Clifford algebra method for network expression, computation, and algorithm construction. (14th August 2013)

• Record Type: Journal Article
• Authors: Yuan, Linwang; Yu, Zhaoyuan; Luo, Wen; Zhang, Jiyi; Hu, Yong
• Is Part Of: Mathematical methods in the applied sciences. Volume 37:Number 10(2014:Jul. 15)
• Page Start: 1428
• Page End: 1435
• Publication Date: 2013-08-14
• Subjects: Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.2904
• Languages: English
• ISSNs: 0170-4214

Abstract

<abstract abstract-type="main" id="mma2904-abs-0001"> <title> <x xml:space="preserve">Abstract</x> </title> <p id="mma2904-para-0001">Clifford algebra is introduced as a theoretical foundation for network topology expression and algorithm construction. Network nodes are coded with basis vectors in a vector space <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghmhf78ds" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:1704214:media:mma2904:mma2904-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>, and the edges and <italic>k</italic>‐walk routes can be expressed by 2‐blades and <italic>k</italic>‐blades, respectively, in the Clifford algebra <italic>Cl</italic>(<italic>n</italic>, 0). The topologies among nodes, edges, and routes of networks can be directly calculated, and the network routes can be extended and traversed with oriented join products. The network algorithm construction processes based on Clifford algebra are instantiated by the single source shortest path algorithm. The experimental results on different scale random networks suggest that Clifford algebra is suited for network expression and relation computation. The Clifford algebra‐based shortest path algorithm is vivid and clear in geometric meaning and has great advantage on temporal and spatial complexity. Copyright © 2013 John Wiley & Sons, Ltd.</p> </abstract>