Dynamic Graphs: PageRank (PG)

Chapter 7: PageRank (PG) in Dynamic Graphs

Introduction

PageRank is an algorithm initially used by Google Search to rank web pages in search engine results. It is widely used in network analysis to determine the importance of nodes within the graph. In dynamic graphs where nodes and edges are frequently added or removed, maintaining an updated PageRank becomes challenging. This chapter focuses on these challenges.

Standard PageRank Algorithm

The classic PageRank algorithm uses the following formula to calculate the rank for a page \( P \):

\[ PR(P) = (1 - d) + d \left( \frac{PR(P1)}{L(P1)} + \frac{PR(P2)}{L(P2)} + \ldots + \frac{PR(Pn)}{L(Pn)} \right) \]

Where \( PR(P) \) is the PageRank of page \( P \), \( d \) is a damping factor, \( L(Pn) \) is the number of outgoing links from a page \( Pn \), and \( P1, P2, \ldots, Pn \) are the pages linking to \( P \).

Dynamic Graphs and PageRank

In dynamic graphs, the following challenges need to be addressed:

• When new nodes or edges are added, the PageRank scores need to be updated.
• When nodes or edges are removed, PageRank scores may decrease or even become invalid.

PageRank in Dynamic Graphs

Strategies for adapting PageRank for dynamic graphs include:

1. Batch Updates: Recompute PageRank after collecting several changes.
2. Incremental Updates: Update PageRank scores as soon as changes occur.

Incremental PageRank Algorithm

  IncrementalPageRank(Graph G_t, PR_Scores, d):
    1. Detect changes (added/removed vertices and edges)
    2. For each affected node:
        3. Recalculate its PageRank based on its neighbors and update PR_Scores

Conclusion

PageRank is an important algorithm for evaluating the importance of nodes within a graph. Dynamic graphs pose challenges to maintaining accurate PageRank scores. Incremental algorithms offer a solution for efficiently updating these scores as the graph evolves.