The problem of quickest change detection for structural changes within correlated, structured distribution is studied. Precisely, we assume that data collected over a network follows a specific instance of undirected graphical models, the Ising model and that the Ising model before and after the change differs only in their structures. We show that for both types of structural change, edge appearance and disappearance, the problem can be suitably transformed into that of Bernoulli random variables. When further restricting the Ising model to have zero mean-­field vectors and forest interaction structure, the transformed Bernoulli problem becomes drastically simpler to solve. Consequently, under the multiple edge appearance setting in Ising forests, our proposed algorithm equipped with only the pre­change structure knowledge is capable of achieving the optimal worst average detection delay and average run length to false alarm trade-off at the large average run length to false alarm regime. Under the one edge disappearance setting in Ising forests, we present an algorithm with sub­optimal worst average detection delay and average run length to false alarm trade-off. We explain the reason for sub­optimality and point out that our method is already optimal under a less strict criteria from the literature. We then turn to improve the runtime complexity of the proposed algorithms. By taking advantage of the correlation propagation nature of correlated networks, we show that instead of doing brute force search for all possible locations of the structural change, monitoring only a few chosen nodes in the network suffice for decent statistical performance. To get the most out of the new procedure, we formulate a node selection optimization problem and provide a simple algorithm to solve it.