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We investigate how the graph topology influences the robustness to noise in undirected linear consensus networks. We measure the structural robustness by using the smallest possible value of steady-state population variance of states under the noisy consensus dynamics with edge weights from the unit interval. We derive tight upper and lower bounds on the structural robustness of networks based on the average distance between nodes and the average node degree. Using the proposed bounds, we characterize the networks with different types of robustness scaling under increasing size. Furthermore, we present a fundamental tradeoff between the structural robustness and the average degree of networks. While this tradeoff implies that a desired level of structural robustness can only be achieved by graphs with a sufficiently large average degree, we also show that there exist dense graphs with poor structural robustness. We then show that random <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-regular graphs (the degree of each node is <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>) with <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula> nodes typically have near-optimal structural robustness among all the graphs with size <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula> and average degree <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> for sufficiently large <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>. We also show that when <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> increases properly with <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula>, random <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-regular graphs maintain a structural robustness within a constant factor of the complete graph&#x0027;s while also having the minimum average degree required for such robustness.

Structural Robustness to Noise in Consensus Networks: Impact of Degrees and Distances, Fundamental Limits, and Extremal Graphs