Weighted Degree Centrality

• Weighted Degree Centrality is a network measure that aggregates the weights of edges incident to each node, reflecting both connectivity and functional significance.
• It operates in linear time with respect to the number of edges, ensuring computational efficiency and stability against perturbations in edge weights.
• The method extends to interval-weighted networks using tunable parameters, making it valuable for practical applications in infrastructure resilience, trade, and mobility analysis.

Weighted Degree Centrality is a fundamental class of centrality measures in network science, designed to quantify the importance of nodes by aggregating the weights of edges incident to each node. Unlike unweighted degree centrality, which merely counts adjacency relationships, weighted degree centrality incorporates edge weights—representing empirical quantities such as traffic volume, interaction strength, freight tonnage, or flow capacity—yielding measures sensitive to functional connectivity and heterogeneity. This centrality has robust mathematical properties, empirical resilience to data perturbations, and admits extensions for uncertainty quantification in interval-weighted networks.

1. Formal Definitions in Weighted Graphs and Generalizations

Let $G = (V, E, W)$ be a weighted graph—possibly directed—with $W: E \rightarrow \mathbb{R}_+$. Weighted degree centrality of a node $v \in V$ is defined as: $C_D(v) = \sum_{u : (v, u) \in E} W(v, u)$ In directed graphs, two variants exist:

• Out-degree centrality: $C_{OD}(v) = \sum_{u : (v,u) \in E} W(v,u)$
• In-degree centrality: $C_{ID}(v) = \sum_{u : (u,v) \in E} W(u,v)$

For undirected graphs, each edge’s weight is counted once per endpoint. In applications such as freight transportation, total flow (both inbound and outbound) is aggregated: $$\mathrm{WD}(i) = \sum_{j \in N_{\mathrm{in}}(i)} \theta_{ji} + \sum_{j \in N_{\mathrm{out}}(i)} \theta_{ij}$$ where $\theta_{ji}$ and $\theta_{ij}$ represent inbound and outbound flows, respectively (Sharmin et al., 31 Dec 2025).

Generalizations to interval-weighted networks (IWN) introduce closed intervals for edge weights, $w_{ij} = [\underline{w}_{ij}, \overline{w}_{ij}]$, yielding interval-valued degree centralities: $$\text{IWD}(i) = \left[ \sum_j \underline{w}_{ij},\; \sum_j \overline{w}_{ij} \right]$$ A tunable version, $\text{IWD}^{(\alpha)}(i)$, interpolates between binary and sum-based degree (Alves et al., 2021).

2. Algorithmic Computation Procedures

In standard settings, computation is linear in the number of incident edges, $O(|E|)$. Each node $v$ sums $W(e)$ for $e$ adjacent to $v$. In interval-weighted contexts, the sum produces two endpoints for each node:

• For each $i \in V$, compute $L_i = \sum \underline{w}_{ij}$ and $U_i = \sum \overline{w}_{ij}$.
• Apply $\alpha$-tuning by raising endpoints to exponent $\alpha$ and scaling by $k_i^{1-\alpha}$, with $k_i$ the binary degree.

Ranking by interval-valued centrality is handled via order extensions (e.g., midpoint-then-radius criteria).

Simulation frameworks for network disruption employ static initial WDC computation, sequential node ranking, and progressive intervention—scaling incident edge weights to model node degradation (Sharmin et al., 31 Dec 2025).

3. Stability, Robustness, and Continuity Properties

Weighted degree centrality exhibits mathematical stability under weight perturbations. Let $d_{(V,E)}(G,H) = \sum_{e \in E} |W_G(e) - W_H(e)|$ define a metric between graphs with common topology but perturbed weights. The following holds (Segarra et al., 2014): $|C^G_D(v) - C^H_D(v)| \leq d_{(V,E)}(G,H)$ This tight bound (stability constant $K_G=1$) implies WDC is maximally stable: small changes in weights yield proportionally bounded changes in node scores. Empirically, numerical tests on random and real-world weighted networks confirm that WDC rankings are minimally displaced under edge noise, consistently outperforming closeness, eigenvector, betweenness, and even stabilized variants of betweenness centrality.

This robustness ensures preservation of top-$k$ rankings with high probability, even under moderate noise, making WDC suitable where edge weights are estimated or subject to error.

4. Extensions to Interval-Weighted Networks

Interval-Weighted Degree centrality (IWD), as formalized in (Alves et al., 2021), extends classical measures to settings with uncertain or temporally aggregated edge weights:

• For each node, the centrality is an interval $[\text{lower bound}, \text{upper bound}]$ capturing extreme scenarios.
• The tunable $\alpha$ generalizes between pure topology ($\alpha=0$, yielding $[k_i, k_i]$) and pure strength ($\alpha=1$).
• Algorithmic complexity remains $O(|E|)$.

Illustrative examples span both synthetic (toy graphs) and real-world mobility and trade networks. Intervals encode uncertainty (e.g., commuter flows, yearly trade) and allow immediate nesting of the classical strength metric if weight intervals collapse to points.

Ordering of intervals for ranking central nodes requires care; midpoint-radius schemes are typical but yield only partial orderings.

5. Practical Applications in Infrastructure and Transport Resilience

Weighted degree centrality underpins targeted intervention strategies in critical infrastructure networks. In intermodal freight systems, nodes ranked by WDC (flow-weighted total tonnage) are identified as vulnerability points (Sharmin et al., 31 Dec 2025). Simulation results show:

• Sequential partial degradation of high-WDC nodes produces sharper drops in network efficiency than betweenness- or random-targeted removals.
• For the U.S. freight network, reducing the capacity of top WDC nodes to 60% functionality impaired global throughput more than fully eliminating path-bottlenecks, demonstrating outsized functional impact of large-volume hubs.
• Structural resilience indicators (giant connected component size) confirm WDC-based removals fragment networks faster than unweighted degree-based selections.

A plausible implication is that flow-weighted centrality provides a more direct proxy for system-wide vulnerability than measures based solely on topology or shortest-path positions.

6. Limitations and Open Challenges

WDC inherits certain limitations that merit methodological caution, particularly for advanced analyses:

• Ranking via interval-valued scores is coarser than with real-valued centralities, necessitating explicit ordering schemes or interval order extensions.
• Algebraic operations (e.g., inverse, subtraction) complicate translation of IWD to path- or cost-based centrality measures.
• Endpoint sensitivity in intervals can lead to inaccurate rankings in cases of highly skewed uncertainty; percentile or trimmed interval approaches may mitigate excessive pessimism or optimism.
• For certain global connectivity tasks, WDC may underemphasize pathway criticality vis-à-vis betweenness, necessitating combined strategies.

7. Domain-Specific Applications and Guidelines

Weighted degree centrality, and its interval-weighted and flow-weighted generalizations, are applicable across domains where edge weights reflect empirical quantities subject to variability or error:

• Air transport connectivity (seat-sharing graphs, as in (Segarra et al., 2014)).
• Regional mobility (commuter flows (Alves et al., 2021)).
• International trade (annual export/import variability (Alves et al., 2021)).
• Freight infrastructure and supply chain resilience (Sharmin et al., 31 Dec 2025).

Practical guidelines suggest preferring WDC in noisy or uncertain data environments, prioritizing local connectivity for immediate network-level assessments, and supplementing with alternative measures when global path-based influence is critical.

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Summary Table: Centrality Definitions and Key Features

Measure	Mathematical Definition	Key Stability/Uncertainty Feature
Weighted Degree Centrality	$\sum_{u : (v,u) \in E} W(v,u)$	Stable ($K_G=1$); robust to noise
Interval-Weighted Degree (IWD)	$$\left[ \sum_j \underline{w}_{ij}, \sum_j \overline{w}_{ij} \right]$$	Encodes uncertainty; interval ranking
Flow-Weighted Degree	$$\sum_{j \in N_{\mathrm{in}}(i)} \theta_{ji} + \sum_{j \in N_{\mathrm{out}}(i)} \theta_{ij}$$	Critical for functional resilience

Weighted Degree Centrality combines computational tractability, resilience to weight perturbations, and capacity for uncertainty quantification, making it foundational for robust network analysis in empirical, infrastructure, and uncertain environments.