The Proxy Boundary

In network analysis, node degree — the number of connections — is the simplest centrality measure. It is cheap to compute, easy to interpret, and widely used as a proxy for systemic importance. The eigenvector centrality is the “correct” measure: it weights connections by the importance of the nodes they connect to, capturing cascading influence. But it is expensive to compute and requires global network information.

The question is: when is degree good enough? When does the cheap proxy match the expensive truth?

The answer (arXiv:2601.00807) depends on network structure in specific, characterizable ways. In assortative networks — where high-degree nodes connect to high-degree nodes — degree and eigenvector centrality agree. The proxy works because the network’s wiring reinforces the simple measure: popular nodes are connected to popular nodes, so counting connections captures cascading importance automatically.

In networks with strong community structure or core-periphery organization, the proxy fails. A node at the boundary between communities can have moderate degree but high eigenvector centrality, because it bridges clusters. A peripheral node can have high degree within its local neighborhood but low systemic importance. The network’s architecture creates “regions of spectral safety” where degree is trustworthy, and regions of spectral danger where it misleads.

The practical consequence: whether you can use the simple measure depends on where in the network you’re looking and what the network looks like. There is no global answer — the proxy’s validity is local and structural. The same degree value means different things in different positions, and the network’s topology determines which positions make the proxy reliable.

This is a general principle about proxies. A proxy works when the structure of the system reinforces the correlation between the proxy and the target. It fails when the structure creates positions where the two diverge. The question is never “is this proxy good?” but “where in the system is this proxy good?”