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The eccentricity of a node in a graph is defined as the length of a longest shortest path starting at that node. The eccentricity distribution over all nodes is a relevant descriptive property of the graph, and its extreme values allow the derivation of measures such as the radius, diameter, center and periphery of the graph. This paper describes two new methods for computing the eccentricity distribution of large graphs such as social networks, web graphs, biological networks and routing networks.We first propose an exact algorithm based on eccentricity lower and upper bounds, which achieves significant speedups compared to the straightforward algorithm when computing both the extreme values of the distribution as well as the eccentricity distribution as a whole. The second algorithm that we describe is a hybrid strategy that combines the exact approach with an efficient sampling technique in order to obtain an even larger speedup on the computation of the entire eccentricity distribution. We perform an extensive set of experiments on a number of large graphs in order to measure and compare the performance of our algorithms, and demonstrate how we can efficiently compute the eccentricity distribution of various large real-world graphs. [ABSTRACT FROM AUTHOR]