On the initial value of PageRank

TL;DR

This paper analytically proved that when the initial value of vertices is either proportional to their degrees or set to zero, the PageRank values of the vertices become directly proportional to their degrees.

Abstract

Google employs PageRank to rank web pages, determining the order in which search results are presented to users based on their queries. PageRank is primarily utilized for directed networks, although there are instances where it is also applied to undirected networks. In this paper, we have applied PageRank to undirected networks, showing that a vertex's PageRank relies on its initial value, often referred to as an intrinsic, non-network contribution. We have analytically proved that when the initial value of vertices is either proportional to their degrees or set to zero, the PageRank values of the vertices become directly proportional to their degrees. Simulated and empirical data are employed to bolster our research findings. Additionally, we have investigated the impact of initial values on PageRank localization.

Figures (16)

• Figure 1: (Colour online) Theoretical versus numerical analysis on Zachary's karate club network zachary1977 for different initial values. When the initial value is (a) the inverse of the vertex degree, (b) equal for all vertices, (c) proportional to the vertex degree, (d) proportional to the square of the vertex degree.
• Figure 2: (Colour online) PageRank on Erdös-Réyni (ER) random network for different initial values. When the initial value is (a) inverse of the vertex degree, (b) equal for all vertices, (c) proportional to the vertex degree, (d) proportional to the square of the vertex degree. Random network model proposed by Erdös-Réyni Bollobas01. Here the ER random network is generated with $1000$ vertices and the probability of connecting two vertices is $0.01$.
• Figure 3: (Colour online) PageRank on the small-world network for different initial values. When the initial value is (a) the inverse of the vertex degree, (b) equal for all vertices, (c) proportional to the vertex degree, (d) proportional to the square of the vertex degree. Small-world network proposed by Watts-Strogatz WS98. Here the small-world network is generated by rewiring regular ring lattice of size $1000$ and average degree $10$ with rewiring probability $0.4$.
• Figure 4: (Colour online) PageRank stability on the scale-free network for different initial values. When the initial value is (a) inverse of the vertex degree, (b) equal for all vertices, (c) proportional to the vertex degree, (d) proportional to the square of the vertex degree. Scale-free network proposed by Barabási-Albert BA99. Here the scale-free network is generated with size $1000$ and the size of the seed network is $m_0=5$ and a new vertex is added with existing $m=5$ vertices.
• Figure 5: (Colour online) PageRank stability on the duplication-divergence network for different initial values. When the initial value is (a) inverse of the vertex degree, (b) equal for all vertices, (c) proportional to the vertex degree, (d) proportional to the square of the vertex degree. The duplication-divergence model proposed by I. Ispolatov et al.ispolatov2005duplication. Here the duplication-divergence network is generated with size $1000$ with the link retention probability $\sigma = 0.4$.

Theorems & Definitions (2)

• Remark 1
• Remark 2