In past decades, a lot of studies have been carried out on complex networks and heat conduction in regular lattices. However, very little attention has been paid to the heat conduction in complex networks. In this work, we study (both thermal and electric) energy transport in physical networks rewired from 2D regular lattices. It is found that the network can be transferred from a good conductor to a poor conductor, depending on the rewired network structure and coupling scheme. Two interesting phenomena were discovered: (i) the thermal-siphon effect—namely the heat flux can go from a low-temperature node to a higher-temperature node and (ii) there exits an optimal network structure that displays small thermal conductance and large electrical conductance. These discoveries reveal that network-structured materials have great potential in applications in thermal-energy management and thermal-electric-energy conversion.