Normalized Betweenness Centralization

• Normalized Betweenness Centralization is a measure that quantifies the concentration of shortest-path control by normalizing raw betweenness against the maximal star graph benchmark.
• It scales values to the unit interval, enabling fair comparisons across networks of different sizes, with 1 indicating maximal centralization and 0 representing uniform path distribution.
• NBC complements other centralization metrics like normalized closeness and degree centralization, offering unique insights into path-based dominance and network control.

Normalized betweenness centralization (NBC) is a global measure of network centralization that quantifies the concentration of shortest-path control among nodes in an undirected graph. NBC builds on Freeman’s betweenness-based centralization and normalizes it to the unit interval $[0,1]$ by referencing the star graph as a maximally centralized benchmark. This normalization facilitates comparison across networks of different sizes and structures, providing a rigorous index of path-based dominance that complements alternative centralization metrics based on degree and closeness. NBC is distinguished by its sensitivity to network topology and its compliance with five out of six foundational axioms, as demonstrated in recent axiomatic and numerical assessments (Saberi et al., 26 Nov 2025).

1. Formal Definition and Normalization

Let $G=(V,E)$ denote a simple undirected graph with $|V|=n$ nodes. The raw betweenness centrality $b_i$ of node $i$ is the total number of shortest paths between all other pairs that pass through $i$. Freeman’s original centralization is

$C_B(G) = \sum_{i=1}^n [b_{\text{max}} - b_i]$

where $b_{\text{max}} = \max_j b_j$. To enable normalization across varying network sizes, each node’s betweenness is rescaled:

$C_B(i) = \frac{b_i}{\frac{(n-1)(n-2)}{2}}$

so that $0 \leq C_B(i) \leq 1$, and let $C_B^* = \max_i C_B(i)$. NBC is defined as

$$\mathrm{NBC}(G) = \frac{\sum_{i=1}^n [C_B^* - C_B(i)]}{\max_{H:|V(H)|=n} \sum_{i=1}^n [C_B^* - C_B(i)]} = \frac{\sum_{i=1}^n [C_B^* - C_B(i)]}{n-1}$$

where the denominator $n-1$ corresponds to the star graph. By construction, $\mathrm{NBC}(G) = 1$ if and only if $G$ is a star, and $\mathrm{NBC}(G) = 0$ if and only if all nodes have equal normalized betweenness.

2. Axiomatic Assessment

NBC has been evaluated against six centralization postulates:

Postulate (Palak & Nguyen, 2021)	Formal Statement	NBC Status	Justification
P1 (Zero-calibration)	$|V|=1$, $G$ complete, or $|E|=0,\, C(G)=0$	✓	All paths degenerate/equal
P2 (Max-star)	$G$ star $\implies C(G)=1$	✓	Center dominates paths
P3 (Isomorphism invariance)	$G_1 \cong G_2 \implies C(G_1)=C(G_2)$	✓	Path counts preserved
P4 (Non-saturated $<$ 1)	No node degree $n-1 \implies C(G)\lt 1$	✓	No unique path controller
P5 (Saturating decreases)	Adding saturated nodes $\nrightarrow\uparrow C(G)$	✓	Reduced uniqueness
P6 (Saturating increases)	Creating first saturated node $\nrightarrow\downarrow C(G)$	✗	Possible decrease in NBC

NBC fulfills five axioms: zero-calibration, max-star, isomorphism invariance, non-saturated below unity, and saturating decreases. It fails P6 due to observed decreases in NBC near the saturation threshold in specific small-network configurations.

3. Numerical Behavior on Canonical Synthetic Graphs

Systematic evaluation of NBC across six canonical topologies as $n$ increases reveals consistent responses to structural variation:

• Star: NBC$\,=\,1$ for all $n$ (maximal centralization).
• Ring: NBC$\,=\,0$ (uniform path accessibility).
• Complete: NBC$\,=\,0$ (no path bottlenecks).
• Perturbed Star (rewire): NBC$\rightarrow 1$ as $n\rightarrow\infty$, smoothly recovers maximal value.
• Perturbed Ring (add chord): NBC$\rightarrow 0$ as $n\rightarrow\infty$, with diminishing asymmetry.
• Perturbed Complete (remove edge): NBC$\rightarrow 0$, reflecting near-uniformity.

NBC achieves full consistency across all test cases, demonstrating reliable boundary behavior and appropriate sensitivity to minor structural perturbations.

4. Comparative Perspective: NBC, NCC, and NDC

NBC is one of three empirically and axiomatic-robust centralization measures identified (Saberi et al., 26 Nov 2025). Its function and interpretation are distinct from normalized closeness centralization (NCC) and normalized degree centralization (NDC), which measure geodesic accessibility and direct link concentration, respectively. NBC highlights path-based bottlenecks, making it ideal for networks where control over communication flows, transport, or organizational information is the central focus. NCC is preferable when efficiency of reach is critical, as in contact tracing or emergency response, while NDC excels in cases of biological hub proteins or infrastructure. The joint application of NBC, NCC, and NDC is recommended to provide multidimensional insight: although they agree on overall high- vs. low-centralization, they often diverge in network rankings, revealing complementary facets of "hubness" not captured by any single measure.

5. Empirical Applications

NBC has been applied to a range of real-world networks to elucidate path-dominance phenomena:

Network	NBC	NCC	NDC	Interpretation
Facebook Ego (n=4039)	0.480	0.367	0.248	Bottlenecks without singular high-degree hubs
Zachary Karate Club (n=34)	0.405	0.298	0.399	Instructor dominates ties and paths
High-school friendship (timesteps)	0.22 → 0.52 → 0.10 → 0.36	—	—	Fluctuating emergence/dissolution of path-based hubs

NBC distinctly captures the transient dominance of specific nodes over shortest paths, as observed in temporal high-school friendship dynamics. The decoupling between NBC and degree/closeness centralization across cases (such as the Facebook ego network) suggests utility in scenarios where indirect influence or information control is of interest, beyond raw connectivity or proximity.

6. Conceptual Significance and Use Guidelines

NBC provides a path-centric quantification of centralization, rigorously normalized and empirically validated. Its compliance with five foundational axioms and robust numerical behavior recommend it for network domains where control over shortest-path flows is diagnostically relevant. When used in tandem with NCC and NDC, NBC enables comprehensive profiling of hub dominance, accounting for the multidimensional nature of centralization in empirical networks. Application to temporal networks and diverse domains substantiates its interpretive value and practical reliability, with the caveat that local violations of the saturating increases axiom (P6) may arise in specific configurations.

In summary, normalized betweenness centralization serves as a theoretically and empirically sound index for global network centralization, providing precise, comparable, and interpretable measurement of path-based dominance alongside related degree- and closeness-centric metrics (Saberi et al., 26 Nov 2025).