Topological energy of the distance matrix

The network structure and spectrum of adjacency matrix such as loops, community structure and degree distribution have a close relationship. A direct analysis has been performed to describe the characteristics of network structure, where entropy and graph energy are commonly used indicators. In particular, graph energy has drawn significant research attention owing to its importance as invariant and close relationship with network structure. However, there are various definitions of graph energy and entropy.

The concept of energy has been expanded to provide a more acceptable definition of different types of energy. Moreover, some types of graph entropy have been expressed as energy to describe graphs in some applications like financial markets. Although there have been significant research efforts to define energy, most previous studies have defined energy in terms of discrete graph structures, which is still inadequate. A fundamental issue is the definition of energy on a point set with metric structure with basic invariance. Three common types of invariances are rotation, translation and scaling invariance. However, satisfying these invariances is impossible because of the extensive nature of the energy matrix defined by singular values. Additionally, it is impossible to generalize the distance energy as it is defined on the distance matrix of the graph.

Herein, Dr. Chun-Xiao Nie from Zhejiang Gongshang University studied the topological graph energy defined on the distance matrix. A topological data analysis technique was employed to define the energy of the matrix. Specifically, a graph sequence was generated by filtering the distance matrix using the order complex in the topological data analysis, followed by defining the energy and entropy on the graph sequence. Their work is currently published in the journal, Communications in Nonlinear Science and Numerical Simulation.

In their approach, the average value or integral was calculated in appropriate intervals to define the global entropy and energy indicators. The definition of the distance matrix was restricted and some simple transformations were carried out to transform the similarity matrix into a dissimilarity matrix. The distance matrix was used as the underlying concept to clearly show the impact of linear and nonlinear transformations on energy. Finally, the relationship between the topological entropy and energy was analyzed.

The author demonstrated the capability of extended energy and entropy in capturing the effects of nonlinear transformation, such as changes in the point set induced by the transformation of the nonlinear coordinate. A series of point sets generated by chaotic systems were used to demonstrate how the differences in attractors were captured by energy values. The point set generated by Henon map exhibited high similarity, leading to high sensitivity to the initial value. For the Rossler system, however, significant changes in the energy values were observed.

Examples of chaotic systems and financial markets were provided to illustrate the distance matrix energy concept. Financial market data analysis showed that the energy and entropy curves could effectively describe the complexity of financial markets. Furthermore, the analysis revealed that even after properly defining the energy, some significant transformations, such as mapping, failed to change the energy value or topological structure of the data set.

In summary, the robustness of matrix energy was verified by analyzing the attractors of different systems. The presented method generally provided distance matrix indicators and allowed the observation of a point set with metric structure from the graph energy perspective. In a statement to Advances in Engineering, Dr. Chun-Xiao Nie noted that the study provided useful insights that would provide a proper definition of entropy and graph energy to advance their applications in different fields, such as financial market analysis.