# WNCG Seminar: A structural result for Personalized PageRank and its algorithmic consequences

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 n2 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.

This is joint work with Daniel Vial, University of Michigan.