Reply to: Influence of the reinsertion algorithm on the performance of the Vertex Entanglement method

replying to: A. Ghavasieh Communications Physics Matters Arising https://doi.org/10.1038/s42005-025-02429-y (2024)

Empirical networks possess considerable heterogeneity of node connections, leading to a small subset of vertices playing pivotal roles in network structure and function. Consequently, identifying these key vertices represents a critical challenge in network science.

Recent advancements in applying quantum-based methodologies to network analysis have introduced innovative approaches for addressing this challenge. Notably, vertex entanglement (VE)¹ has been proposed, drawing inspiration from the quantum resource theory and network information theory, particularly network entanglement (NE)². Network entanglement, a pioneering concept introduced by Ghavasieh et al., is innovative but encounters several conceptual and practical limitations. In our recently published paper¹, we have extensively acknowledged previous works, particularly that of Ghavasieh et al.², and have endeavoured to clearly distinguish between existing contributions and the novel aspects of our research.

Nomenclature consistency

In the original manuscript¹, we referred to Collective Network Entanglement (CNE) when considering the method of Ghavasieh et al.². The nomenclature Collective Network Entanglement was adopted because Ghavasieh et al.² originally introduced this term to denote the specific case in which the average network entanglement (NE) across all nodes reaches its minimum at a critical hyperparameter value (β = β_c). Thus, using CNE rather than NE accurately emphasizes that our comparisons were conducted under this well-defined condition, ensuring clarity and avoiding ambiguity regarding general NE implementations.

Theoretical difference

Ghavasieh et al.² quantify the importance of a single node x by simply detaching it from the original network G, defining network entanglement as ${M}_{\beta }(x)=[{S}_{\beta }({G}_{x}^{{\prime} })+{S}_{\beta }(\delta {G}_{x})]-{S}_{\beta }(G)$². Here, S_β( ⋅ ) denotes the spectral entropy parameterized by β, ${G}_{x}^{{\prime} }$ represents the network remainder after detaching node x, while δG_x denotes the star network consisting of the perturbed node x and its local connections. It is noted that ${S}_{\beta }(\delta {G}_{x}) = \frac{1}{Z\ln 2}\left[\ln Z\left(1+{e}^{-\beta }({k}_{x}-1)+{e}^{-\beta ({k}_{x}+1)}\right)+\beta {e}^{-\beta }({k}_{x}-1)+\beta ({k}_{x}+1){e}^{-\beta ({k}_{x}+1)}\right]$, where $Z=1+({k}_{x}-1){e}^{-\beta }+{e}^{-\beta ({k}_{x}+1)}$. Therefore, the term S_β(δG_x) = f(k_x, β) depends solely on the node degree k_x and β, introducing computational complexity while failing to capture the network’s global properties.

On the other hand, the extent to which a quantum state is entangled can be characterized by the distance or angle to the nearest unentangled state. This geometric measure of entanglement is well-established in various settings. Building on this, vertex entanglement (VE) provides a more refined entanglement metric for network science, and it is defined as follows

$${E}_{\tau }(v)={S}_{\tau }\left({G}_{v}\right)-{S}_{\tau }(G),$$

(1)

where G_v, referred to as the v-control network, is resulted from the local perturbation of v for the original network G. This definition not only simplifies the computational complexity but also offers greater flexibility by allowing a variety of perturbation strategies without excessively disrupting the network structure. Furthermore, vertex entanglement aligns more closely with the definition of quantum entanglement in its mathematical form.

Technical assessment

We show the evolution of the Largest Connected Component (LCC) and Second Largest Connected Component (SLCC) during the dismantling processes in Fig. 1. Notably, the reinsertion procedure³ is an integral component of the VE-based network dismantling framework. To further clarify and quantify the results, we report the positions and sizes of the SLCC peaks in Table 1. VE generally reaches the SLCC peak earlier than NE variants, except in the PH network. Although NE_mid peaks first in this case, its low SLCC value (20) indicates that the network was not effectively fragmented into comparable-sized components.

Fig. 1: Dismantling process of different methods.

a The size of the largest connected component (LCC), also referred to as the giant connected component (GCC) in some references, is commonly used as a key measure for evaluating network robustness under attacks. Compared with the other methods, the number of vertices attacked and the area beneath the curve for vertex entanglement (VE, red line) are comparably small, indicating superior dismantling performance. Panel (b) illustrates how the size of the second-largest connected component (SLCC) evolves during the dismantling process. Generally, the peak of the SLCC marks the critical dismantling point where the network transitions from a single large component to multiple smaller ones. Typically, an earlier peak indicates better dismantling performance.

Table 1 Peak position (P. Pos.) and corresponding peak size (P. Size) of the second largest connected component (SLCC)

Analytical analysis from Ghavasieh et al.² shows that the network entanglement ${M}_{\beta }(x)\approx {\log }_{2}({k}_{x}+1)$ when β → 0 and ${M}_{\beta }(x)\approx \beta \bar{k}{\log }_{2}{C}_{x}^{{\prime} }$ when β → ∞, indicating that NE_small and NE_large are not novel metrics to some extent. Therefore, we prioritized NE_mid and compared it to the VE in the original manuscript. Ghavasieh et al.² provide a computational method in which they claim that the average NE of all nodes in the network reaches its minimum at a specific value of β = β_c, which they define as collective network entanglement (CNE). Their study further suggests that β_c serves as a suitable characteristic scale for network dismantling. Accordingly, in the original manuscript, we employed NE with β = β_c for comparisons and retained the term collective network entanglement (CNE) as originally introduced. Furthermore, to balance theoretical rigor with experimental performance, we extended the replication of CNE beyond β_c by uniformly sampling 100 values from the intervals (0, 1/λ₂) and (1/λ₂, 10/λ₂), where λ₂ denotes the second smallest eigenvalue of the graph Laplacian. We then selected the β value that minimized the number of attacked nodes as the mid-scale parameter. This approach optimizes NE’s network dismantling performance, directly refuting Ghavasieh et al.’s claim that we reported only the worst-performing curve for NE. It is also worth noting that comparing vertex entanglement (which uses a consistent hyperparameter selection strategy) with network entanglement (which employs varied hyperparameters, namely, NE_small, NE_large, and NE_mid) might be somewhat unfair. Nevertheless, in response to Ghavasieh et al.’s concerns, we include these additional benchmarks as requested, as illustrated in Fig. 1.

As for other benchmark reproduction in the original manuscript¹, we strictly adhered to the original methodology in our experiments. For example, MinSum³, GNDR⁴, and BPD⁵ incorporate reinsertion in their methods, whereas NE² does not; we followed the strategy outlined in these original works to ensure methodological consistency. We believe it is essential to respect the original methodology. However, we acknowledge the interest in determining the role of reinsertion on the performance of the methods prompted by the Matters Arising by Ghavasieh et al. To convincingly show the differences in performance between CNE and VE, we now evaluate various versions of NE with reinsertion (termed NE_mid + R and CNE + R) alongside VE. While reinsertion plays a key role in the performance of VE over CNE, the results in Fig. 1 illustrate that VE demonstrates comparable or even superior dismantling performance compared to the benchmark methods, hence confirming the accuracy of our original claims of performance.

Ghavasieh et al. claim that they could not reproduce the results and that VE offers no practical or significant advantage over NE and other state-of-the-art methods. We respectfully disagree with these assessments. The methodology and code of VE are publicly available, and the results are fully reproducible, as demonstrated in Fig. 1. Moreover, a recent comprehensive study on network robustness has reproduced and collected various network dismantling methods, including both NE² and VE¹, further validating the reproducibility of VE and the accuracy of our original NE implementation. Figure 2 carefully compares our original implementations to these publicly available versions. While minor differences naturally arise due to variations in hyperparameter selection, our extensive parameter search confirms that such differences are trivial and do not significantly affect conclusions regarding performance. Notably, the parameters selected for NE in our original manuscript¹ actually demonstrate better dismantling performance than those provided in the repository⁶, further supporting the robustness of our results.

Fig. 2: Comparison of different algorithm implementations.

The solid lines represent results obtained using the implementation provided in the open-source project⁶, while the dashed lines correspond to the implementation used in VE¹. Although differences in hyperparameter selection exist, their impact on dismantling performance is trivial.

In conclusion, while we appreciate the engagement with our work and the critical perspectives offered, we stand by the validity and significance of our contributions. VE is not simply a rebranding of NE but represents a meaningful advancement in the field of network science. We hope this response clarifies our intentions and the scientific basis for our work, and we look forward to continued dialogue and exploration in this important area of research.