In the interactions of a social group, people usually update and express their opinions through the observational learning behaviors. The formed directed networks are adaptive which are influenced by the evolution of opinions; while in turn modify the dynamic process of opinions. We extend the Hegselmann-Krause (HK) model to investigate the coevolution of opinions and observational networks (directed Erdös-Rényi network). Directed links can be broken with a probability if the difference of two opinions exceeds a certain confidence level ε, but new links can form randomly. Simulation results reveal that both the static networks and adaptive networks have three types: more than one cluster (fragmented) with small ε, consensus with a certain probability with moderate ε, always consensus with large ε. Also, on both networks, the tendencies of average of opinion clusters, consensus probability and average of convergence rounds are similar, and the fewest of average of opinion clusters satisfies the rough 1/(2 ε)-rule. On static networks, final opinions are influenced by percolation properties of networks; but on directed adaptive networks, it is basically determined by the rewiring probability, which increases the average degree of networks. When rewired probability is larger than zero, the results of adaptive networks are getting better than static networks. However, after the final average in-and out-degree of both networks exceeds a threshold, there is little improvement on the results.
Hexagonal grids use a hierarchical subdivision tessellation to cover the entire plane or sphere. Due to the 6-fold rotational symmetry, hexagonal grids have some advantages (e.g. isoperimetry, equidistant neighbors, and uniform connectivity) over quadrangular and triangular girds, which makes them suitable to tackle tasks of geospatial information processing and intelligent decision-making. In this paper, we first introduce some applications based on the hexagonal grids. Then, we introduce the planer and spherical hexagonal grids and analyze the group representations for them, we review geometric deep learning, some Convolutional Neural Networks (CNNs) for hexagonal grids, and group-based equivariant convolution. Next in importance, we propose the HexagonNet for hexagonal grids, and define a new convolution operator and pooling operator. Finally, in order to evaluate the effectiveness of the proposed HexagonNet, we perform experiments on two tasks: aerial scene classification on the aerial image dataset (AID), and 3D shape classification on the ModelNet40 dataset. The experimental results verify the practical applicability of the HexagonNet given some fixed parameter budgets.
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