New PageRank model adds time-dependent memory for dynamic rankings

PageRank, the algorithm that powers Google Search, traditionally computes a static ranking based on link structure. A new paper from researchers at (presumably) Universidad Rey Juan Carlos proposes a dynamical version where the personalization vector—the bias toward certain pages—evolves as a time-dependent weighted average of its past values. This is modeled using an integro-differential equation with a memory function, building on earlier work by Gleich and Rossi. The authors prove existence and uniqueness of solutions and show that the solution remains a valid probability vector at all times, preserving key PageRank properties.

For strongly connected networks, the team demonstrates that most memory functions (e.g., exponential decay) lead to convergence to a stationary state independent of initial conditions. However, when the memory function takes an exponential-oscillatory form (like e^{at} cos(bt)), the PageRank dynamics become asymptotically periodic—the ranking oscillates over time. This is a novel finding: oscillatory memory can fundamentally alter ranking evolution, potentially capturing periodic trends in web content or user behavior. The work has implications for time-sensitive search, adaptive personalization, and understanding how ranking algorithms respond to changing data.