[Paper Review] Ranking pages and the topology of the web

Argimiro Arratia, Carlos Marijuán|arXiv (Cornell University)|May 9, 2011

Web Data Mining and Analysis10 references3 citations

TL;DR

This paper proposes a topological framework for optimizing PageRank by analyzing hierarchical web structures such as rooted trees, cyclical trees, and PR–digraph trees. It derives closed-form PageRank formulas for the root of these structures and establishes a hierarchy showing that certain topologies—particularly PR–digraph trees—maximize PageRank by enabling efficient, parallel computation through strongly connected components with single inter-component links.

ABSTRACT

This paper presents our studies on the rearrangement of links from the structure of websites for the purpose of improving the valuation of a page or group of pages as established by a ranking function as Google's PageRank. We build our topological taxonomy starting from unidirectional and bidirectional rooted trees, and up to more complex hierarchical structures as cyclical rooted trees (obtained by closing cycles on bidirectional trees) and PR--digraph rooted trees (digraphs whose condensation digraph is a rooted tree that behave like cyclical rooted trees). We give different modifications on the structure of these trees and its effect on the valuation given by the PageRank function. We derive closed formulas for the PageRank of the root of various types of trees, and establish a hierarchy of these topologies in terms of PageRank. We show that the PageRank of the root of cyclical and PR--digraph trees basically depends on the number of vertices per level and the number of cycles of distinct lengths among levels, and we give a closed vector formula to compute PageRank.

Motivation & Objective

• To understand how web topology influences PageRank values, especially for individual pages or groups of pages.
• To develop a mathematical framework for structuring websites to maximize PageRank using graph-theoretic principles.
• To accelerate PageRank computation by exploiting hierarchical and strongly connected substructures in web graphs.
• To formalize the theoretical basis for SEO practices involving link structure manipulation.
• To establish a hierarchy of topological structures based on their PageRank-optimizing potential.

Proposed method

• Model the web as a directed graph (digraph) with vertices as pages and arcs as hyperlinks.
• Define and analyze unidirectional and bidirectional rooted trees as foundational topological structures.
• Introduce cyclical rooted trees by adding arcs to close paths in bidirectional trees, forming cycles.
• Propose PR–digraph trees as a generalization, where the condensation digraph is a rooted tree of strongly connected components (SCCs), each being a PR–digraph.
• Derive closed-form PageRank formulas for the root of various tree types, including cyclical and PR–digraph trees.
• Leverage results from [6] to compute PageRank independently on SCCs when inter-component links are single, enabling parallel and layered computation.

Experimental results

Research questions

• RQ1 How does the topological structure of a website—specifically its tree-like or cyclic hierarchy—affect the PageRank of its root page?
• RQ2 Which topological configurations maximize the PageRank of a target page under fixed link constraints?
• RQ3 Can closed-form PageRank expressions be derived for complex hierarchical structures such as cyclical and PR–digraph trees?
• RQ4 How can the PageRank computation be accelerated by exploiting the hierarchical and strongly connected structure of web graphs?
• RQ5 What conditions ensure that a digraph qualifies as a PR–digraph tree, and how does this structure support efficient ranking computation?

Key findings

• The root of a unidirectional rooted tree has a PageRank value that can be expressed in closed form, depending on the tree's depth and branching factor.
• Cyclical rooted trees, formed by closing paths in bidirectional trees, yield higher PageRank than simple rooted trees due to feedback loops.
• PR–digraph trees, which generalize cyclical structures with single inter-component links, allow independent PageRank computation on each SCC, enabling parallel processing.
• The paper establishes a hierarchy of topologies by PageRank value, showing that PR–digraph trees are among the most effective for maximizing root PageRank.
• The underlying digraph of a PR–digraph tree is a cyclical tree, which allows the use of the same PageRank formula as for cyclical trees, ensuring theoretical consistency.
• By treating each SCC as a unit and leveraging single-link inter-component connectivity, the PageRank of the entire structure can be computed layer-by-layer from the leaves to the root, significantly improving computational efficiency.