We review our recent work on the synchronization of a network ofdelay-coupled maps, focusing on the interplay of the network topology and thedelay times that take into account the finite velocity of propagation ofinteractions. We assume that the elements of the network are identical ($N$logistic maps in the regime where the individual maps, without coupling, evolvein a chaotic orbit) and that the coupling strengths are uniform throughout thenetwork. We show that if the delay times are sufficiently heterogeneous, foradequate coupling strength the network synchronizes in a spatially homogeneoussteady-state, which is unstable for the individual maps without coupling. Thissynchronization behavior is referred to as ``suppression of chaos by randomdelays'' and is in contrast with the synchronization when all the interactiondelay times are homogeneous, because with homogeneous delays the networksynchronizes in a state where the elements display in-phase time-periodic orchaotic oscillations. We analyze the influence of the network topologyconsidering four different types of networks: two regular (a ring-type and aring-type with a central node) and two random (free-scale Barabasi-Albert andsmall-world Newman-Watts). We find that when the delay times are sufficientlyheterogeneous the synchronization behavior is largely independent of thenetwork topology but depends on the networks connectivity, i.e., on the averagenumber of neighbors per node.
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