Imagine a random walk on the graph. Starting from an unlabeled node i, the random walker moves to a neighbor node j with probability proportional to the edge weight between i and j. The random walk repeats until the walker hits a labeled node. Let h(i)=P(hit label 1 | start from i) be the probability that the random walk, starting from node i, hits a labeled node with label 1. Here the labeled data is viewed as an "absorbing boundary" for the random walk. Then the harmonic function is exactly h(i). The harmonic function is used for semi-supervised learning. See our ICML03 paper for details.
The following Java animation shows the random walk. Reload the page to change the initial labeled points.