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We present exact analysis of the physical properties of bimodal networksspecified by the two peak degree distribution fully incorporating thedegree-degree correlation between node connection. The structure of thecorrelated bimodal network is uniquely determined by the Pearson coefficient ofthe degree correlation, keeping its degree distribution fixed. The percolationthreshold and the giant component fraction of the correlated bimodal networkare analytically calculated in the whole range of the Pearson coefficient from$-1$ to $1$ against two major types of node removal, which are the randomfailure and the degree-based targeted attack. The Pearson coefficient fornext-nearest-neighbor pairs is also calculated, which always takes a positivevalue even when the correlation between nearest-neighbor pairs is negative.From the results, it is confirmed that the percolation threshold is amonotonically decreasing function of the Pearson coefficient for the degrees ofnearest-neighbor pairs increasing from $-1$ and $1$ regardless of the types ofnode removal. In contrast, the node fraction of the giant component for bimodalnetworks with positive degree correlation rapidly decreases in the early stageof random failure, while that for bimodal networks with negative degreecorrelation remains relatively large until the removed node fraction reachesthe threshold. In this sense, bimodal networks with negative degree correlationare more robust against random failure than those with positive degreecorrelation.

Robustness analysis of bimodal networks in the whole range of degree correlation