We investigate how the graph topology influences the robustness to noise in undirected linear consensus networks. We consider the expected steady state population variance of states as the measure of vulnerability to noise. We quantify the structural robustness of a network by using the smallest value this measure can attain under edge weights from the unit interval. Our main result shows that the average distance between nodes and the average node degree define tight upper and lower bounds on the structural robustness. Using these bounds, we characterize the networks with different types of robustness scaling. We also present a fundamental trade-off between the structural robustness and the sparsity of networks. We then show that random regular graphs typically have near-optimal structural robustness among the graphs with same size and average degree. Some simulation results are also provided.