The structure of local optima in multi-objective combinatorial optimization and their impact on algorithm performance are not yet properly understood. In this paper, we are interested in the representation of multi-objective landscapes and their multi-modality. More specifically, we revise and extend the network of Pareto local optimal solutions (PLOS-net), inspired by the well-established local optima network from single-objective optimization. We first define a compressed PLOS-net which allows us to enhance its perception while preserving the important notion of connectedness between local optima. We then study an alternative visualization of the (compressed) PLOS-net that focuses on good-quality solutions, improves the distinction between connected components in the network, and generalizes well to landscapes with more than 2 objectives. We finally define a number of network metrics that characterize the PLOS-net, some of them being strongly correlated with search performance. We visualize and experiment with small-size multi-objective nk-landscapes, and we disclose the effect of PLOS-net metrics against well-established multi-objective local search and evolutionary algorithms.