Normalized Closeness Centralization

Updated 3 December 2025

Normalized closeness centralization is a metric that quantifies the disparity in node accessibility by normalizing the deviation of closeness scores against the star graph maximum.
It standardizes measurements to a [0,1] scale, facilitating direct comparisons across networks of varying sizes, densities, and structures.
The measure applies to both one-mode and bipartite networks, though it only meets five of six axioms, indicating nuanced limitations in specific small-graph scenarios.

Normalized closeness centralization (NCC) is a global network index designed to rigorously quantify the extent to which the shortest-path accessibility of nodes is dominated by one or a few central “hubs.” Distinct from degree or betweenness-based measures, NCC captures the efficiency disparity in node-to-node reachability by leveraging the extremal properties of star and nearly regular graphs. The metric is normalized to the [0,1] interval through division by analytically derived maxima, enabling direct comparison across different network sizes and densities for both one-mode (simple graphs) and two-mode (bipartite) networks (Saberi et al., 26 Nov 2025, Krnc et al., 2016).

1. Mathematical Foundations and Normalization

Let $G=(V,E)$ be a simple undirected connected graph with $n=|V| \geq 3$ . The closeness centrality of node $i$ is given by

$CC(i) = \frac{n-1}{\sum_{j \neq i} d(i,j)}$

where $d(i,j)$ is the shortest path length from $i$ to $j$ . Freeman’s unnormalized closeness centralization measures the deviation from uniform closeness:

$C_C(G) = \sum_{i=1}^n [CC^* - CC(i)], \qquad CC^* = \max_i CC(i)$

The maximum possible $C_C(G)$ for a given $n$ is achieved by the star graph $S_n$ :

$\max C_C(G) = \frac{(n-1)(n-2)}{2n-3}$

Normalized closeness centralization is

$NCC(G) = \frac{\sum_{i=1}^n [CC^* - CC(i)]}{(n-1)(n-2)/(2n-3)}$

or, equivalently,

$NCC(G) = \frac{(2n-3)\sum_{i=1}^n [CC^* - CC(i)]}{(n-1)(n-2)}$

This normalization strictly bounds $NCC(G)$ to $[0,1]$ , with $NCC(S_n)=1$ (maximal) and $NCC(K_n)=0$ (complete graph, minimal).

For bipartite (two-mode) graphs $G=(U,V,E)$ with $|U|=n_0$ , $|V|=n_1$ , centralization and normalization follow analogous definitions, with the closed-form extremal structure and value given by a special depth-2 tree rooted at a node in $U$ (Krnc et al., 2016).

2. Comparative Interpretation and Rationale

NCC’s normalization by the analytical maximum for any $n$ -node graph (one-mode) or any $(n_0,n_1)$ -sized bipartite graph ensures the metric is independent of network order or density. This guarantees consistent interpretation: an NCC of $0.6$ signifies 60% of the possible theoretical centralization for any network of comparable type, regardless of size. The comparability holds across sparsity/density regimes and for networks of fundamentally different scale (Saberi et al., 26 Nov 2025).

This property makes NCC uniquely suitable for comparative network analysis, both across empirical data sets and structural network models subject to changing size or density.

3. Compliance with Centralization Postulates

Palak & Nguyen (2021) define six postulates (P1a–P1c, P2–P6) specifying desirable properties for global centralization metrics. For NCC (one-mode):

P1 (Minimal centralization): Satisfied (NCC $=0$ for trivial, complete, or empty graphs).
P2 (Star maximality): Satisfied (NCC $=1$ uniquely for stars).
P3 (Isomorphism invariance): Satisfied (NCC invariant to relabeling).
P4 (Strictly below 1 if no saturated node): Satisfied.
P5 (Non-increase on saturating further nodes when one is already saturated): Satisfied.
P6 (Non-decrease on creating the first saturated node): Violated—counterexamples exist in small graphs where NCC decreases upon creating the first saturated node.

Thus, NCC satisfies five of the six foundational axioms, similar to normalized betweenness (NBC) and normalized degree centralization (NDC), indicating strong, though not strictly complete, axiomatic validity (Saberi et al., 26 Nov 2025).

4. Numerical and Structural Properties

Evaluated on canonical graphs, NCC displays:

Star graphs: $NCC=1$ for all $n$ .
Rings and complete graphs: $NCC=0$ for all $n$ .
Single-edge perturbations: NCC decays smoothly from 1 to 0 as hub edges are removed/added or as edge density increases, approaching correct asymptotics.

For random graphs:

Erdős–Rényi $G(n,p)$ : NCC is moderate (0.2–0.3) in sparse/just-connected regimes, decreases to zero in dense limit as path lengths equalize.
Scale-free (Barabási–Albert): NCC stabilizes at nonzero values (approx. 0.4–0.7, parameter-dependent) even as $n \rightarrow \infty$ , due to persistent hub-induced path-length reductions.

NCC thus robustly identifies hub-driven geodesic centralization and is insensitive to trivial scaling or overall edge-density effects once normalized (Saberi et al., 26 Nov 2025).

5. Application to Bipartite (Two-Mode) Networks

For bipartite graphs $G=(U,V,E)$ with prescribed part sizes, the normalized closeness centralization is defined via Freeman’s bipartite closeness and normalized by the explicit extremal value, realized by the depth-2 tree $H(u;n_0,n_1)$ constructed as follows:

The root $u \in U$ is adjacent to every $v \in V$ .
Each $v$ is attached to either $\lfloor \frac{n_0-1}{n_1} \rfloor$ or $\lceil \frac{n_0-1}{n_1} \rceil$ further $U$ -nodes such that all of $U$ is exhausted.

The closed-form normalization guarantees

$0 \leq CC^\mathrm{norm}(G) \leq 1$

with $CC^\mathrm{norm}(G)=1$ if and only if $G \cong H(u;n_0,n_1)$ .

In empirical two-mode networks (e.g., affiliations, co-occurrence, participation data), high normalized centralization reflects a dominant accessibility “core,” while near-zero values indicate uniformly distributed reachability (Krnc et al., 2016).

6. Empirical Observations and Interpretive Range

In (Saberi et al., 26 Nov 2025), NCC is computed for diverse real-world networks:

Network	Nodes	NCC	Interpretation
Brain Network (HCP)	45	0.288	Modest: no single dominating region
Condensed Matter Collaboration	21363	0.217	Very low: distributed collaboration
Enron Email	36692	0.269	Moderate: few slightly more accessible nodes
Facebook Ego	4039	0.367	High: connectors/ego node dominate distance
Food Web Michigan	39	0.701	Very high: accessibility bottlenecks
Yeast Protein Interaction	2375	0.206	Low: alternative routes, no hub domination
Zachary’s Karate Club	34	0.298	Moderate: some nodes mediate social reach

NCC values in the empirical range $[0.2,0.7]$ represent the spectrum from network-wide equality of accessibility (low NCC) to acute hub-centric reachability (high NCC). In each case, NCC quantifies how much observed average node-to-node path lengths are shortened by dominant nodes in the network.

NCC provides a distinct perspective:

Closeness Centralization (NCC): Geodesic efficiency—captures the reduction in average path lengths due to hub(s).
Betweenness Centralization (NBC): Path control—emphasizes nodes that intercept most shortest paths.
Degree Centralization (NDC): Raw degree disparity—captures concentration of direct connections.

All three metrics satisfy five of six foundational postulates and are normalized to [0,1] using extremal graph constructions, ensuring cross-network comparability (Saberi et al., 26 Nov 2025).

NCC is particularly recommended when the research question centers on global accessibility or efficiency of reach. In practice, joint profiling with NBC and NDC is advocated for a comprehensive, multidimensional view of network “hubness” relevant for dynamic processes, structural resilience, and diffusion phenomena.

Researchers should limit NCC application to connected graphs or, in practice, restrict computation to the largest connected component. For bipartite scenarios, normalization requires explicit computation of the tree extremal bound for the observed part sizes.

References

"Evaluating Global Measures of Network Centralization: Axiomatic and Numerical Assessments" (Saberi et al., 26 Nov 2025).
"Closeness Centralization Measure for Two-mode Data of Prescribed Sizes" (Krnc et al., 2016).