When calculating fractional-order differential dynamics systems numerically， there are difficulties of long-time memory storage by discretizing differential equation directly. In order to solve this problem， firstly， the differential equation was integrated once and then discretized. At the same time， a recurrence formula was given and its applicable conditions were discussed. Some common non-linear problems were calculated by this formula. The results of the above were consistent with those of other numerical methods. As whether there is chaotic motion in two-dimensional fraction-order continuous dynamics system is not concluded， this recurrence formula was used to study the two-dimensional continuous coupled Logistic model. It is found that there is only the limit cycle generated by the equilibrium point through Hopf bifurcation in this system without chaotic motion. Finally， the Lyapunov exponent criterion for the motion of two-dimensional fractional-order continuous Logistic system was given.

In order to measure the similarity of multi-relational nodes and mine the community structure with multi-relational nodes， a community mining algorithm based on multi-relationship of nodes， called LSL-GN， was proposed. Firstly， based on node similarity and node reachability， LHN-ISL， a similarity measurement index for multi-relational nodes， was described to reconstruct the low-density model of the target network， and the community division was completed by combining with GN （Girvan-Newman） algorithm. The LSL-GN algorithm was compared with several classical community mining algorithms on Modularity （Q value）， Normalized Mutual Information （NMI） and Adjusted Rand Index （ARI）. The results show that LSL-GN algorithm achieves the best results in terms of three indexes， indicating that the community division quality of LSL-GN is better. The “User-Application” mobile roaming network model was divided by LSL-GN algorithm into community structures based on basic applications such as Ctrip， Amap and Didi Travel. These results of community division can provide strategic reference information for designing personalized package services.

Considering high dimensional data image in face recognition application and Euclidean distance cannot accurately reflect the similarity between samples, a Mahalanobis distance based Local Marginal Fisher Analysis (MLMFA) dimensionality reduction algorithm was proposed. A Mahalanobis distance could be ascertained from the existing samples. Then, the Mahalanobis distance was used to choose neighbors and to reduce the dimensionality of new samples. Meanwhile, to describe the intra-class compactness and the inter-class separability, intra-class “similarity” graph and inter-class “penalty” graph were constructed by using Mahalanobis distance, and local structure of data set was preserved well. With the proposed algorithm being conducted on YALE and FERET, MLMFA outperforms the algorithms based on traditional Euclidean distance with maximum average recognition rate by 1.03% and 6% respectively. The results demonstrate that the proposed algorithm has very good classification and recognition performance.