Directed Eigenvector Centrality
Also known as: directed EC, asymmetric eigenvector centrality, right eigenvector centrality, left eigenvector centrality
Directed eigenvector centrality extends the classic eigenvector centrality to directed graphs by scoring each node according to the centrality of the nodes that point to it (in-direction) or that it points to (out-direction). A node earns a high score not merely by having many connections but by being connected to other highly central nodes, capturing asymmetric influence in citation networks, social hierarchies, and information flows.
Key highlights
- Captures recursive, quality-weighted influence: connecting to well-connected nodes amplifies your own score.
- Separates prestige (who cites you) from influence (whom you cite) via right and left eigenvectors.
- Computationally efficient via power iteration even on large sparse directed graphs.
- Natural generalisation of the widely understood eigenvector centrality concept to directed settings.
- Applicable to weighted directed graphs by using the weighted adjacency matrix directly.
Intuition
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How it works
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When to use it
Use directed eigenvector centrality when ties are inherently asymmetric — citations, follower/following, funding flows, or food webs — and you want to rank nodes by the quality, not just the quantity, of their incoming or outgoing connections. It is especially appropriate for identifying authoritative sources in citation or web networks. Avoid it on networks with many sink nodes or without a dominant strongly connected component, because the eigenvector may be zero for large portions of the graph; in those cases PageRank (which adds a teleportation correction) is more robust. Also avoid on small networks (fewer than ~30 nodes) where eigenvalue instability makes scores unreliable.
Strengths & limitations
- Captures recursive, quality-weighted influence: connecting to well-connected nodes amplifies your own score.
- Separates prestige (who cites you) from influence (whom you cite) via right and left eigenvectors.
- Computationally efficient via power iteration even on large sparse directed graphs.
- Natural generalisation of the widely understood eigenvector centrality concept to directed settings.
- Applicable to weighted directed graphs by using the weighted adjacency matrix directly.
- Ill-defined or zero scores for sink nodes and nodes outside the largest strongly connected component.
- Sensitive to network boundary decisions: which nodes and edges are included strongly affects scores.
- Does not account for path length or geodesic distance, only recursive neighbourhood influence.
- Left and right eigenvector interpretations require explicit documentation to avoid confusion in reporting.
Common pitfalls
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Applications
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Frequently asked
What is the difference between directed eigenvector centrality and PageRank?
Both rank nodes by the quality of their in-neighbours, but PageRank adds a damping factor (teleportation probability) that redistributes score from sink nodes back to the whole graph, making it well-defined on any directed graph. Directed eigenvector centrality is the cleaner theoretical formulation but requires a dominant strongly connected component to be meaningful.
Should I use the right or the left eigenvector?
The right eigenvector scores nodes by the centrality of nodes pointing to them (prestige/authority). The left eigenvector scores nodes by the centrality of nodes they point to (influence/hub). Report both when in-direction and out-direction roles are both theoretically relevant.
My network has isolated or sink nodes — will scores be valid?
Sink nodes (zero out-degree) and isolated components often receive zero or near-zero eigenvector scores regardless of their position, which is a structural artefact. Restrict analysis to the largest strongly connected component or switch to PageRank before interpreting rankings.
How does directed eigenvector centrality differ from in-degree centrality?
In-degree counts raw incoming ties. Directed eigenvector centrality weights each incoming tie by the score of its source, so a single endorsement from a high-prestige node counts more than many endorsements from low-prestige nodes. Use it when the quality of connections matters, not just the count.
Is directed eigenvector centrality appropriate for weighted networks?
Yes — replace the binary adjacency matrix with the weighted adjacency matrix. Scores then reflect both the number and strength of directed connections, which is especially useful in co-authorship, trade, or communication networks where tie weights carry substantive meaning.
Sources
- 1.Bonacich, P. (1987). Power and centrality: A family of measures. American Journal of Sociology, 92(5), 1170–1182.
- 2.Eigenvector centrality. Wikipedia.
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Cite this page
ScholarGate. (2026, June 3). Directed Eigenvector Centrality. ScholarGate. https://scholargate.app/network-analysis/directed-eigenvector-centrality