Personalized PageRank dimensionality and algorithmic implications

04/09/2018
by   Daniel Vial, et al.
0

Many systems, including the Internet, social networks, and the power grid, can be represented as graphs. When analyzing graphs, it is often useful to compute scores describing the relative importance or distance between nodes. One example is Personalized PageRank (PPR), which assigns to each node v a vector whose i-th entry describes the importance of the i-th node from the perspective of v. PPR has proven useful in many applications, such as recommending who users should follow on social networks (if this i-th entry is large, v may be interested in following the i-th user). Unfortunately, computing n such PPR vectors (where n is the number of nodes) is infeasible for many graphs of interest. In this work, we argue that the situation is not so dire. Our main result shows that the dimensionality of the set of PPR vectors scales sublinearly in n with high probability, for a certain class of random graphs and for a notion of dimensionality similar to rank. Put differently, we argue that the effective dimension of this set is much less than n, despite the fact that the matrix containing these vectors has rank n. Furthermore, we show this dimensionality measure relates closely to the complexity of a PPR estimation scheme that was proposed (but not analyzed) by Jeh and Widom. This allows us to argue that accurately estimating all n PPR vectors amounts to computing a vanishing fraction of the n^2 vector elements (when the technical assumptions of our main result are satisfied). Finally, we demonstrate empirically that similar conclusions hold when considering real-world networks, despite the assumptions of our theory not holding.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
03/28/2022

Personalized Graph Summarization: Formulation, Scalable Algorithms, and Applications

Are users of an online social network interested equally in all connecti...
research
08/25/2022

Local Intrinsic Dimensionality Measures for Graphs, with Applications to Graph Embeddings

The notion of local intrinsic dimensionality (LID) is an important advan...
research
04/01/2023

Entry-wise dissipation for singular vector perturbation bounds

Consider a random, symmetric perturbation of a symmetric, low rank matri...
research
02/13/2023

A simple statistic for determining the dimensionality of complex networks

Detecting the dimensionality of graphs is a central topic in machine lea...
research
07/22/2022

A flexible PageRank-based graph embedding framework closely related to spectral eigenvector embeddings

We study a simple embedding technique based on a matrix of personalized ...
research
05/15/2011

Generating Similar Graphs From Spherical Features

We propose a novel model for generating graphs similar to a given exampl...
research
01/20/2021

Fast Evaluation for Relevant Quantities of Opinion Dynamics

One of the main subjects in the field of social networks is to quantify ...

Please sign up or login with your details

Forgot password? Click here to reset