^{k}

^{*}

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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 ℝ^{k}

Branching networks can be frequently observed in nature, such as river systems [

Unlike the aforementioned approaches, Lee

In contrast to the engineering examples mentioned above, the BLS entropy and its profile could be used to characterize and analyze the spatial distribution of elements of a system. 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. This is because the spatial distribution is likely to indicate intraspecific and interspecific interactions, such as competition, predation and reproduction [

Recently, along with the increase of accessibility and accuracy in remote sensing technology, large-scale analysis and space-time data collection, it has been known that the spatial distribution is strongly scale-dependent in many systems (see [

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. However, although the statistical method could be reliably used in the issues mentioned above, the mathematical properties of the BLS entropy should be extensively explored on a preferential basis to provide a solid ground for applications to a wide range of spatial systems. One of the basic mathematical properties is: what is the value of the BLS entropy for networks consisting of an infinite number of nodes? This question is directly related to the performance and efficiency of the statistical methods based on the BLS entropy profile in the application problems. Jeon and Lee [^{k}

We define the Branch Length Similarity (BLS) entropy as the property of simple branching networks composed of _{i}_{ij}_{i}_{j}_{ij}_{i}_{j}|_{ij}_{i}

Applying this notion to the nodes placed on an arbitrary bounded domain in ℝ^{k}

^{k}_{n}, of any node in_{R}^{k}_{0}_{0} is the position vector of _{n}_{n}_{n}_{0}_{n}_{n}

Assume that there are distinct _{n}_{j}_{n}_{j}_{j}_{j}_{n}_{n→∞}

Now, consider the second term of _{j}_{n}, R_{n}/

Theorem 1. can be extended to the unbounded domain in ℝ^{k}

^{k}. Then, the BLS entropy, S_{n}, of the node in

_{n}_{n}_{n}_{n}_{j}_{n}_{j}_{n}_{n→∞} δ

From

We calculate the BLS entropies of the nodes on the bounded regions by increasing the number of nodes. To see the effect of the domain shape and the distribution of the nodes, we consider two regions (rectangle and triangle) and uniform and random distributions.

^{2} squares, there are ^{2} nodes on _{ij}^{k}^{2n} for an integer, _{∞}^{∞}_{∞}_{x∈R} |_{Nj}_{∞}/|1 − _{Nj−1}|_{∞}. ^{2k}/ ln 2^{2(k+1)} =

_{N}|_{∞}^{k}

In this paper, we showed that the BLS entropy of any network in ℝ^{k}

One important point is: what is the optimal

Consequently, our result is meaningful in that it not only shows the convergence rate of the BLS entropy on the networks in ℝ^{k}

This work was supported by the National Institute for Mathematical Sciences.

The authors declare no conflicts of interest.

The

The Branch Length Similarity (BLS) entropy profiles of the uniformly distributed network on the rectangle,

Convergence rates for each test: _{j}

The BLS entropy profiles of the randomly distributed network on the rectangle,

The BLS entropy profiles of the randomly distributed network on the triangle,

The convergence rate of the uniformly distributed network on the rectangle,

_{N}_{∞} |
|||

_{1} = 8^{2} = (2^{3})^{2} |
2.6452E-2 | - | - |

_{2} = 16^{2} = (2^{4})^{2} |
2.1490E-2 | 0.81241 | 0.75000 |

_{3} = 32^{2} = (2^{5})^{2} |
1.7523E-2 | 0.81540 | 0.80000 |

_{4} = 64^{2} = (2^{6})^{2} |
1.4693E-2 | 0.83849 | 0.83333 |

_{5} = 128^{2} = (2^{7})^{2} |
1.2613E-2 | 0.85844 | 0.85714 |

The convergence rate of the randomly distributed network on the rectangle,

_{N}_{∞} |
|||
---|---|---|---|

_{1} = 3^{4} |
3.3683E-2 | - | - |

_{2} = 3^{5} |
2.4627E-2 | 0.73114 | 0.75000 |

_{3} = 3^{6} |
1.9495E-2 | 0.79161 | 0.80000 |

_{4} = 3^{7} |
1.6536E-2 | 0.84822 | 0.83333 |

_{5} = 3^{8} |
1.4265E-2 | 0.86266 | 0.85714 |

_{6} = 3^{9} |
1.2516E-2 | 0.87739 | 0.87500 |

The convergence rate of the randomly distributed network on the triangle,

_{N}_{∞} |
|||
---|---|---|---|

_{1} = 3^{4} |
3.5863E-2 | - | - |

_{2} = 3^{5} |
2.6943E-2 | 0.75128 | 0.75000 |

_{3} = 3^{6} |
2.1883E-2 | 0.81220 | 0.80000 |

_{4} = 3^{7} |
1.8489E-2 | 0.84490 | 0.83333 |

_{5} = 3^{8} |
1.5982E-2 | 0.86441 | 0.85714 |

_{6} = 3^{9} |
1.3968E-2 | 0.87398 | 0.87500 |