The stability of networked systems is considered under time-delayed diffusive coupling. Necessary conditions for stability are given for general directed and weighted networks with both positive and negative weights. Exact stability conditions are obtained for undirected networks with nonnegative weights, and it is shown that the largest eigenvalue of the graph Laplacian determines the effect of the connection topology on stability. It is further shown that the stability region in the parameter space shrinks with increasing values of the largest eigenvalue, or of the time delay of the same parity. In particular, unstable fixed points of the individual maps can be stabilized for certain parameter ranges when they are coupled with an odd time delay, provided that the connection structure is not bipartite. Furthermore, signal propagation delays are compared to signal processing delays and it is shown that delay-induced stability cannot occur for the latter. Connections to continuous-time systems are indicated.
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