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Complex data can often be represented in terms of random graphs or networks. Important features of real world networks can be described by a special class of random graphs called small-world networks. Small-world networks emerge in many contexts, from systems biology to distributed technological systems. Here we ask how the functional and structural properties of specialized real world networks are reflected in convergence-divergence properties of their edges and nodes. We introduced a novel metric called edge convergence degree and studied it on small-world networks generated according to different rules. The obtained results were compared with Erdős-Renyi random networks. We found that convergence degree sensitively distinguishes different models of random networks we studied.