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The Internet has been evolving into a more heterogeneous internet work with diverse new applications imposing more stringent bandwidth and QoS requirements. Already such new applications such as YouTube, Hulu, and Netflix are consuming a large fraction of the total bandwidth. We argue that, in order to engineer future internets such that they can adequately cater to their increasingly diverse and complex set of applications while using resources efficiently, it is critical to be able to characterize the load that emerging and future applications place on the underlying network. In this paper, we investigate entropy as a metric for characterizing per-flow network traffic complexity. While previous work has analyzed aggregated network traffic, we focus on studying isolated traffic flows. Per-application flow characterization caters to the need of network control functions such as traffic scheduling and admission control at the edges of the network. Such control functions necessitate differentiating network traffic on a per-application basis. The "entropy fingerprints" that we get from our entropy estimator summarize many characteristics of each application's network traffic. Not only can we compare applications on the basis of peak entropy, but we can also categorize them based on a number of other properties of the fingerprints.