Hence if any of the mappings is identified with any

topological property, we can as well associate the other mapping with the same property and vice versa.

Section 2 discusses the methodology used in this paper, the data description, and network topological property. Section 3 reveals the hierarchical structured features of the visibility graphs.

Network Topological Property. From Table 1, we can find that all indexes of average degree, average clustering coefficient, networks diameter, and path lengths are highly similar on four exchange rate networks, though we can observe relatively large heterogeneous network topology among Figures 1(b), 1(d), 1(f), and 1(h).

The steering methods presented in [19, 20] are shown to verify the topological property as well.

This is called the topological property [12] which is formulated more precisely in Appendix B.

In conclusion, our results of the graph theory analysis of the FA-weighted brain structural network in MDD patients support the concept that the brain structural network is a large and complex network with an optimal economical small-world

topological property. Although MDD patients showed an overall intact small-world topology, significantly altered global topological organization and regional characteristic of the nodes in the frontal, parietal and temporal areas of the brain structural network were found.

A

topological property P is called discretely reflexive in a class A if a space X from the class A has P if and only if the closure of every discrete subspace of X has P.

The fact that a figure is made up of several disconnected pieces is a

topological property; such a figure is distinct from one consisting of a single piece, for it is not permitted to glue the parts together when they are compared.

Depending on the nature or type of the

topological property, this property may be reflexive in that [e.sub.ij] [not equal to] [e.sub.ji].

Our study shows that schizophrenia is characterized by aberrant intrinsic dysconnectivity from a new insight of

topological property. The decreased putamen DC value and increased left superior frontal gyrus DC value in all patients and the relationship between symptoms and DC value are the best evidence.

The results of this study demonstrate that the

topological property of large-scale functional brain networks in AD patients is frequency dependent.

Finally, we integrate the

topological property (k-core index) and the health index to identify the influential nations in the observed ecological network.