Spatial and Temporal Entropies in the Spanish Football League: A Network Science Perspective We quantified the spatial and temporal entropy related to football teams and their players by means of a passbased interaction. First, we calculated the spatial entropy associated to the positions of all passes made by a football team during a match, obtaining a spatial entropy ranking of Spanish teams during the 20172018 season. Second, we investigated how the playerxe2x80x99s average location in the field is related to the amount of entropy of his passes. Next, we constructed the temporal passing networks of each team and computed the deviation of their network parameters along the match. For each network parameter, we obtained the permutation entropy and the statistical complexity of its temporal fluctuations. Finally, we investigated how the permutation entropy (and statistical complexity) of the network parameters was related to the total number of passes made by a football team. Our results show that (i) spatial entropy changes according to the position of players in the field, and (ii) the organization of passing networks change during a match and its evolution can be captured measuring the permutation entropy and statistical complexity of the network parameters, allowing to identify what parameters evolve more randomly.

Scaleresolved analysis of brain functional connectivity networks with spectral entropy Abstract   Functional connectivity is derived from interregional correlations in spontaneous fluctuations of brain activity, and can be represented in terms of complete graphs with continuous (realvalued) edges. The structure of functional connectivity networks is strongly affected by signal processing procedures to remove the effects of motion, physiological noise and other sources of experimental error. However, in the absence of an established ground truth, it is difficult to determine the optimal procedure, and no consensus has been reached on the most effective approach to remove nuisance signals without unduly affecting the network intrinsic structural features. Here, we use a novel informationtheoretic approach, based on von Neumann entropy, which provides a measure of information encoded in the networks at different scales. We also define a measure of distance between networks, based on information divergence, and optimal null models appropriate for the description of functional connectivity networks, to test for the presence of nontrivial structural patterns that are not the result of simple local constraints. This formalism enables a scaleresolved analysis of the distance between a functional connectivity network and its maximally random counterpart, thus providing a means to assess the effects of noise and image processing on network structure. We apply this novel approach to address a few open questions in the analysis of brain functional connectivity networks. Specifically, we demonstrate a strongly beneficial effect of network sparsification by removal of the weakest links, and the existence of an optimal threshold that maximizes the ability to extract information on largescale network structures. Additionally, we investigate the effects of different degrees of motion at different scales, and compare the most popular processing pipelines designed to mitigate its deleterious effect on functional connectivity networks. We show that network sparsification, in combination with motion correction algorithms, dramatically improves detection of large scale network structure.