Networks, whose junctions are free to move along the edges, such as two-dimensional soap froths and membrane tubular networks of endoplasmic reticulum are intrinsically unstable. This instability is a result of a positive tension applied to the network elements. A paradigm of networks exhibiting stable polygonal configurations in spite of the junction mobility, are networks formed by bundles of Keratin Intermediate Filaments (KIFs) in live cells. A unique feature of KIF networks is a, hypothetically, negative tension generated in the network bundles due to an exchange of material between the network and an effective reservoir of unbundled filaments. Here we analyze the structure and stability of two-dimensional networks with mobile three-way junctions subject to negative tension. First, we analytically examine a simplified case of hexagonal networks with symmetric junctions and demonstrate that, indeed, a negative tension is mandatory for the network stability. Another factor contributing to the network stability is the junction elastic resistance to deviations from the symmetric state. We derive an equation for the optimal density of such networks resulting from an interplay between the tension and the junction energy. We describe a configurational degeneration of the optimal energy state of the network. Further, we analyze by numerical simulations the energy of randomly generated networks with, generally, asymmetric junctions, and demonstrate that the global minimum of the network energy corresponds to the irregular configurations.