判斷隨機漫步在有限（或無限）的圖上是transient或recurrent可以藉由討論在有限（或無限）的圖上制定一電子電路（electric network），我們得出一重要結論：此隨機漫步是transient 若且為若此電子電路的有效電導為正。我們可以簡單的在有限的圖上計算有效電導，但對於計算無限圖上的有效電導並不容易，所以我們更進一步的推廣上述結論，利用功率（energy）來判別transient和recurrent。 第二部分則是探討一些tree上的問題，為了更加了解tree的結構，我們探討branching number的定義及計算方式。 ;We consider random walks on graphs, we investigate whether a random walk is transient or recurrent. This problem can be solved by developing electric networks on graphs, and we obtain the main consequence of our discussion: the random walk is transient if and only if the effective conductance of the network is positive. Base on this result, furthermore, we study energy for the calculation of effective conductance. The second part, we study the problems on trees such as the percolation on Galton-Watson tree and random spanning trees. Branching numbers can be calculated by using the definition of Hausdorff dimension and capacity.
