A Mapping Theorem in “Local” Topology

1953 ◽  
Vol s1-28 (3) ◽  
pp. 269-278
Author(s):  
H. B. Griffiths
Keyword(s):  
Author(s):  
Jochen Schiewe

AbstractMaps that correctly represent the geographic size and shape of regions, taking into account scaling and generalization, have the disadvantage that small regions can easily be overlooked or not seen at all. Hence, for some map use tasks where small regions are of importance, alternative map types are needed. One option is the so-called equal area unit maps (EAUMs), where every enumeration unit has the same area size, possibly also the same basic shape such as squares or hexagons. The geometrical distortion of EAUMs, however, leads to a more difficult search for regions as well as a falsification of topological relationships and spatial patterns. To describe these distortions, a set of analytical measures is proposed. But it turns out that the expressiveness of these measures is rather limited. To better understand and to model the influence of distortions, two user studies were conducted. The study on the search in EAUMs (also with the aim of reconstruct the search strategies of the users) revealed how important it is to consider the local topology (e.g. corner or border positions of regions) during the generation process. With regard to pattern identification, it could be shown that EAUMs significantly increase the detection rate of local extreme values. On the other hand, global lateral gradients or geostatistical hot spots often get blurred or even lost. As a consequence, a task-oriented selection of map types and further developments are recommended.


2021 ◽  
pp. 1-18
Author(s):  
YOTAM SMILANSKY ◽  
YAAR SOLOMON

Abstract We prove that in every compact space of Delone sets in ${\mathbb {R}}^d$ , which is minimal with respect to the action by translations, either all Delone sets are uniformly spread or continuously many distinct bounded displacement equivalence classes are represented, none of which contains a lattice. The implied limits are taken with respect to the Chabauty–Fell topology, which is the natural topology on the space of closed subsets of ${\mathbb {R}}^d$ . This topology coincides with the standard local topology in the finite local complexity setting, and it follows that the dichotomy holds for all minimal spaces of Delone sets associated with well-studied constructions such as cut-and-project sets and substitution tilings, whether or not finite local complexity is assumed.


2018 ◽  
Vol 32 (25) ◽  
pp. 1850279 ◽  
Author(s):  
Hanzhang Kong ◽  
Qinma Kang ◽  
Chao Liu ◽  
Wenquan Li ◽  
Hong He ◽  
...  

Community detection in complex network analysis is a quite challenging problem spanning many applications in various disciplines such as biology, physics and social network. A large number of methods have been developed for this problem, among which the label propagation algorithm (LPA) has attracted much attention because of its advantages of nearly-linear running time and easy implementation. Nevertheless, the random updating order and tie-breaking strategy in LPA make the algorithm unstable and may even lead to the formation of a monster community. In this paper, an improved LPA called LPA-INTIM is proposed for solving the community detection problem. Firstly, an intimacy matrix is constructed using local topology information for measuring the intimacy between nodes. And then, the node importance is calculated to ensure that nodes are updated in a specific order. Finally, the label influence is evaluated for updating node label during the label propagation process. In addition, we introduce a novel tightness function to improve the stability of the proposed algorithm. By the comparison with the methods presented in the literatures, experimental results on real-world and synthetic networks show the efficiency and effectiveness of our proposed algorithm.


Sign in / Sign up

Export Citation Format

Share Document