| 1. |
- Abola, Benard, et al.
(författare)
-
A Variant of Updating PageRank in Evolving Tree Graphs
- 2021
-
Ingår i: Applied Modeling Techniques and Data Analysis 1. - : John Wiley & Sons, Inc. Hoboken, NJ, USA. - 9781786306739 - 9781119821564 ; , s. 3-22
-
Bokkapitel (refereegranskat)abstract
- A PageRank update refers to the process of computing new PageRank valuesafter a change(s) (addition or removal of links/vertices) has occurred in real-lifenetworks. The purpose of updating is to avoid re-calculating the values from scratch.To efficiently carry out the update, we consider PageRank to be the expected numberof visits to a target vertex if multiple random walks are performed, starting at eachvertex once and weighing each of these walks by a weight value. Hence, it mightbe looked at as updating a non-normalized PageRank. We focus on networks of treegraphs and propose an approach to sequentially update a scaled adjacency matrix afterevery change, as well as the levels of the vertices. In this way, we can update thePageRank of affected vertices by their corresponding levels.
|
|
| 2. |
- Abola, Benard, et al.
(författare)
-
Chapter 2. Nonlinearly Perturbed Markov Chains and Information Networks
- 2021
-
Ingår i: Applied Modeling Techniques and Data Analysis 1. - Hoboken, NJ : John Wiley & Sons. - 9781786306739 - 9781119821564 ; , s. 23-55
-
Bokkapitel (refereegranskat)abstract
- This chapter is devoted to studies of perturbed Markov chains, commonly used for the description of information networks. In such models, the matrix of transition probabilities for the corresponding Markov chain is usually regularized by adding aspecial damping matrix, multiplied by a small damping (perturbation) parameter ε. In this chapter, we present the results of detailed perturbation analysis of Markov chains with damping component and numerical experiments supporting and illustrating the results of this perturbation analysis.
|
|
| 3. |
- Abola, Benard, 1971-, et al.
(författare)
-
PageRank in evolving tree graphs
- 2018
-
Ingår i: Stochastic Processes and Applications. - Cham : Springer. - 9783030028244 ; , s. 375-390
-
Bokkapitel (refereegranskat)abstract
- 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.
|
|
| 4. |
|
|
| 5. |
- Abola, Benard, 1971-, et al.
(författare)
-
Updating of PageRank in Evolving Tree graphs
- 2020
-
Ingår i: Data Analysis and Applications 3. - : John Wiley & Sons. - 9781786305343 - 9781119721871 ; , s. 35-51
-
Bokkapitel (refereegranskat)abstract
- Summary Updating PageRank refers to a process of computing new PageRank values after changes have occurred in a graph. The main goal of the updating is to avoid recalculating the values from scratch. This chapter focuses on updating PageRank of an evolving tree graph when a vertex and an edge are added sequentially. It describes how to maintain level structures when a cycle is created and investigates the practical and theoretical efficiency to update PageRanks for an evolving graph with many cycles. The chapter discusses the convergence of the power method applied to stochastic complement of Google matrix when a feedback vertex set is used. It also demonstrates that the partition by feedback vertex set improves asymptotic convergence of power method in updating PageRank in a network with cyclic components.
|
|
| 6. |
- Biganda, Pitos, 1981-, et al.
(författare)
-
Exploring The Relationship Between Ordinary PageRank, Lazy PageRank and Random Walk with Backstep PageRank for Different Graph Structures
- 2020
-
Ingår i: Data Analysis and Applications 3. - : John Wiley & Sons, Ltd. - 9781786305343 - 9781119721871 ; , s. 53-73
-
Bokkapitel (refereegranskat)abstract
- PageRank is an algorithm for ranking web pages. It is the first and best known webgraph-based algorithm in the Google search engine. The algorithm is simple, robust and reliable to measure the importance of web pages. This chapter presents a comparative review of three variants of PageRank, namely ordinary PageRank (introduced by Brin and Page as a measure of importance of a web page), lazy PageRank and random walk with backstep PageRank. It compares the variants in terms of their convergence and consistency in rank scores for different graph structures with reference to PageRank’s parameters, damping factor and backstep parameter. The chapter also shows that ordinary PageRank can be formulated from the other two variants by some proportionality relationships.
|
|
| 7. |
- Biganda, Pitos, 1981-, et al.
(författare)
-
PageRank and Perturbed Markov Chains
- 2021
-
Ingår i: Applied Modeling Techniques and Data Analysis 1. - : John Wiley & Sons, Inc. Hoboken, NJ, USA. - 9781786306739 - 9781119821564 ; , s. 57-74
-
Bokkapitel (refereegranskat)abstract
- PageRank is a widely used hyperlink-based algorithm for estimating the relative importance of nodes in networks. Since many real-world networks are large sparse networks, efficient calculation of PageRank is complicated. Moreover, we need to overcome dangling effects in some cases as well as slow convergence of the transition matrix. Primitivity adjustment with a damping (perturbation) parameter is one of the essential procedures known to ensure convergence of the transition matrix. If the perturbation parameter is not small enough, the transition matrix loses information due to the shift of information to the teleportation matrix. We formulate the PageRank problem as a first- and second-order Markov chains perturbation problem. Using numerical experiments, we compare convergence rates for different values of perturbation parameter on different graph structures and investigate the difference in ranks for the two problems.
|
|
| 8. |
- Biganda, Pitos, 1981-, et al.
(författare)
-
Traditional and lazy pageranks for a line of nodes connected with complete graphs
- 2018
-
Ingår i: Stochastic Processes and Applications. - Cham : Springer. - 9783030028244 ; , s. 391-412
-
Bokkapitel (refereegranskat)abstract
- 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.
|
|
| 9. |
- Kakuba, Godwin, et al.
(författare)
-
An algorithm for computing a stationary flow in a binary bifurcation tree
- 2021
-
Ingår i: Applied Numerical Mathematics. - : ELSEVIER. - 0168-9274 .- 1873-5460. ; 159, s. 125-137
-
Tidskriftsartikel (refereegranskat)abstract
- In this work we propose an algorithm for computing a stationary flow in a bifurcation tree. Our idea is to divide the tree into smaller basic blocks, each corresponding to one bifurcation, and solve a sequence of flow problems for the individual blocks. Numerical experiments demonstrate that the algorithm works well. We give a criteria for convergence that can be verified numerically and also an analytical convergence proof for an important special case. The application we have in mind is the computation of the time dependent blood flow in the arterial tree of the human body. The work presented here is for a simplified case but we discuss the extension of our work to the realistic cases. Also potential applications are discussed. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.
|
|
| 10. |
- Muhumuza, Asaph Keikara, et al.
(författare)
-
Connections between the extreme points for Vandermonde determinants and minimizing risk measure in financial mathematics
- 2022
-
Ingår i: <em>Springer Proceedings in Mathematics and Statistics</em>. - : Springer Nature. - 9783031178191 ; , s. 587-623
-
Konferensbidrag (refereegranskat)abstract
- The extreme points of Vandermonde determinants when optimized on surfaces like spheres and cubes have various applications in random matrix theory, electrostatics and financial mathematics. In this study, we apply the extreme points of Vandermonde determinant when optimized on various surfaces to risk minimization in financial mathematics. We illustrate this by constructing the efficient frontiers represented by spheres, cubes and other general surfaces as applies to portfolio theory. The extreme points of Vandermonde determinant lying on such surfaces as efficient frontier would be used to determine the set of assets with minimum risk and maximum returns. This technique can also applied in optimal portfolio selection and asset pricing.
|
|
