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Effects of Backtracking on PageRank

Cory Glover, Tyler Jones, Mark Kempton, Alice Oveson

TL;DR

This work analyzes how backtracking influences PageRank centrality by introducing three variants—non-backtracking PageRank, mu-PageRank, and a newly proposed infty-PageRank. It proves that on regular and bipartite biregular graphs, standard PageRank and its variants are equivalent, and it develops theory for mu-PageRank including its limit as mu grows large. The authors also explore top-node stability between standard and infty-PageRank, and design an infty-PageRank–based clustering method with strong performance on stochastic block models and real networks. Overall, the paper advances understanding of backtracking in PageRank and offers practical clustering tools with potential for fast variant comparisons in large networks.

Abstract

In this paper, we consider three variations on standard PageRank: Non-backtracking PageRank, $μ$-PageRank, and $\infty$-PageRank, all of which alter the standard formula by adjusting the likelihood of backtracking in the algorithm's random walk. We show that in the case of regular and bipartite biregular graphs, standard PageRank and its variants are equivalent. We also compare each centrality measure and investigate their clustering capabilities.

Effects of Backtracking on PageRank

TL;DR

This work analyzes how backtracking influences PageRank centrality by introducing three variants—non-backtracking PageRank, mu-PageRank, and a newly proposed infty-PageRank. It proves that on regular and bipartite biregular graphs, standard PageRank and its variants are equivalent, and it develops theory for mu-PageRank including its limit as mu grows large. The authors also explore top-node stability between standard and infty-PageRank, and design an infty-PageRank–based clustering method with strong performance on stochastic block models and real networks. Overall, the paper advances understanding of backtracking in PageRank and offers practical clustering tools with potential for fast variant comparisons in large networks.

Abstract

In this paper, we consider three variations on standard PageRank: Non-backtracking PageRank, -PageRank, and -PageRank, all of which alter the standard formula by adjusting the likelihood of backtracking in the algorithm's random walk. We show that in the case of regular and bipartite biregular graphs, standard PageRank and its variants are equivalent. We also compare each centrality measure and investigate their clustering capabilities.
Paper Structure (12 sections, 4 theorems, 33 equations, 8 figures, 1 table, 1 algorithm)

This paper contains 12 sections, 4 theorems, 33 equations, 8 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Standard, Non-Backtracking, and $\mu$-PageRank
  3. Standard PageRank
  4. Non-backtracking PageRank
  5. $\mu$-PageRank
  6. Analysis of $\mu$-PageRank
  7. $\mu$-PageRank of Regular Graphs and Biregular Bipartite Graphs
  8. Limit as $\mu\rightarrow\infty$
  9. Monotonicity
  10. Top Nodes in $\infty$-PageRank and Standard PageRank
  11. Clustering with $\infty$-PageRank
  12. Conclusion

Key Result

Theorem 2.1

In Definition 2.2, The PageRank of $G$ is $\pi=T\hat{\pi}$.

Figures (8)

  • Figure 1: $\mu$-PageRank Values over $\mu\in[0,100]$
  • Figure 2:
  • Figure 3:
  • Figure 4: The top row is the football network and the bottom row is a random stochastic block network. The networks in the left column are the correctly labeled networks and the right are clustered networks. The plot on the right shows the algorithm accuracy as measured by NMI as a function of the clustering strength in the network. The blue band shows the 95% confidence interval.
  • Figure 5: Standard versus $\infty$-PageRank Distributions
  • ...and 3 more figures

Theorems & Definitions (18)

  • Definition 2.1: PageRank
  • Definition 2.2: Edge PageRank arrigo2019non
  • Theorem 2.1: arrigo2019non Corollary 1
  • Definition 2.3: Non-backtracking PageRank arrigo2019non
  • Definition 2.4: $\mu$-PageRank
  • Remark 1
  • Remark 2
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • ...and 8 more