As social networks are constantly changing and evolving, methods to analyze dynamic social networks are becoming more important in understanding social trends. However, due to the restrictions imposed by the social network service providers, the resources available to fetch the entire contents of a social network are typically very limited. As a result, analysis of dynamic social network data requires maintaining an approximate copy of the social network for each time period, locally. In this paper, we study the problem of dynamic network and text fetching with limited probing capacities, for identifying and maintaining influential users as the social network evolves. We propose an algorithm to probe the relationships (required for global influence computation) as well as posts (required for topic-based influence computation) of a limited number of users during each probing period, based on the influence trends and activities of the users. We infer the current network based on the newly probed user data and the last known version of the network maintained locally. Additionally, we propose to use link prediction methods to further increase the accuracy of our network inference. We employ PageRank as the metric for influence computation. We illustrate how the proposed solution maintains accurate PageRank scores for computing global influence, and topic-sensitive weighted PageRank scores for topic-based influence. The latter relies on a topic-based network constructed via weights determined by semantic analysis of posts and their sharing statistics. We evaluate the effectiveness of our algorithms by comparing them with the true influence scores of the full and up-to-date version of the network, using data from the micro-blogging service Twitter. Results show that our techniques significantly outperform baseline methods (80 percent higher accuracy for network fetching and 77 percent for text fetching) and are superior to state-of-the-art techniques from the literature (21 percent higher accuracy).