Abstract
Branching network is one of the most universal phenomena in living or non-living systems, such as river systems and the bronchial trees of mammals. To topologically characterize the branching networks, the Branch Length Similarity (BLS) entropy was suggested and the statistical methods based on the entropy have been applied to the shape identification and pattern recognition. However, the mathematical properties of the BLS entropy have not still been explored in depth because of the lack of application and utilization requiring advanced mathematical understanding. Regarding the mathematical study, it was reported, as a theorem, that all BLS entropy values obtained for simple networks created by connecting pixels along the boundary of a shape are exactly unity when the shape has infinite resolution. In the present study, we extended the theorem to the network created by linking infinitely many nodes distributed on the bounded or unbounded domain in Rk for k ≥ 1. We proved that all BLS entropies of the nodes in the network go to one as the number of nodes, n, goes to infinite and its convergence rate is 1 - O(1= ln n), which was confirmed by the numerical tests.
Highlights
Branching networks can be frequently observed in nature, such as river systems [1,2], the arterial and bronchial trees of mammals [3] and phylogenetic trees [4]
As the extension study of [20], we explored another theorem for the Branch Length Similarity (BLS) entropy on the network, which is created by linking infinitely many nodes distributed in the domain in Rk
We showed that the BLS entropy of any network in Rk increases at every node and, converges to one as the number of nodes, N, increases, and we confirmed it by the numerical tests on the rectangle and triangle
Summary
Branching networks can be frequently observed in nature, such as river systems [1,2], the arterial and bronchial trees of mammals [3] and phylogenetic trees [4]. They performed topological and morphometric analyses [9], which can be applied to all branching networks that are organized into a hierarchy These approaches have been extended to economic and social systems [10], which are composed of abstractly defined nodes representing the elements of the system and branches representing the interaction between them. In the area of ecology, ecologists have explored statistical methods to characterize the spatial distribution of the ecological elements, such as population density, to infer the existence of underlying processes, such as movement or responses to environmental heterogeneity. Through the investigation of the network properties, such as the connectivity and concentration, one can understand various aspects of the systems at the multi-scale level In this viewpoint, the statistical method based on the BLS entropy and its profile, providing a way to make the network and a measure to characterize the network, could be an effective alternative approach. As the extension study of [20], we explored another theorem for the BLS entropy on the network, which is created by linking infinitely many nodes distributed in the domain in Rk
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