Degree Centrality-Driven Node Removal

• Degree centrality-driven node removal is a network strategy that targets nodes with the highest direct connections to disrupt overall connectivity.
• The method ranks nodes by degree and adjusts removal priorities based on cost constraints, revealing shifts in vulnerability across network topologies.
• Its applications include epidemic control, infrastructure protection, and network resilience design, offering practical insights for real-world disruptions.

Degree centrality-driven node removal refers to the strategy of identifying and eliminating nodes from a network in decreasing order of their degree centrality, with the goal of maximally impairing network structure or function. This approach, widely examined in network science, forms the basis for both theoretical analyses of network robustness and many real-world strategies for disruption, immunization, or vulnerability assessment across biological, infrastructure, and digital domains. Degree centrality, while a simple local measure, has profound implications for connectivity and vulnerability—especially in networks with heterogeneous degree distributions.

1. Foundations of Degree Centrality-Driven Node Removal

Degree centrality, defined as the number of direct connections (edges) a node has, ranks nodes solely on their immediate neighborhoods. In node removal strategies, nodes with the highest degree are targeted first:

$c_{i}^{DC} = \frac{\text{deg}(i)}{N-1}$

where $\text{deg}(i)$ is the degree of node $i$ and $N$ is the total number of nodes.

This approach is motivated by the central role that high-degree "hub" nodes often play in maintaining network connectivity and facilitating flows (data, energy, infection, etc.) across various kinds of networks. Removal of such nodes tends to have a disproportionately large impact on network integrity, compared to random node loss.

2. Cost-Constrained and Intelligent Attacks

The effectiveness and optimization of degree centrality-based removal are highly contingent on the cost structure assigned to node elimination. The fundamental result is that, if the cost to remove a node increases rapidly with its degree, the classical vulnerability of scale-free networks to targeted attacks may be reversed (1005.4283).

The general cost model considered is:

$\psi(\xi, k) = \kappa (1 - \xi) \phi(k)$

where $\xi$ denotes the fraction of node functionality removed, $k$ is node degree, $\phi(k)$ is a non-decreasing cost function (with special choices like $\phi(k) = k^\zeta (k-1)$), and $\kappa$ normalizes the total budget. The attacker, subject to $N^{-1} \sum_{i} \psi(\xi_i, k_i) \leq 1$, seeks to minimize the network's process integrity $\Gamma[p, q]$.

The optimal attack protocol is to:

1. Rank degrees according to the damage-per-cost ratio $\frac{k(k-1)}{\phi(k)}$.
2. Target degrees maximizing this ratio, exhausting resources in order.
3. If $\phi(k)$ is constant or grows weakly, high-degree nodes are removed first.
4. For rapidly increasing $\phi(k)$ (e.g., $\sim k^2$), the attack shifts away from hubs to intermediate or even low-degree nodes.

This framework highlights that network vulnerability hinges not only on degree heterogeneity, but crucially on the attacker's resource constraints and the cost scaling with degree.

3. Network Topology and Robustness Under Removal

Network resilience to degree-centrality-driven node removal depends on the interplay between the degree distribution and node removal cost. Several patterns emerge:

• Power-law (scale-free) networks are highly robust to random failure but, under degree-independent removal costs, fragile to hub-targeted removal. When node removal costs scale rapidly with degree, power-law networks become more resilient—even outperforming homogeneous (Poissonian) topologies (1005.4283).
• Phase transitions in network structure can be induced by node removal. For example, in generalized preferential attachment models with random node deletions, a sufficiently high removal rate or randomization of attachment leads to a shift from a scale-free (power-law) to an exponential (homogeneous) degree distribution (1201.4044). In the exponential regime, effective hubs disappear, reducing attack efficacy (see summary table below).

Regime	Degree Distribution	Implication for Attacks
Power-law, low removal	Scale-free	Hubs exist, attacks effective
Exponential, high removal	Exponential (no hubs)	Degree-based attack much less effective

Under general, sequential node removal with a tunable hub-bias parameter $\theta$, the transition of the remnant network degree distribution from power-law to Poisson is quantitatively mapped via relative entropy; only sufficiently strong and extensive hub-targeting ($\theta \ll 0$, large $f$) leads to loss of scale-free structure (2205.03887).

4. Vulnerability Assessment and Network Function

The impact of degree centrality-driven node removal has been quantified using several measures:

• Connectivity Breakdown: Rapid reduction in the size of the giant connected component and fragmentation into smaller subgraphs is typical, particularly in heterogeneous degree networks (1312.4707, 2303.16596).
• Flow Disruption: In practical networks (e.g., ISP topologies), removal of high-degree nodes decreases aggregate network flow and increases fragmentation nearly as effectively as attacks guided by computationally intensive global metrics (betweenness, PageRank) (1312.4707).
• Process Integrity: Process-specific measures, such as ability to support communication or metabolic flow, are minimized directly by targeting nodes with the best damage-per-cost tradeoff.

Empirical studies show that, in real ISP and infrastructure networks, degree centrality-based removal often matches or approximates more sophisticated global methods both in degrading flow capacity and overall connectivity (1312.4707). The destructive power of degree-based attacks is especially pronounced when centrality indices show high overlap at the top-ranked nodes.

5. Comparative Effectiveness and Limitations

While degree centrality-guided removal is highly effective, especially in hub-dominated networks, its limitations become apparent when:

• Node removal costs increase rapidly with degree (thus, high-degree nodes are protected).
• The network topology is engineered or evolved to mitigate the structural role of hubs (e.g., modular core-periphery with few high-degree nodes).
• Viral, cascading, or function-specific processes are strongly shaped by nonlocal structural properties rather than direct connectivity.

Optimized attack strategies that incorporate cost, global connectivity, or redundancy may outperform naïve degree-based approaches. For example, spectral methods using Laplacian one-norm minimization or bridge-based strategies targeting neighborhood fragmentation can yield much greater disruption per node removed under certain circumstances (1403.2024, 2309.16197).

6. Applications and Design Consequences

Degree centrality-driven node removal underpins diverse applications:

• Epidemic Control: Targeting or immunizing highest-degree nodes is theoretically optimal for fragmenting the giant component and restricting large disease outbreaks in configuration-model networks (2303.16596).
• Network Defense and Infrastructure Planning: Understanding the conditions under which degree-based removal is most/least effective informs hardening and redundancy investment, especially when hub nodes are costly or impractical to protect.
• Robust Network Design: Network designers can leverage insights from degree distribution and removal cost structure to select architectures (e.g., core-periphery, modular, or regular) best suited for anticipated attack/defense scenarios (1005.4283).

Node Removal Cost $\phi(k)$	Optimal Attack	Resilient Network (Under Cost)
constant	Target hubs	Poissonian/homogeneous
linear ($\sim k$)	Highest-degree first	Some benefit from fat tails
superlinear ($\sim k^2$)	Lower/intermediate $k$	Power-law with few protected hubs

7. Theoretical and Empirical Insights

Recent work rigorously characterizes the structural transition induced by sequential degree-targeted removals, providing explicit formulae and bounds (for example, on giant component size) for random graphs with specified degree distributions (2303.16596). Analytical frameworks leveraging local convergence, configuration models, and relative entropy allow precise predictions of network disintegration pathways and identify the sharp distinction between targeted and random failures.

A key conclusion is that the canonical fragility of scale-free networks to degree-based targeted attacks is contingent; only if attackers can afford the cost of removing hubs does this vulnerability materialize. Otherwise, degree heterogeneity can confer superior resilience.

References Table: Cost Scaling and Robustness

Node Removal Cost Structure	Targeting Optimal?	Resilience Mechanism	Reference
Constant	Hubs targeted, effective	Degree-homogeneity preferred	(1005.4283)
Degree-dependent (linear)	Still hub-focused	Fat-tailed, core–periphery	(1005.4283)
Rapidly increasing (quadratic)	Attack shifted to lower $k$	Hubs effectively protected	(1005.4283)

Conclusion

Degree centrality-driven node removal remains a cornerstone of network vulnerability analysis, with its destructive impact determined by the interplay between degree distribution, node removal cost, and resource constraints. Analytical and empirical results collectively demonstrate that degree-based attacks are devastating in hub-dominated networks under low attack cost, but this is neither universal nor inevitable. When node protection scales appropriately, or when network topology is adapted, vulnerability may be sharply reduced or even inverted, positioning scale-free architectures as optimal in constrained adversarial settings. Practitioners must account for the cost structure, resource limits, and process-specific network requirements to accurately assess and mitigate risks associated with targeted node removal.