Here's a simple simulation from the book The Armchair Universe: An Exploration of Computer Worlds by A.K. Dewdney.
The squares in a grid are randomly colored blue and red. These squares represent the political persuasion of a city block. One color might represent a Democrat, the other a Republican. For each turn, a city block is selected randomly. It's political persuasion is randomly set to one of it's adjacent neighbors.
While simple-minded, an interesting behavior develops from the initial setup. First, large blocks of like voters develop within the grid. Then the blocks migrate around the grid. For a while two blocks will struggle for dominance. Finally, the two-part system collapses as everyone ends up voting the same way.