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