Matematično ozadje algoritma PageRank

magistrsko delo

Gregor Spačal (Author), Aleksander Malnič (Mentor), Rok Požar (Co-mentor)

Abstract

PageRank je Googlov algoritem za rangiranje spletnih strani po pomembnosti. Strani lahko glede na vrednost PageRanka hierarhično uredimo in tako omogočimo boljše rezultate iskanja na spletu. V magistrskem delu obravnavamo pomen, lastnosti in delovanje spletnega iskanja. Predstavljamo slabosti spletnega iskanja, ki so se pojavljale pred nastankom Googla. Eno najpomembnejših vprašanj je zagotovo to, ali lahko delovanje PageRanka formalno matematično utemeljimo ter katere matematične koncepte in teorije za to potrebujemo. Za konec predstavimo primer implementacije algoritma v obliki spletne aplikacije in s tem pokažemo njegovo delovanje na preprostem primeru spleta. V magistrskem delu torej predstavimo matematično ozadje delovanja algoritma PageRank, za kar potrebujemo teoretično znanje s področja linearne algebre in teorije grafov. Poleg formalnega matematičnega opisa ilustriramo tudi njegovo delovanje na primerih. Splet modeliramo z usmerjenim grafom, ki mu priredimo neko matriko. Rezultat algoritma PageRank pa je lastni vektor te matrike pri lastni vrednosti 1, ki ga izračunamo s pomočjo potenčne iterativne metode. Obravnavamo tudi težave, ki lahko nastanejo pri računanju: ali je rezultat potenčne metode vedno smiseln, ali je lahko rešitev za dani primer več in ali je rezultat odvisen od začetnih parametrov. Glavni namen tega magistrskega dela je podati konkreten in širok vpogled v spletno iskanje s poukarkom na PageRanku, tako iz zgodovinskega, računalniškega kot matematičnega vidika, in poiskati ustrezne zglede za ilustracijo delovanja, težav in rešitev potenčne iterativne metode. Algoritem PageRank je tako predstavljen celostno na nivoju, primernem za tiste, ki jim spletno iskanje, linearna algebra in teorija grafov niso tuji, vendar se z nekaterimi zahtevnejšimi pojmi in koncepti še niso spoznali.

Keywords

algoritem PageRank;spletno iskanje;matematika algoritma PageRank;Google;rangiranje spletnih strani;

Data

Language:	Slovenian
Year of publishing:	2016
Typology:	2.09 - Master's Thesis
Organization:	UL PEF - Faculty of Education
Publisher:	[G. Spačal]
UDC:	004(043.2)
COBISS:	11335497
Views:	898
Downloads:	175
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	The mathematics behind PageRank algorithm
Secondary abstract:	PageRank is Google's algorithm for ranking web pages by relevance. Pages can then be hierarchically sorted in order to provide better search results. The MSc thesis considers functioning, relevance, general properties of web search and its weaknesses before the appearance of Google. One of the most important questions is, if we can formally explain the mathematics behind PageRank algorithm and what mathematical knowledge is necessary. Finally, we present an example of its implementation in a form of a web application, to demonstrate how PageRank works on a form of simplified web. The MSc theses presents the mathematics behind PageRank algorithm. To this end, we need linear algebra and graph theory. Beside formal mathematical description of the algorithm, we also provide examples to illustrate how it works. Web is modeled as a directed graph, to which we assign a certain matrix. The result of PageRank, performed on this matrix, is the eigenvector, corresponding to matrix's eigenvalue 1. The eigenvector is calculated with power iteration method. We consider problems, that can occur during the calculation: does the result of the power iteration method always make sense, can there be more than one solution for a given example and does the result depend on the starting parameters. The major objective of this thesis is to provide a wide and concrete insight into web search, emphasising PageRank, considering historical, mathematical and computer science viewpoint. We wish to provide relevant examples to demonstrate how the algorithm works. With these examples we also try to demonstrate problems as well as solutions that can occur during calculation with the power iteration method. PageRank is presented comprehensively in a way suitable for those familiar with basic knowledge in web search, linear algebra and graph theory, yet still in need of an introduction to some advanced concepts.
Secondary keywords:	computer science;računalništvo;
File type:	application/pdf
Type (COBISS):	Master's thesis/paper
Thesis comment:	Univ. v Ljubljani, Pedagoška fak., Poučevanje, Predmetno poučevanje
Pages:	58 str.
ID:	9228537