Fast algorithm to identify minimal patterns of synchrony through fibration symmetries in large directed networks.

Recent studies have revealed the interplay between the structure of network circuits with fibration symmetries and the functionality of biological networks within which they have been identified. The presence of these symmetries in complex networks predicts the phenomenon of cluster synchronization, which produces patterns of a synchronized group of nodes. Here, we present a fast, and memory efficient, algorithm to identify fibration symmetries in networks. The algorithm is particularly suitable for large networks since it has a runtime of complexity O(Mlog⁡N) and requires O(M+N) of memory resources, where N and M are the number of nodes and edges in the network, respectively. The algorithm is a modification of the so-called refinement paradigm to identify circuits that are symmetrical to information flow (i.e., fibers) by finding the coarsest refinement partition over the network. Finally, we show that the algorithm provides an optimal procedure for identifying fibers, overcoming current approaches used in the literature.

The price of a vote: Diseconomy in proportional elections.

The increasing cost of electoral campaigns raises the need for effective campaign planning and a precise understanding of the return of such investment. Interestingly, despite the strong impact of elections on our daily lives, how this investment is translated into votes is still unknown. By performing data analysis and modeling, we show that top candidates spend more money per vote than the less successful and poorer candidates, a relation that discloses a diseconomy of scale. We demonstrate that such electoral diseconomy arises from the competition between candidates due to inefficient campaign expenditure. Our approach succeeds in two important tests. First, it reveals that the statistical pattern in the vote distribution of candidates can be explained in terms of the independently conceived, but similarly skewed distribution of money campaign. Second, using a heuristic argument, we are able to explain the observed turnout percentage for a given election of approximately 63% in average. This result is in good agreement with the average turnout rate obtained from real data. Due to its generality, we expect that our approach can be applied to a wide range of problems concerning the adoption process in marketing campaigns.

Statistical patterns of visual search for hidden objects.

The movement of the eyes has been the subject of intensive research as a way to elucidate inner mechanisms of cognitive processes. A cognitive task that is rather frequent in our daily life is the visual search for hidden objects. Here we investigate through eye-tracking experiments the statistical properties associated with the search of target images embedded in a landscape of distractors. Specifically, our results show that the twofold process of eye movement, composed of sequences of fixations (small steps) intercalated by saccades (longer jumps), displays characteristic statistical signatures. While the saccadic jumps follow a log-normal distribution of distances, which is typical of multiplicative processes, the lengths of the smaller steps in the fixation trajectories are consistent with a power-law distribution. Moreover, the present analysis reveals a clear transition between a directional serial search to an isotropic random movement as the difficulty level of the searching task is increased.

Nonlocal product rules for percolation.

Despite original claims of a first-order transition in the product rule model proposed by Achlioptas et al. [Science 323, 1453 (2009)], recent studies indicate that this percolation model, in fact, displays a continuous transition. The distinctive scaling properties of the model at criticality, however, strongly suggest that it should belong to a different universality class than ordinary percolation. Here we introduce a generalization of the product rule that reveals the effect of nonlocality on the critical behavior of the percolation process. Precisely, pairs of unoccupied bonds are chosen according to a probability that decays as a power law of their Manhattan distance, and only that bond connecting clusters whose product of their sizes is the smallest becomes occupied. Interestingly, our results for two-dimensional lattices at criticality shows that the power-law exponent of the product rule has a significant influence on the finite-size scaling exponents for the spanning cluster, the conducting backbone, and the cutting bonds of the system. In all three cases, we observe a clear transition from ordinary to (nonlocal) explosive percolation exponents.