In this thesis, I study fluctuations and rare events of time-additive observables of discrete-time Markov chains on finite state spaces. The observable of interest is the mean node connectivity visited by a random walk running on instances of an Erdős-Rényi (ER) random graph. I implement and analyze the Adaptive Power Method (APM) which converges to the driven process, a biased random walk defined through a control parameter that simulates trajectories corresponding to rare events of the observable in the original dynamics. The APM demonstrates good convergence and accurately produces the desired quantities from a single trajectory. Due to the bulk-dangling-chain structure in the ER graph, the driven process seems to undergo a dynamical phase transition (DPT) for infinitely large graphs, meaning the behavior of the trajectories changes abruptly as the control parameter is varied. Observations show that the random walk visits two distinct phases, being de-localized in the bulk or localized in the chain. Through two simpler models capturing the bulk-dangling-chain property of the ER graph I study how the DPT occurs as the graph size increases. I observe that the trajectories of the driven process near the transition show intermittent behavior between the two phases. The diverging time scale of the DPT is found to be the average time that the random walk spends in a phase before it transitions to the other one. On the ER graph the trajectories are also intermittent but the form of the time scaling remains open due to computational limits on the graph size.