Computing Nash equilibrium in multi-agent games is a longstanding challenge at the interface of game theory and computer science. It is well known that a general normal form game in N players and k strategies requires exponential space simply to write down. This Curse of Multi-Agents prompts the study of succinct games which can be written down efficiently. A canonical example of a succinct game is the graphical game which models players as nodes in a graph interacting with only their neighbors in direct analogy with markov random fields. Graphical games have found applications in wireless, financial, and social networks. However, computing the nash equilbrium of graphical games has proven challenging. Even for polymatrix games, a model where payoffs to an agent can be written as the sum of payoffs of interactions with the agent's neighbors, it has been shown that computing an epsilon approximate nash equilibrium is PPAD hard for epsilon smaller than a constant. The focus of this work is to circumvent this computational hardness by considering average case graph models i.e random graphs. We provide a quasipolynomial time approximation scheme (QPTAS) for computing an epsilon approximate nash equilibrium of polymatrix games on random graphs with edge density greater than poly(k, 1/epsilon, ln(N))$ with high probability. Furthermore, with the same runtime we can compute an epsilon-approximate Nash equilibrium that epsilon-approximates the maximum social welfare of any nash equilibrium of the game. Our primary technical innovation is an "accelerated rounding" of a novel hierarchical convex program for the nash equilibrium problem. Our accelerated rounding also yields faster algorithms for Max-2CSP on the same family of random graphs, which may be of independent interest.