This article proposes a novel chunk-based caching scheme known as the Progressive Popularity-Aware Caching Scheme (PPCS) to improve content availability and eliminate the cache redundancy issue of Information-Centric Networking (ICN). Particularly, the proposal considers both entire-object caching and partial-progressive caching for popular and non-popular content objects, respectively. In the case that the content is not popular enough, PPCS first caches initial chunks of the content at the edge node and then progressively continues caching subsequent chunks at upstream Content Nodes (CNs) along the delivery path over time, according to the content popularity and each CN position. Therefore, PPCS efficiently avoids wasting cache space for storing on-path content duplicates and improves cache diversity by allowing no more than one replica of a specified content to be cached. To enable a complete ICN caching solution for communication networks, we also propose an autonomous replacement policy to optimize the cache utilization by maximizing the utility of each CN from caching content items. By simulation, we show that PPCS, utilizing edge-computing for the joint optimization of caching decision and replacement policies, considerably outperforms relevant existing ICN caching strategies in terms of latency (number of hops), cache redundancy, and content availability (hit rate), especially when the CN's cache size is small.