The toughening of sparse elastic networks, such as hydrogels, foams, or meshes against fracture is one of the most important problems in materials science. However, the principles of toughening have not yet been established despite urgent engineering requirements and several efforts made by materials scientists. Here we address the above-mentioned problem by focusing on the topology of a network. We perform fracture experiments for two-dimensional periodic lattices fabricated from rubber strings and connecters with well-defined topological structures. We find that systematic increase in the largest coordination number while maintaining the average coordination number (=4) as constant leads to significant improvement in toughness. We reproduce the observed toughening behavior through numerical simulations and confirm that the stress concentration in the vicinity of a crack tip can be controlled by the topology of the network. This provides a new strategy for creating tough sparse elastic networks, especially hydrogels.