Choice of invariant function for vitality centrality

Determine which invariant function should be chosen to score a whole network when defining a vitality-based centrality measure, so that the resulting node ranking reflects the process or phenomenon being measured.

Background

The paper defines vitality centrality by selecting an invariant function that scores the whole network and measuring how much that score decreases when a node is removed. In the paper's dependency-network setting, the invariant is N, the number of nodes left operational after failure propagation. The related-work discussion notes that alternative invariants, such as network efficiency or process-dependent spreading outcomes, encode different assumptions about how activity or failure moves through a network.

The unresolved issue is therefore to identify an appropriate invariant function in general. Choosing an invariant whose assumptions do not match the system can produce an incorrect importance ranking, making the selection of the invariant a substantive open problem for vitality-based centrality analysis.

References

Which invariant to choose is then the open question, and Borgatti gives the reason it matters: every centrality makes an assumption about how the thing being measured moves through the network, and an index whose assumption is wrong for the system at hand ranks the wrong nodes.

— Series and Parallel Dependency Links: Vitality for Interdependent Networks of Networks  (2610.21843 - Curaba et al., 15 Oct 2026) in Section 2, Related Work, paragraph “Vitality, and where its invariant comes from”