Eigenvector Centrality

A network node score recursively weighted by the scores of connected nodes and selected from a leading adjacency eigenvector.

Version

v1 · 2026-10-03 · History

Domain-specific #

13180

Domain group

Formal Sciences

Origin domain

Mathematics

Subdomain

Network Analysis → Mathematics

Core Idea

Eigenvector centrality gives each network node a score proportional to the scores of its connected nodes: for a nonnegative adjacency matrix \(A\), the conventional normalized nonnegative leading eigenvector satisfies \(Ax=\lambda x\) (with orientation adjusted explicitly for directed graphs). A connection to a high-score node contributes more than an otherwise equal connection to a low-score node. The score is a network-relative spectral measure, not a direct measure of actual influence or worth.[^{ref-73bd502c78c5][}ref-d30a60784dfa]

Scope of Application

The same equation can score social-relation networks and weighted voxel-similarity networks. The edge meaning changes across these settings. A connected undirected nonnegative network supports a unique strictly positive Perron score up to scale; a strongly connected directed network has a corresponding result under its chosen left/right convention. Disconnected or reducible graphs can yield zeros or nonunique rankings. Katz, PageRank and Bonacich's broader parameterized family are related measures, not interchangeable aliases.[^{ref-73bd502c78c5][}ref-48109d0556fc][^ref-d30a60784dfa]

Clarity

The centrality equation makes recursive prominence precise and separates it from degree, which merely counts immediate ties. It also makes modeling commitments visible: what do edges mean, which way do directed edges contribute, and is the graph connected enough to support a positive unique ranking? A voxel's centrality is not thereby a causal physiological role, just as a social position's score is not automatically bargaining power.[^{ref-73bd502c78c5][}ref-48109d0556fc]

Manages Complexity

Many weighted ties become one score vector, making a large connection pattern inspectable. Lohmann et al. applied this to voxel-level similarity data and compared it with degree maps. The compression is useful but leaves out other network properties and all substantive information absent from the edge definition. Regularizing a disconnected graph may help produce comparable rankings but changes the pure spectral measure.[^{ref-48109d0556fc][}ref-d30a60784dfa]

Abstract Reasoning

The equation \(x_i=\lambda^{-1}\sum_j A_{ij}x_j\) requires scores to be solved jointly: each node depends on neighbors whose scores depend on further neighbors. Perron–Frobenius supports positive scale-unique scores only under irreducibility. If a graph splits into components, the global dominant eigenvector may vanish on some components or be nonunique when dominant blocks tie; local node degree alone cannot repair that global ambiguity.[^{ref-73bd502c78c5][}ref-d30a60784dfa]

Knowledge Transfer

The roles—specified nodes and edges, nonnegative adjacency, recursive scores and leading-eigenvector normalization—transfer literally from Bonacich's social-network status to Lohmann's voxel-similarity map. The substantive interpretation does not transfer automatically. A generic eigenpair is not by itself centrality; no iterative computation makes Fixed Point a necessary parent.[^{ref-73bd502c78c5][}ref-48109d0556fc][^{ref-9cb942c6d756][}ref-9cb942c6d756-2]

[^ref-73bd502c78c5]: Phillip Bonacich, “Power and Centrality: A Family of Measures”, American Journal of Sociology 92(5), 1987, pp.1170–1182, especially Eq. (1)–(4). [^ref-48109d0556fc]: Gabriele Lohmann et al., “Eigenvector Centrality Mapping for Analyzing Connectivity Patterns in fMRI Data of the Human Brain”, PLOS ONE 5(4):e10232, 2010. [^ref-d30a60784dfa]: NetworkX project, eigenvector centrality official documentation, Notes and NumPy implementation sections. [^ref-9cb942c6d756]: Encyclopedia of Abstractions, live prime_abstractions/v2/network.md, Core Idea and Structural Signature, inspected 2026-10-01. [^ref-9cb942c6d756-2]: Encyclopedia of Abstractions, live prime_abstractions/v2/eigenvalue_and_eigenvector.md, Core Idea and Structural Signature, inspected 2026-10-01.

Relationships to Other Abstractions

Current abstraction Eigenvector Centrality Domain-specific

Parents (2) — more general patterns this builds on

• Eigenvector Centrality presupposes Eigenvalue And Eigenvector Prime

  The score is necessarily a leading eigenvector of the adjacency operator with an associated eigenvalue.

• Eigenvector Centrality presupposes Network Prime

  Adjacency-based centrality presupposes a defined network relation.

Hierarchy paths (3) — routes to 3 parentless roots

• Eigenvector Centrality → Eigenvalue And Eigenvector → Linearity

Show alternative paths (2)

Neighborhood in Abstraction Space

Eigenvector Centrality sits in a sparse region of the domain-specific corpus (74^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Graph Structures & Combinatorial Objects (44 abstractions)

Nearest neighbors

• Join Count Statistic — 0.84
• Metric dimension (graph theory) — 0.83
• Hadwiger number — 0.83
• Narrative network — 0.83
• Generalized blockmodeling of binary networks — 0.83

Computed from structural-signature embeddings · 2026-10-08