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In this thesis, we study random graphs using tools from Random Matrix Theory and probability to tackle key problems in complex networks and Big Data. First we study graph anomaly detection. Consider an Erdős-Rényi (ER) graph with edge probability q and size n containing a planted subgraph of size m and probability p. We derive a statistical test based on the eigenvalue and eigenvector properties of a suitably defined matrix to detect the planted subgraph. We analyze the distribution of the derived test statistic using Random Matrix Theoretic techniques. Next, we consider subgraph recovery in this model in the presence of side-information. We analyse the effect of side-information on the detectability threshold of Belief Propagation (BP) applied to the above problem. We show that BP correctly recovers the subgraph even with noisy side-information for any positive value of an effective SNR parameter. This is in contrast to BP without side-information which requires the SNR to be above a certain threshold. Finally, we study the asymptotic behaviour of PageRank on a class of undirected random graphs called fast expanders, using Random Matrix Theoretic techniques. We show that PageRank can be approximated for large graph sizes as a convex combination of the normalized degree vector and the personalization vector of the PageRank, when the personalization vector is sufficiently delocalized. Subsequently, we characterize asymptotic PageRank on Stochastic Block Model (SBM) graphs, and show that it contains a correction term that is a function of the community structure.