Weighted graphs can be easily interpreted in several ways: as a stochastic process or electric network. From these different interpretations many different properties of graphs can be derived and used to further describe its geometry. The fundamental properties of random walks are transience and recurrence. Electric networks represent completely different approach on graph interpretation regarding random walks and enable us to interpret transience and recurrence through the concept of effective resistance. Variational problems are natural upgrade on the theory of electric networks and enable us to further understand connection of graph geometry to transience and recurrence. For example, how cutting and shorting the edges affect effective resistance. Using these approaches we derive new techniques for proving transience and recurrence. Furthermore, different types of graphs can be connected using concept of rough isometries that preserve certain relations between graphs. The main idea of this thesis is to investigate the properties of graphs connected to random walks: transience and recurrence and prove their stability under rough isometries. Usefulness of this property is following: for a graph that is transient or recurrent, transience or recurrence comes easily for all graphs that are roughly isometric to him, meaning that rough isometry preserves some kind of graph structure.