In this paper, we consider the problem of model reduction of consensus networks. We propose a new method of model reduction based on removing edges that close cycles in the network graph. The agent dynamics of the consensus network is given by a symmetric multivariable input-state-output system. In the network, the agents exchange relative output information with their neighbors. We assume that the network graph is connected, unweighted, and undirected. The network used to approximate the original system is defined on the same number of nodes as the original graph, but its edge set is a strict subset of the original edge set. Explicit expressions and upper bounds for the approximation errors are formulated in terms of the signed path vectors of the removed edges and the eigenvalues of the Laplacian matrices of the original and reduced network graphs.