Biased random walk on a graph

Imagine you are playing a game where you have a bunch of toys, and you are in a big room with lots of different places to go. You can move around by taking one step at a time, and you can randomly decide which direction to go in. It's like you close your eyes and point to a place to go.

When you take a step, you can either go to a nearby toy or to a toy that's further away. But you are more likely to go to toys that are closer to you than toys that are far away. This means you are a little bit biased towards the nearby toys.

Now, imagine that instead of toys you have a graph, which is just a bunch of points connected by lines. You can still take a step at a time and choose which point to go to. But now, some points are more likely to be chosen than others. This is called a biased random walk on a graph.

The bias in this case might be something like a weight or a score assigned to each point. Points with higher weights or scores are more likely to be chosen than points with lower weights or scores. This bias can affect how the walker moves through the graph and which points it visits more often.

Overall, a biased random walk on a graph means that you are randomly choosing where to go, but you have a preference for certain points over others. This can be used to model lots of different things, like how diseases spread through a network of people or how information travels through a social media network.