of epidemic. The second, inspired by a gypsy moth outbreak in the late 1980s in NY leads to chaotic behavior. I will concentrate on what is true rather than why, so the talk should be accessible to a wide audience.