Equivalence of Personalized PageRank and Successor Representations

TL;DR

Problem: unify memory retrieval and planning in the hippocampus. Approach: demonstrate that Personalized PageRank and the successor representation are isomorphic as the stationary distribution of a graph-walk, formalized through $\pi$ and the fixed-point equations. Key mapping: from $\pi = \alpha P \pi + (1-\alpha)p^*$ to $\pi^* = (I - \alpha P)^{-1}(1-\alpha)p^*$, identify $\gamma$ with $\alpha$, and $M = (I - \gamma \mathcal{T})^{-1}$ with $r = (1-\alpha)p^*$, placing the SR value function as $V(x) = M r(x)$. Significance: provides a unified, graph-based computational primitive for retrieval and planning across arbitrary graphs, with concrete neural and algorithmic predictions for hippocampal activity patterns.

Abstract

The hippocampus appears to implement two core but highly distinct functions in the brain: long term memory retrieval and planning and spatial navigation. Naively, these functions appear very different algorithmically. In this short note, we demonstrate that two powerful algorithms that have each independently been proposed to underlie the hippocampal operation for each function -- personalized page-rank for memory retrieval, and successor representations for planning and navigation, are in fact isomorphic and utilize the same underlying representation -- the stationary distribution of a random walk on a graph. We hypothesize that the core computational function of the hippocampus is to compute this representation on arbitrary input graphs.