PageRank is a metric that assigns importance to the vertices of a graph based on its neighbors and their scores. Recently, there has been increasing interest in computing PageRank on dynamic graphs, where the graph structure evolves due to edge insertions and deletions. However, traditional barrier-based approaches for updating PageRanks encounter significant wait times on certain graph structures, leading to high overall runtimes. Additionally, the growing trend of multicore architectures with increased core counts has raised concerns about random thread delays and failures. In this study, we propose a lock-free algorithm for updating PageRank scores on dynamic graphs. First, we introduce our Dynamic Frontier (DF) approach, which identifies and processes vertices likely to change PageRanks with minimal overhead. Subsequently, we integrate DF with our lock-free and fault-tolerant PageRank (Alg. DFLF), incorporating a helping mechanism among threads between its two phases. Experimental results demonstrate that Alg. DFLF not only eliminates waiting times at iteration barriers but also withstands random thread delays and crashes. On average, it is 4.6× faster than lock-free Naive-dynamic PageRank (Alg. NDLF).