Partition and Colored Distances in Graphs Induced to Subsets of Vertices and Some of Its Applications

If G is a graph and P is a partition of V(G), then the partition distance of G is the sum of the distances between all pairs of vertices that lie in the same part of P. A colored distance is the dual concept of the partition distance. These notions are motivated by a problem in the facility location network and applied to several well-known distance-based graph invariants. In this paper, we apply an extended cut method to induce the partition and color distances to some subsets of vertices which are not necessary a partition of V(G). Then, we define a two-dimensional weighted graph and an operator to prove that the induced partition and colored distances of a graph can be obtained from the weighted Wiener index of a two-dimensional weighted quotient graph induced by the transitive closure of the Djoković–Winkler relation as well as by any partition that is coarser. Finally, we utilize our main results to find some upper bounds for the modified Wiener index and the number of orbits of partial cube graphs under the action of automorphism group of graphs.

On the use of deep learning for the improvement of swh and wave spectra from CFOSAT

<p>The SWIM (Surface Waves Investigation and Monitoring) carried by the CFOSAT (Chinese-French Oceanography Satellite) is designed to obtain the nadir wave height and directional spectrum. This work first introduces the efficiency of deep learning technique to improve wave height and spectrum observation from CFOSAT. A set of deep learning neural networks (DNN) are established and trained to improve the accuracy of SWIM nadir wave height. According to the assessment based on independent buoy observations, the DNN reduces the root mean square error (RMSE) of significant wave height by 32.2% (from 0.26 m to 0.17 m), and the scatter index by 25.7% (from 14 % to 10 %). The result shows that the bias is significantly decreased from 0.11 m to -0.02 m. To correct the SWIM wave spectra, 6 months of NDBC frequency spectra have been used to obtain 19 sets of DNN, each of them is corresponding to one effective frequency of SWIM accordingly. Each set of DNN contains 14 hidden layers with the input as the energy from the different beams 6°, 8° and 10°. Then, the DNN forms a new combined spectrum based on wave spectra of all beams of the SWIM instrument. The independent assessment shows that the wave spectra which come from the 19 sets of DNN significantly reduced the relative error (RE) by 10% to 46% in comparison with beam 10°, which has the best accuracy performance among all the beams. The deep learning technique is also used as a quality control procedure before the assimilation of SWIM wave data. The Siamese convolution neural network (CNN) connected with a deep learning neural network (DNN) is applied to perform such comparison and verification for the SWIM directional spectra. The SWIM directional wave spectra are considered as the 2-dimensional “energy-pictures” with a matrix dimension of 17 frequencies and 9 directions. The Siamese CNN network is made up of 2 pairs of convolution layer and pooling layer, in which the 32 groups of convolution kernels are used to generate one-dimensional features from the directional wave spectra. Both wave spectra of SWIM instrument and wave model are inputted into the same Siamese CNN network, being transformed into 2 sets of features accordingly. Then the features would go to the DNN to generate the index of the similarity. This gives a quantitative description of how different between the SWIM directional spectra and the ones from wave model. Through the training of 26014 pairs of directional wave spectra, the Siamese CNN and DNN have showed a consistency of 78% to 86% with the “partition distance” method in the independent testing data, but with a much faster computation speed. We revealed that the Siamese technique increases the number of wave spectra passing through the verification than the “partition distance” method. This will ensure larger impact of the assimilation of SWIM data in the analysis and forecast period. The case study by running the wave model MFWAM using Siamese network and “partition distance” in assimilation is investigated.</p>