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This paper introduces a node-asynchronous communication protocol in which an agent in a network wakes up randomly and independently, collects states of its neighbors, updates its own state, and then broadcasts back to its neighbors. This protocol differs from consensus algorithms and it allows distributed computation of an arbitrary eigenvector of the network, in which communication between agents is allowed to be directed. (The graph operator is still required to be a normal matrix). To analyze the scheme, this paper studies a random asynchronous variant of the power iteration. Under this random asynchronous model, an initial signal is proven to converge to an eigenvector of eigenvalue 1 (a fixed point) even in the case of operator having spectral radius larger than unity. The rate of convergence is shown to depend not only on the eigenvalue gap but also on the eigenspace geometry of the operator as well as the amount of asynchronicity of the updates. In particular, the convergence region for the eigenvalues gets larger as the updates get less synchronous. Random asynchronous updates are also interpreted from the graph signal perspective, and it is shown that a non-smooth signal converges to the smoothest signal under the random model. When the eigenvalues are real, second order polynomials are used to achieve convergence to an arbitrary eigenvector of the operator. Using second order polynomials the paper formalizes the node-asynchronous communication model. As an application, the protocol is used to compute the Fiedler vector of a network to achieve autonomous clustering.