Weighted graphs with presumed cluster structure are challenging to many existing graph drawing methods, even though ways of visualizing such graphs would be much needed in complex networks research. In the field of dimension reduction, t-distibuted stochastic neighbor embedding (t-SNE) has proven successful in visualizing clustered data. Here, we extend t-SNE into graph-SNE (GSNE). Our method builds on the sensitivity of random walks to cluster structure in graphs. We use random walks to define a neighborhood probability that realizes the properties behind the success of t-SNE in visualizing clustered data sets: Gaussian-like behavior of neighborhood probabilities, adaptation to local edge density, and an adjustable granularity scale. We show that GSNE correctly visualizes artificial graphs where ground-truth cluster structure is known. Using real-world networks, we show that GSNE is able to produce meaningful visualizations that display plausible cluster structure which is not captured by state-of-the-art visualization methods.