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Abola, Benard

Open this publication in new window or tab >>A Variant of Updating Page Rank in Evolving Tree graphs### Abola, Benard

### Biganda, Pitos

### Engström, Christopher

### Anguzu, Collins

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_0_j_idt188_some",{id:"formSmash:j_idt184:0:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_some",multiple:true}); ### Mango, John Magero

### Kakuba, Gudwin

### Silvestrov, Sergei

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_0_j_idt188_otherAuthors",{id:"formSmash:j_idt184:0:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_otherAuthors",multiple:true}); Show others...PrimeFaces.cw("SelectBooleanButton","widget_formSmash_j_idt184_0_j_idt188_j_idt202",{id:"formSmash:j_idt184:0:j_idt188:j_idt202",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_j_idt202",onLabel:"Hide others...",offLabel:"Show others..."}); 2019 (English)In: Proceedings of 18th Applied Stochastic Models and Data Analysis International Conference with the Demographics 2019 Workshop, Florence, Italy: 11-14 June, 2019 / [ed] Christos H. Skiadas, ISAST: International Society for the Advancement of Science and Technology , 2019, p. 31-49Conference paper, Published paper (Refereed)
##### Abstract [en]

##### Place, publisher, year, edition, pages

ISAST: International Society for the Advancement of Science and Technology, 2019
##### Keywords

PageRank, randomwalk, graph, networks.
##### National Category

Probability Theory and Statistics Computational Mathematics
##### Research subject

Mathematics/Applied Mathematics
##### Identifiers

urn:nbn:se:mdh:diva-47083 (URN)978-618-5180-33-1 (ISBN)
##### Conference

ASMDA2019, 18th Applied Stochastic Models and Data Analysis International Conference
#####

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##### Funder

Sida - Swedish International Development Cooperation Agency
Available from: 2020-02-20 Created: 2020-02-20 Last updated: 2020-02-24Bibliographically approved

Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Department of Mathematics, School of Physical Sciences, Makerere University, Kampala, Uganda.

Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Department of Mathematics, College of Natural and Applied Sciences, University of Dar es Salaam,Tanzania.

Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.

Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Department of Mathematics, Makerere University, Kampala, Uganda.

Department of Mathematics, Makerere University, Kampala, Uganda.

Department of Mathematics, Makerere University, Kampala, Uganda.

Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.

PageRank update refers to the process of computing new PageRank values after change(s) (addition or removal of links/vertices) has occurred in real life networks. The purpose of the updating is to avoid recalculating the values from scratch. To efficiently carry out the update, we consider PageRank as the expected number of visits to target vertex if multiple random walks are performed, starting at each vertex once and weighing each of these walks by a weight value. Hence, it might be looked at as updating non-normalised PageRank. In the proposed approach, a scaled adjacency matrix is sequentially updated after every change and the levels of the vertices being updated as well. This enables sets of internal and sink vertices dependent on their roots or parents, thus vector-vector product can be performed sequentially since there are no infinite steps from one vertex to the other.

Open this publication in new window or tab >>Nonlinearly Perturbed Markov Chains and Information Networks### Abola, Benard

### Biganda, Pitos

### Silvestrov, Dmitrii

### Silvestrov, Sergei

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_some",{id:"formSmash:j_idt184:1:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_some",multiple:true}); ### Mango, John

### Kakuba, Gudwin

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_otherAuthors",{id:"formSmash:j_idt184:1:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_otherAuthors",multiple:true}); Show others...PrimeFaces.cw("SelectBooleanButton","widget_formSmash_j_idt184_1_j_idt188_j_idt202",{id:"formSmash:j_idt184:1:j_idt188:j_idt202",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_j_idt202",onLabel:"Hide others...",offLabel:"Show others..."}); 2019 (English)In: Proceedings of 18th Applied Stochastic Models and Data Analysis International Conference with the Demographics 2019 Workshop, Florence, Italy: 11-14 June, 2019 / [ed] Christos H. Skiadas, ISAST: International Society for the Advancement of Science and Technology , 2019, p. 51-79Conference paper, Published paper (Refereed)
##### Abstract [en]

##### Place, publisher, year, edition, pages

ISAST: International Society for the Advancement of Science and Technology, 2019
##### Keywords

Markov chain, Damping component, Information network, Regular perturbation, Singular perturbation, Stationary distribution, Asymptotic expansion, Rate of convergence, Coupling, Ergodic theorem, Triangular array mode
##### National Category

Probability Theory and Statistics
##### Research subject

Mathematics/Applied Mathematics
##### Identifiers

urn:nbn:se:mdh:diva-47082 (URN)978-618-5180-33-1 (ISBN)
##### Conference

ASMDA2019, 18th Applied Stochastic Models and Data Analysis International Conference
#####

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##### Funder

Sida - Swedish International Development Cooperation Agency
Available from: 2020-02-20 Created: 2020-02-20 Last updated: 2020-02-24Bibliographically approved

Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Department of Mathematics, School of Physical Sciences, Makerere University, Kampala, Uganda.

Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Department of Mathematics, College of Natural and Applied Sciences, University of Dar es Salaam,Tanzania.

Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Stockholm University, Sweden.

Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.

Department of Mathematics, Makerere University, Kampala, Uganda.

Department of Mathematics, Makerere University, Kampala, Uganda.

The paper is devoted to studies of perturbed Markov chains commonly used for description of information networks. In such models, the matrix of transition probabilities for the corresponding Markov chain is usually regularised by adding a special damping matrix multiplied by a small damping (perturbation) parameter ε. In this paper, we present results of the detailed perturbation analysis of Markov chains with damping component and numerical experiments supporting and illustrating the results of this perturbation analysis.

Open this publication in new window or tab >>PageRank in evolving tree graphs### Abola, Benard

### Biganda, Pitos

### Engström, Christopher

Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.### Mango, J. M.

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_some",{id:"formSmash:j_idt184:2:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_some",multiple:true}); ### Kakuba, G.

### Silvestrov, Sergei

Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_otherAuthors",{id:"formSmash:j_idt184:2:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_otherAuthors",multiple:true}); Show others...PrimeFaces.cw("SelectBooleanButton","widget_formSmash_j_idt184_2_j_idt188_j_idt202",{id:"formSmash:j_idt184:2:j_idt188:j_idt202",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_j_idt202",onLabel:"Hide others...",offLabel:"Show others..."}); 2018 (English)In: Stochastic Processes and Applications: SPAS2017, Västerås and Stockholm, Sweden, October 4-6, 2017 / [ed] Sergei Silvestrov, Anatoliy Malyarenko, Milica Rančić, Springer, 2018, Vol. 271, p. 375-390Chapter in book (Refereed)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Springer, 2018
##### Series

Springer Proceedings in Mathematics and Statistics, ISSN 2194-1009 ; 271
##### Keywords

Breadth-first search, Forward edge, PageRank, Random walk, Tree, Forestry, Graph theory, Iterative methods, Random processes, Stochastic systems, Trees (mathematics)
##### National Category

Computational Mathematics Probability Theory and Statistics
##### Research subject

Mathematics/Applied Mathematics
##### Identifiers

urn:nbn:se:mdh:diva-41833 (URN)10.1007/978-3-030-02825-1_16 (DOI)2-s2.0-85058567338 (Scopus ID)978-3-030-02824-4 (ISBN)
##### Conference

International Conference on “Stochastic Processes and Algebraic Structures – From Theory Towards Applications”, SPAS 2017; Västerås and Stockholm; Sweden; 4 October 2017 through 6 October 2017; Code 221789
#####

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Available from: 2018-12-27 Created: 2018-12-27 Last updated: 2018-12-31Bibliographically approved

Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Department of Mathematics, School of Physical Sciences, Makerere University, Kampala, Uganda.

Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Department of Mathematics, College of Natural and Applied Sciences, University of Dar es Salaam,Tanzania.

Department of Mathematics, School of Physical Sciences, Makerere University, Kampala, Uganda.

Department of Mathematics, School of Physical Sciences, Makerere University, Kampala, Uganda.

In this article, we study how PageRank can be updated in an evolving tree graph. We are interested in finding how ranks of the graph can be updated simultaneously and effectively using previous ranks without resorting to iterative methods such as the Jacobi or Power method. We demonstrate and discuss how PageRank can be updated when a leaf is added to a tree, at least one leaf is added to a vertex with at least one outgoing edge, an edge added to vertices at the same level and forward edge is added in a tree graph. The results of this paper provide new insights and applications of standard partitioning of vertices of the graph into levels using breadth-first search algorithm. Then, one determines PageRanks as the expected numbers of random walk starting from any vertex in the graph. We noted that time complexity of the proposed method is linear, which is quite good. Also, it is important to point out that the types of vertex play essential role in updating of PageRank.

Open this publication in new window or tab >>Traditional and lazy pageranks for a line of nodes connected with complete graphs### Biganda, Pitos

Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Department of Mathematics, College of Natural and Applied Sciences, University of Dar es Salaam,Tanzania.### Abola, Benard

Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Department of Mathematics, School of Physical Sciences, Makerere University, Kampala, Uganda.### Engström, Christopher

Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.### Mango, J. M.

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_some",{id:"formSmash:j_idt184:3:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_some",multiple:true}); ### Kakuba, G.

### Silvestrov, Sergei

Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_otherAuthors",{id:"formSmash:j_idt184:3:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_otherAuthors",multiple:true}); Show others...PrimeFaces.cw("SelectBooleanButton","widget_formSmash_j_idt184_3_j_idt188_j_idt202",{id:"formSmash:j_idt184:3:j_idt188:j_idt202",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_j_idt202",onLabel:"Hide others...",offLabel:"Show others..."}); 2018 (English)In: Stochastic Processes and Applications: SPAS2017, Västerås and Stockholm, Sweden, October 4-6, 2017 / [ed] Sergei Silvestrov, Anatoliy Malyarenko, Milica Rančić, Springer, 2018, Vol. 271, p. 391-412Chapter in book (Refereed)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Springer, 2018
##### Series

Springer Proceedings in Mathematics and Statistics, ISSN 2194-1009
##### Keywords

Graph, Lazy PageRank, PageRank, Random walk, Random processes, Stochastic systems, Websites, Complete graphs, Diverse methods, Explicit formula, Line graph, Numerical values, Graph theory
##### National Category

Mathematics
##### Research subject

Mathematics/Applied Mathematics
##### Identifiers

urn:nbn:se:mdh:diva-41835 (URN)10.1007/978-3-030-02825-1_17 (DOI)2-s2.0-85058552957 (Scopus ID)978-3-030-02824-4 (ISBN)
##### Conference

International Conference on “Stochastic Processes and Algebraic Structures – From Theory Towards Applications”, SPAS 2017; Västerås and Stockholm; Sweden; 4 October 2017 through 6 October 2017; Code 221789
#####

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Available from: 2018-12-27 Created: 2018-12-27 Last updated: 2018-12-31Bibliographically approved

Department of Mathematics, School of Physical Sciences, Makerere University, Kampala, Uganda.

Department of Mathematics, School of Physical Sciences, Makerere University, Kampala, Uganda.

PageRank was initially defined by S. Brin and L. Page for the purpose of measuring the importance of web pages (nodes) based on the structure of links between them. Due to existence of diverse methods of random walk on the graph, variants of PageRank now exists. They include traditional (or normal) PageRank due to normal random walk and Lazy PageRank due to lazy random walk on a graph. In this article, we establish how the two variants of PageRank changes when complete graphs are connected to a line of nodes whose links between the nodes are in one direction. Explicit formulae for the two variants of PageRank are presented. We have noted that the ranks on a line graph are the same except their numerical values which differ. Further, we have observed that both normal random walk and lazy random walk on complete graphs spend almost the same time at each node.