In this paper we consider the problem of mining influence in a network of topics, where we seek to model direct influences between topics over time, in the form of bursts of topic occurrence - so that influence is measured by topic co-occurrence in high-frequency intervals - and this influence propagates among topics, leading to frequent occurrences of the topics. Although it is clearly significant, this problem has gotten very little attention. To address the problem, we propose a novel model: SemInf. As topics can recur, influence in our model is not constant or single-timestamp ("one-shot", as in social networks), but is instead multi-timestamp, with periods of influence that can span multiple time intervals. More specifically, this model of topic influence captures upward momentum in popularity over all time-stamps in a burst (a period of "elevated occurrence" of topics). A topic hierarchy is used to provide a distance measure among topics and characterize their semantic relatedness. Experiments on biomedical topics give some surprising results, showing both that our model is successful at identifying topics with high impact, and that it can be potentially used as an alternative model of impact in the scientific literature (which can be useful when citation information is not available). We also show that although semantic information helps boost performance of our model, it can work without such information. What's more, we show SemInf can be also generalized to other domains, such as topics in computer science research.