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Availability of large-scale network data for real systems is enabling mathematical and computational methods to systematically model the formation of the networks. Various growth models are proposed to reproduce the structures of the real-world networks. Evaluating how well a model fits the network data is an outstanding challenge, since the structures of networks that have tens of thousands of vertices and edges are highly complex. We here use a trace curve, which is produced by a breadth-first search processing on the network, to characterize the structure of network. Because the trace curve is shaped by both the local and global structure of the network, it can be used to tell the subtle difference between networks. By comparing the curves of model network and real network data, we evaluate the fit of model to the data. The evaluation of fit subsequently can be used to estimate the growth parameters for real network, which are key factors affecting the growth of real system. We illustrate the power of this approach by estimating growth parameters for the Drosophila melanogaster protein interaction network.