Diffusion of Time-Varying Signals in Complex Networks: A   Structure-Dynamics Investigation Focusing the Distance to the Source of   Activation
  The way in which different types of dynamics unfold in complex networks is intrinsically related to the propagation of activation along nodes, which is strongly affected by the network connectivity. In this work we investigate to which extent a time-varying signal emanating from a specific node is modified as it diffuses, at the equilibrium regime, along uniformly random (Erd\H{o}s-R\'enyi) and scale-free (Barab\'asi-Albert) networks. The degree of preservation is quantified in terms of the Pearson cross-correlation between the original signal and the derivative of the signals appearing at each node along time. Several interesting results are reported. First, the fact that quite distinct signals are typically obtained at different nodes in the considered networks implies mean-field approaches to be completely inadequate. It has also been found that the peak and lag of the similarity time-signatures obtained for each specific node are strongly related to the respective distance between that node and the source node. Such a relationship tends to decrease with the average degree of the networks. Also, in the case of the lag, a less intense relationship is verified for scale-free networks. No relationship was found between the dispersion of the similarity signature and the distance to the source. 
Luciano da Fontoura Costa