Normalized Degree Centralization

• Normalized degree centralization is a metric that quantifies the concentration of connectivity by normalizing degree differences on a 0 to 1 scale.
• It distinguishes hub-dominated networks from uniformly connected graphs, allowing for rigorous comparisons across different network sizes.
• Empirical and theoretical evaluations demonstrate that, alongside betweenness and closeness measures, it effectively captures diverse structural aspects of networks.

Normalized degree centralization quantifies the concentration of connectivity within a network, measuring how close a network is to the extremal case in which all edges are incident to a single hub node. This metric, along with normalized betweenness centralization (NBC) and normalized closeness centralization (NCC), forms a triad of global centralization measures that systematically describe hub dominance from distinct structural perspectives. By formally normalizing the degree-based centralization so its values span $[0,1]$ for any graph size, researchers ensure comparability across networks and facilitate rigorous axiomatic and numerical evaluations.

1. Formal Definition and Normalization

Normalized degree centralization (NDC) is constructed to evaluate the extent of degree heterogeneity, bounded between 0 (maximal uniformity: all nodes have equal degree) and 1 (maximal centralization: one node connected to all others, others minimally or not connected). Let $k_i$ denote the degree of node $i$. The raw degree centralization is defined as

$$C_D^{\mathrm{raw}}(G) = \sum_{i=1}^{n} [\max_j k_j - k_i]$$

Saberi and Aref’s normalization ensures network-level NDC lies in $[0,1]$, typically via division by its maximal attainable value (in a star graph of $n$ vertices, where one node has degree $n-1$ and all others have degree 1).

2. Theoretical Axiomatic Assessment

A robust centralization metric is expected to satisfy six axiomatic postulates, as formalized by Palak–Nguyen (2021):

• P1 (Zero baseline): NDC is zero for graphs with $|V|=1$, complete graphs, or empty graphs, as all nodes are structurally equivalent.
• P2 (Star maximality): NDC reaches unity for the star configuration.
• P3 (Isomorphism invariance): NDC is invariant under relabeling.
• P4 (Sub-maximality w/o saturation): NDC is strictly less than 1 if no node is a saturated hub.
• P5 (Non-increase upon adding edges to saturated node): Connectivity concentration cannot increase further when additional links are given to an already saturated node.
• P6 (Non-decrease upon first saturation): Creating a first saturated node should not reduce centralization.

Empirical and theoretical studies indicate that normalized degree centralization satisfies all six postulates for major classes of graphs, affirming its utility for detecting hub-focused structures (Saberi et al., 26 Nov 2025).

3. Numerical Evaluations Across Canonical Graphs

Saberi and Aref’s numerical investigations exhibited NDC’s predictable behavior over prototypical graph families:

Graph Type	Typical NDC Behavior	Limiting Pattern
Star	NDC = 1	All $n$
Complete	NDC = 0	All $n$
Ring	NDC = 0	All $n$
Star+1 edge changes	NDC $\approx$ 1 (slight decrease)	Rapid convergence, $n \to \infty$
Ring+1 chord	NDC $>0$ but decays to 0	Large $n$
Complete–1 edge	NDC small $>0$, decays to 0	Large $n$

A plausible implication is that NDC robustly distinguishes between configurations where hub dominance is structurally imposed (star) and those where equality and redundancy suppress centralization (e.g., rings, complete graphs).

4. Comparative Perspectives with Complementary Measures

NDC isolates degree-based hub concentration, distinguishing it from NBC’s sensitivity to path-based control and NCC’s focus on reachability. The triad enables nuanced characterization:

• NBC (path-based): Sensitive to control of shortest-path flows and bottlenecks.
• NCC (reachability): Highlights inequalities in average distances to central nodes.
• NDC (degree-based): Identifies centralization via sheer number of connections.

Joint measurement facilitates discrimination of networks where dominance arises by different structural mechanisms. When NBC, NCC, and NDC concur, the network likely exhibits robust hub centralization; divergence among them uncovers more subtle forms of topological organization (Saberi et al., 26 Nov 2025). This suggests a multidimensional centralization profile is preferable for comprehensive network analysis.

5. Empirical Application: Real-World Network Evidence

Analysis across several real networks underscores NDC’s interpretive value. Two cases are illustrative:

• Facebook Ego Network ($n=4039$, $m=88,234$): NDC = 0.248 (moderate), NBC = 0.480 (high), NCC = 0.367 (intermediate). This suggests that while few nodes possess disproportionately many ties, structural control of shortest-path traffic is notably concentrated; average proximity to the hub is not extreme. Interpretation: Certain users act as bridges rather than pure degree hubs.
• Brain Functional Connectome (HCP, $n=45$, $m=242$): NDC = 0.291 (moderate), NBC = 0.062 (low), NCC = 0.288 (moderate). Here, degree and closeness centralization indicate moderately connected regions, but path-based centralization is minimal, signifying rich alternative routes among nodes.

These patterns demonstrate that NDC, in concert with NBC and NCC, reveals the nuanced interplay between direct connectivity, path dominance, and accessibility in real networks.

6. Practical Implications and Considerations

Normalized degree centralization is especially suited to contexts such as protein–protein interaction networks, neurological hub analysis, and any domain where the raw number of links implies critical importance. Its robustness under various perturbations and its compliance with key theoretical postulates recommend its use for both exploratory and comparative studies in network science. A plausible implication is that deploying NDC alongside path- and closeness-based metrics allows for the detection of network organizational principles not evident from a single centralization facet. The measure’s bounded normalization further simplifies cross-network and cross-domain comparisons, reinforcing its methodological utility (Saberi et al., 26 Nov 2025).