Personalized PageRank dimensionality and algorithmic implications

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.