Latent Geometry-Driven Network Automata for Complex Network Dismantling

1. Introduction

Complex networks are the backbone of our modern world, from the biological pathways within a cell to global financial and transportation systems (Newman 2003). While the interconnected nature of these systems is often a source of efficiency and strength, it also introduces profound vulnerabilities. A localized failure can be absorbed, or it can trigger a cascade of disruptions leading to a systemic collapse. Understanding this fragility is crucial, as the consequences are far-reaching: targeted disruptions can compromise cellular function in metabolic networks, dictate the spread of a virus through a social fabric, or cause catastrophic blackouts in power grids and failures in financial markets (Albert et al. 2000; Artime et al. 2024). The formal study of these vulnerabilities is known as network dismantling. It addresses a fundamental question: what is the most efficient way to fragment a network by removing a minimal set of nodes or links, to disrupt its structural integrity and functional capacity? Answering this question is essential not only for predicting the impact of malicious attacks but, more importantly, for designing robust and resilient systems that can withstand them. The task of dismantling serves a dual purpose. It determines whether a system is robust and how to reinforce desirable networks, for example preventing system failure in a flight network or security compromises in internet infrastructure. Conversely, it reveals how to disrupt undesirable systems, severing communications in terrorist cells or halting the spread of an epidemic. Efficient network dismantling is challenging because identifying the minimal set of nodes for optimal disruption is an NP-hard problem: no known algorithm can solve it efficiently for large networks (Artime et al. 2024), forcing the field to rely on heuristic approximations. This difficulty arises not only from the prohibitively large solution space but also from the structural complexity of real-world networks, which exhibit heterogeneous, fat-tailed connectivity (Barabási & Albert 1999; Broido & Clauset 2019; Serafino et al. 2021; Voitalov et al. 2019), modular and community structures (Newman 2012), hierarchies (Clauset et al. 2008; Ravasz & Barabási 2003), higher-order structures (Battiston et al. 2021; Lambiotte et al. 2019), and a latent geometry (Boguñá et al. 2021; Krioukov et al. 2010; Muscoloni et al. 2017; Serrano et al. 2008; Wu et al. 2015).

Node Betweenness Centrality (NBC) is a network centrality measure (Freeman 1977) that quantifies the importance of a node in terms of the fraction of the shortest paths that pass through it. NBC-based attack, where nodes are removed in order of their betweenness centrality, is considered one of, if not the best, method for network dismantling (Engsig et al. 2024; Holme et al. 2002; Motter & Lai 2002; Servedio et al. 2025). However, like many other dismantling techniques, it requires global knowledge of the entire network topology, and its high computational cost limits its scalability to large networks. These limitations are shared by many other state-of-the-art dismantling methods, which additionally rely on black-box machine learning models, and are rarely validated across large, diverse sets of real-world networks (see Table 1, Table A4, and Table A3).

Latent geometry has been recognized as a key principle for understanding the structure and complexity of real-world networks. Recent works in network science suggest that the latent geometry of complex networks could explain critical network characteristics such as small-worldness, degree heterogeneity, clustering, and navigability, and drives critical processes like efficient information flow (Boguñá et al. 2009, 2021; Kleinberg 2000; Krioukov et al. 2010; Muscoloni & Cannistraci 2018a, 2019; Serrano et al. 2008; Wu et al. 2015). Work by Muscoloni et al. (2017) revealed that betweenness centrality is a global latent geometry estimator: it approximates node distances in an underlying geometric space. They also introduced Repulsion-Attraction network automata rule 2 (RA2), a local latent geometry estimator that uses only first-neighbor connectivity. RA2 performed comparably to NBC in tasks such as network embedding and community detection, despite relying solely on local information. This raises the first question: can latent geometry, whether estimated globally or locally, guide effective network dismantling? If complex systems run on a latent manifold, estimating it may offer a more efficient way to disrupt connectivity. The second question concerns efficiency. While both NBC and RA2 have $O\left(Nm\right)$ complexity (N as the number of nodes and m the number of links), RA2 is significantly faster in practice because its local computations avoid NBC’s large computational overhead. This motivates exploring whether local latent geometry estimators can match the dismantling performance of global methods like NBC while offering lower running time.

Motivated by these questions, we introduce the Latent Geometry-Driven Network Automata (LGD-NA) framework. Our first and primary contribution is the principle of (1) Latent Geometry-Driven (LGD) dismantling, where methods estimate effective node distances on a network’s latent manifold to expose critical structural information. Specifically, our (2) LGD-NA framework uses local network automata rules to approximate these geometric distances; a node’s summed distance to its neighbors estimates how critical it is for dismantling. Within this framework, we discovered that a (3) simple common neighbors-based rule, which we term Common Neighbor Dissimilarity (CND), is highly effective, achieving performance close to the state-of-the-art method, NBC. We prove the effectiveness of our approach through (4) comprehensive experimental validation on an ATLAS of 1,475 real-world networks across 32 complex systems domains, the largest and most diverse collection to date, showing that LGD-NA consistently outperforms all other existing dismantling algorithms, including machine learning and spectral Laplacian-based methods. To enable dismantling at large scales, we implement (5) GPU-acceleration for LGD-NA, yielding remarkable running time advantages over methods like NBC. Finally, using the explainability of our CND measure, we introduce a new method for (6) engineering network robustness, substantially reducing the effectiveness of the best dismantling methods. We further validate the practical utility of our dismantling framework and robustness engineering method by demonstrating their impact on domain-specific functional metrics, including neuronal firing rates in the Drosophila Connectome, flight map efficiency, epidemic sizes, and communication reachability in terrorist cells.

2. Related Work

• Latent Geometry of Complex Networks.

Many real-world networks are shaped by latent geometric manifolds of the complex systems that govern their topology and dynamics. These hidden geometries explain essential structural features such as small-worldness, degree heterogeneity, clustering, and community structure (Boguñá et al. 2021; Muscoloni & Cannistraci 2018a, 2018b; Muscoloni et al. 2017; Serrano et al. 2008; Wu et al. 2015; Zuev et al. 2015). The underlying metric space is not only descriptive but functional: it facilitates efficient routing and navigation with limited global knowledge (Boguñá et al. 2009; Kleinberg 2000; Krioukov et al. 2010; Muscoloni & Cannistraci 2019). Such properties emerge consistently across diverse systems, including biological, social, technological, and socio-ecological networks (Boguñá et al. 2021; Wu et al. 2015). Latent geometries also enable predictive modeling of dynamical processes such as network growth (Muscoloni & Cannistraci 2018a, 2018b; Papadopoulos et al. 2012), and epidemic spreading (Brockmann & Helbing 2013).

• Latent Geometry Estimators.

Latent geometry estimators assign edge weights to approximate linked nodes’ pairwise distances in the hidden geometric manifold. Among them, network automata rules based on the Repulsion-Attraction (RA) criterion use only local topological information to infer proximity in the latent space (Muscoloni et al. 2017). RA is grounded in the theory of network navigability (Boguñá et al. 2009), which posits that nodes with many non-overlapping neighbors tend to occupy distant regions in the latent space. Edges between such nodes receive higher dissimilarity scores due to strong repulsion, while those with many common neighbors are scored lower due to attraction. RA1 and RA2 are network automata rules for approximating linked nodes’ pairwise distances on the latent manifold of a complex network. These rules are categorized as network automata because they adopt only local information to infer the score of a link in the network without the need for pre-training of the rule. Note that RA1 and RA2 are predictive network automata that differ from generative network automata, which are rules created to generate artificial networks (Barabási & Albert 1999; Muscoloni & Cannistraci 2018b; Papadopoulos et al. 2012). They were introduced to serve as pre-weighting strategies for approximating angular distances associated with node similarities in hyperbolic network embeddings. RA2 performed slightly better than RA1, so for this reason we will only consider RA2 in this study. RA2 defines dissimilarity between nodes i and j as:

$$\mathrm{RA}2(i,j)={\displaystyle \frac{1+e_i+e_j+e_i·e_j}{1+{\mathrm{CN}}_{ij}}}.$$

where ${\mathrm{CN}}_{ij}$ is the number of common neighbors of nodes i and j, and $e_i$ and $e_j$ are the external degrees of i and j, representing the count of neighbors of i and j that are not involved in the common neighbors interactions. In the same work, Muscoloni et al. also showed that betweenness centrality is a global latent geometry estimator. By comparing it with RA2, they demonstrated that both global (betweenness centrality) and local (RA) estimators can effectively capture latent geometry, achieving strong results in network embedding and community detection. See Table A1 for a comparison of estimators and Figure A1 for illustrative examples. See also Appendix C, where we validate the ability of these latent-geometry estimators in identifying node importance and estimate link distances in networks with a known geometry.

• Topological centrality measures.

Degree, betweenness centrality, and their variants have all been used in the majority of dismantling studies (Artime et al. 2024), with betweenness centrality having been found to be the most effective strategy when applying dynamic dismantling, meaning the scores are recomputed after every step. Degree centrality ranks nodes by their number of neighbors, and betweenness centrality (Freeman 1977) counts how frequently a node lies on shortest paths. Other centrality variants include eigenvector centrality (Bonacich 1972), which gives higher scores to nodes connected to other influential nodes. PageRank (Page et al. 1999), based on a random walk model, favors nodes that receive many and high-quality links. Beyond these classical measures, several centrality indices have been developed specifically to capture aspects of network resilience. Fitness centrality (Servedio et al. 2025), adapted from economic complexity theory, evaluates node importance through the capabilities of neighbors while penalizing connections to weak nodes. DomiRank (Engsig et al. 2024) centrality models a competitive dynamic in which nodes gain or lose dominance, or importance, based on the relative strength of their neighbors. Resilience centrality (Zhang et al. 2020), derived from a dynamical systems reduction, quantifies how a node’s removal alters the system’s resilience. See Table A3 for more information.

• Statistical and Machine Learning Network Dismantling.

We focus on network dismantling for targeted attacks, where the goal is to fragment a network as efficiently as possible by removing selected nodes. Message passing-based methods such as Belief Propagation-guided Decimation (BPD) (Mugisha & Zhou 2016) and Min-Sum (MS) (Braunstein et al. 2016) use message-passing algorithms to decycle the network and then fragment the resulting forest with a tree-breaker algorithm, while CoreHD (Zdeborová et al. 2016) achieves decycling by iteratively removing the highest-degree nodes from the 2-core of the network and also includes a tree-breaker algorithm. Decycling and dismantling are, in fact, closely related tasks, as a tree (or a forest) can be dismantled almost optimally (Braunstein et al. 2016). Generalized Network Dismantling (GND) (Ren et al. 2019) targets nodes that maximize an approximated spectral partitioning. Collective Influence (CI) (Morone et al. 2016) targets nodes with maximal influence on their neighborhoods, and Explosive Immunization (EI) (Clusella et al. 2016), uses explosive percolation dynamics. Machine learning-based methods include Graph Dismantling with Machine Learning (GDM) (Grassia et al. 2021), which trains graph neural networks to predict optimal attack strategies in a supervised manner. FINDER (Fan et al. 2020b) uses reinforcement learning instead to autonomously learn dismantling strategies without needing labeled data. CoreGDM (Grassia & Mangioni 2023) combines ideas from CoreHD and GDM as it attacks the 2-core of the network but uses machine learning models trained on optimal dismantling solutions to guide node removal. See Table A4 for more information.

3. Latent Geometry-Driven Network Automata

We introduce the Latent Geometry-Driven Network Automata (LGD-NA) framework. LGD-NA adopts a parameter-free network automaton rule, such as RA2, to estimate latent geometric linked node pairwise distances and to assign edge weights based on these geometric distances. Then, it computes for each node its network centrality as a sum of the weights of adjacent edges. The higher this sum, the more a node dominates numerous and far-apart regions of the network, becoming a prioritized candidate for a targeted attack in the network dismantling process. This prioritized node is then removed from the network, and the procedure is iteratively repeated until the network is dismantled. See Figure 1 for a full breakdown of the LGD-NA framework.

3.1. Latent Geometry-Driven Dismantling

Our first contribution is Latent Geometry-Driven (LGD) dismantling, where any function can be used to estimate edge weights that represent effective distances between nodes, capturing the network’s underlying latent geometry. These inferred weights are used to construct a dissimilarity-weighted network, encoding a hidden geometric structure beneath the observable topology and allowing the dismantling process to prioritize nodes according to their geometric centrality in the latent manifold. Latent geometric structures have been shown not only to explain key properties of complex networks, but also to support the understanding of dynamical processes such as navigation, routing, and epidemic spreading. Building on the idea that network geometry captures essential structural and dynamical properties of complex systems, LGD dismantling is guided by a geometric intuition about how nodes connect distant regions in the latent space. If two nodes are connected to many different nodes but have little overlap in their neighborhoods, they are likely to be far apart in the network’s latent space. An edge between them, therefore, connects distant regions of the network. A node that has many such edges is central to holding the network together, as it links otherwise separate areas. We propose that removing those geometrically central nodes is an effective way to fragment the network. Muscoloni et al. (2017) also offered evidence that betweenness centrality can be used as a latent geometry estimator, hence, NBC is a global topology centrality measure which can be used for latent geometry-driven dismantling.

3.2. LGD Network Automata

Our second contribution is the introduction of a parameter-free network automaton framework for LGD dismantling. In this framework, node importance is estimated by aggregating edge geometric weights into node strengths, and the network is dismantled iteratively by removing the nodes with the highest strength and all their edges. The underlying intuition is that nodes that connect to many external, non-overlapping regions are geometrically central and thus more structurally important, leading to higher strength values. Formally, we begin with an undirected, unweighted network without isolated components. A network automaton rule, such as RA2, that is able to adopt local topology to estimate latent geometry, is applied to assign a weight ${\nu }_{ij}$ to the edge between node i and node j, representing the estimated geometric distance between the two nodes. We get a dissimilarity-weighted network from these edge weights. The strength $s_i$ of node i is then calculated by summing the geometric weights of all its edges, that is, the weights to all its neighbors in the set $\mathcal{N}\left(i\right)$:

$$s_i=\sum _{j\in \mathcal{N}\left(i\right)}{\nu }_{ij}.$$

In this paper, we adopt three types of LGD network automata rules. The first rule is RA2, which was proposed by Muscoloni et al. (2017) for hyperbolic network embedding purposes. The second rule is proposed in this study as an ablation test of the RA2 rule. It is the denominator of the RA2, which we call common neighbors dissimilarity (CND), defined as:

$${\nu }_{ij}\to \mathrm{CND}(i,j)={\displaystyle \frac{1}{1+{\mathrm{CN}}_{ij}}}.$$

where ${\mathrm{CN}}_{ij}$ is the number of common neighbors between nodes i and j. Here, the lower the number of common neighbors two interacting nodes have, the more geometrically distant they are, and thus a higher edge weight is assigned between these two nodes. The rationale for proposing a network automaton rule based only on the common neighbors denominator term of RA2 is to account for the mere attraction between a node and its neighbors. Neglecting the repulsion part associated with the external links (the numerator of RA2) makes sense in a dismantling task because any time we compute the common neighbors of a seed node with one of its neighbors, we indirectly account for the exclusion of nodes that are not in the topological neighborhood of the seed node. For completeness, we also investigate a third rule as an ablation test of RA2 in which we consider only the external links term in the RA2 numerator, expecting that the mere RA2 numerator should also work, but not as well as the common neighbor-based denominator. Indeed, a previous study offers evidence that common neighbors are among the topological features most associated with community organization and mesoscale network geometry (Bianconi et al. 2014).

4. Experiments

4.1. Evaluation Procedure

We evaluate all dismantling methods using a widely accepted procedure in the field of network dismantling (Artime et al. 2024). For each method, nodes are removed sequentially according to the order it defines. After each removal, we track the normalized size of the Largest Connected Component (LCC), defined as the ratio $\mathrm{LCC}\left(x\right)/\left|N\right|$, where $\left|N\right|$ is the total number of nodes in the original network and $\mathrm{LCC}\left(x\right)$ is the number of nodes in the largest component after x removals. This process continues until the LCC falls below a predefined threshold. A commonly accepted threshold in dismantling studies is 10% of the original network size. To quantify dismantling effectiveness, we compute the Area Under the Curve (AUC) of the LCC trajectory throughout the removal process, which records the normalized LCC size at each step. A lower AUC indicates a more efficient dismantling, as it reflects an earlier and sharper disruption of network connectivity. The AUC is computed using Simpson’s rule. See Figure A2, Figure A6, Figure A10, and Figure A15 for visual illustrations of the LCC curve.

4.2. Optional Reinsertion Step

After reaching the dismantling threshold, we optionally perform a reinsertion step to reduce the dismantling cost, defined as the number of removals. Nodes are sequentially reinserted back into the network, one at a time, until the LCC of the remaining network just meets or exceeds the predetermined dismantling threshold. Reinsertion can significantly improve dismantling performance; recent work shows that simple heuristics with reinsertion can match or outperform complex algorithms that include reinsertion by default (Fan et al. 2020a). As a result, we enforce two constraints to ensure the reinsertion step does not override the original dismantling method: (1) reinsertion cannot reinsert all nodes to recompute a new dismantling order, and (2) reinsertion must use the reverse dismantling order as a tiebreak. If a method includes reinsertion by default, we also evaluate its performance without reinsertion for a fair comparison. See Table A5 for more information.

4.3. ATLAS Dataset

Our fourth contribution is the breadth and diversity of real-world networks tested in our experiments, demonstrating the generality and robustness of LGD-NA across domains and scales. We build an ATLAS of 1,475 real-world networks across 32 complex systems domains, which is the largest and most diverse collection of real-world networks to date used for testing in network dismantling studies. We first test all methods across networks of up to 5,000 nodes and 205,000 edges without reinsertion ($n=1,296$), and 38,000 edges with reinsertion ($n=1,237$). To assess the practical running time of the best performing methods, we evaluate NBC and RA2 on even larger networks of up to 23,000 nodes and 507,000 edges ($n=1,475$). Current state-of-the-art dismantling algorithms have been evaluated on no more than 57 real-world networks (see Table 1), with most algorithms tested on fewer than a dozen. Our experiments cover 1,475 networks, representing a substantial expansion. A key aspect of our ATLAS dataset is the diversity of network types (see Table 2). We test across 32 different complex systems domains, ranging from protein-protein interaction (PPI) to power grids, international trade, terrorist activity, ecological food webs, internet systems, brain connectomes, and road maps. Since fields vary in both the number of networks and their characteristics, we evaluate dismantling methods using a mean field approach, ensuring that fields with more networks do not dominate the overall evaluation. Also, because dismantling performance varies in scale across fields, we compute a mean field ranking to make results comparable across domains.

4.4. LGD-NA Performance and Comparison to Other Methods

We compare our LGD-NA framework against the best-performing dismantling algorithms in the literature. Main results are visualized in Figure 2, and full quantitative results, including side-by-side comparisons of absolute AUC and mean-field ranks for all methods and fields, are reported in Table A21 through Table A33 in the Appendix.

First, we find that all latent geometry network automata, NBC, RA2, and its variants, achieve top dismantling performance, both with and without reinsertion. These findings show that estimating the latent geometry of a network effectively reveals critical nodes for dismantling, confirming our first contribution. For each method, we evaluate three reinsertion strategies and report the best result. We show in Figure A5 that using different reinsertion methods does not change the mean field ranking of the dismantling methods, and in Figure A7 and Figure A8 that the improvement in performance varies across fields and reinsertion methods (see Figure A6 for an illustrative example). We also adopt a dynamic dismantling process for the network automata rules and all centrality measures, where we recompute the scores after each dismantling step, as it consistently outperforms the static variant (see Figure A9 for an example of the improvements for CND and Figure A10 for an illustrative example). Second, we find that local network automata rules RA2, CND, and ${\mathrm{RA}2}_{\mathrm{num}}$, which adopt only the local network topology around a node, are highly effective. In particular, RA2 and its variants consistently outperform all other non-latent geometry-driven dismantling algorithms, including those relying on global topological measures or machine learning. This confirms our second contribution. See Figure A2 for illustrative examples where the local network automata rules outperform NBC. In addition, Appendix C validates the ability of our latent-geometry-based network automata rules in identifying node importance and estimating latent geometric distances. Third, we find that the simplest RA2 variant, based solely on inverse common neighbors, which we refer to as common neighbor dissimilarity (CND), achieves the best performance among all local network automata rules. This is our third contribution and demonstrates that even minimal local topology-based information can effectively approximate latent geometry useful for effective dismantling. NBC strictly dominates as the top-ranking method across all fields. However, among the second-best performers, the LGD-NA methods lead in the majority of domains: CND ranks second in Internet networks, RA2 in Biomolecular and Brain networks, and RA2_num in Covert networks. The only fields where non-LGD-NA methods rank second are Foodweb (Fitness Centrality), Infrastructure (GDM), and Social networks (GND). LGD-NA consistently outperforms all other non-latent geometry-driven dismantling algorithms, including those relying on spectral Laplacian-based methods and machine learning. The only measure that still outperforms LGD-NA is the NBC metric (which is also latent-geometry-driven), applied to dynamic dismantling. These results strongly demonstrate the practical reliability of our latent geometry-driven dismantling framework, LGD-NA.

4.5. GPU Acceleration of LGD-NA for Large-Scale Dismantling

We implement GPU acceleration for all three LGD-NA variants by reformulating the required computations as matrix operations. On large networks, this enables a significant speedup in running time. When comparing RA2 and NBC, on the largest network, GPU-accelerated RA2 is 130 times faster than its CPU counterpart, highlighting the inefficiency of matrix multiplication on CPU. It is also over 63 times faster than NBC running on CPU, thanks to our GPU-optimized implementation. Note that NBC on CPU remains faster than RA2 on CPU, again due to the limitations of CPU-based matrix operations. We report only the CPU running time for NBC, as its GPU implementation did not yield any speedup (see Table A13). While some studies report GPU implementations of NBC with improved performance (Bernaschi et al. 2016; Fan et al. 2017; McLaughlin & Bader 2018; Pande & Bader 2011; Sariyüce et al. 2013; Shi & Zhang 2011), these are often limited by hardware-specific optimizations, data-specific assumptions (e.g., small-world, social, or biological networks), and the use of heuristics that are tailored to specific settings rather than offering general solutions. Moreover, publicly available code is rare, making these approaches difficult to reproduce or integrate. Overall, NBC is not naturally suited for GPU implementation, as it does not rely on matrix multiplication, but is based on computing shortest path counts between all node pairs. Overall, while NBC achieves better dismantling performance, its high computational cost makes it impractical for large-scale use. In contrast, thanks to our GPU-optimized implementation, our local latent geometry estimators based on network automata rules are the only viable option for efficient dismantling at scale. Here, we look at the details of our matrix operations for the LGD-NA measures. First, the common neighbors matrix is computed as

$${\mathbf{CN}}_{\mathrm{L}2}=\mathbf{A}\circ \left({\mathbf{A}}^2\right)$$

where $\mathbf{A}$ is the adjacency matrix and ∘ denotes element-wise multiplication. Here, ${\mathbf{A}}^2$ counts the number of paths of length two (i.e., common neighbors) between all node pairs. The Hadamard product with $\mathbf{A}$ ensures that values are only retained for existing edges. Next, we compute the number of external links a node has relative to each of its neighbors. Given the degree matrix $\mathbf{D}$, the external degree matrix is:

$${\mathbf{E}}_{\mathrm{L}2}=\mathbf{A}\circ \left(\mathbf{D}-{\mathbf{CN}}_{\mathrm{L}2}-\mathbf{A}\right)$$

Each entry $(i,j)$ of ${\mathbf{E}}_{\mathrm{L}2}$ represents the external degree of node i with respect to node j: the number of neighbors of i that are neither connected to j nor directly connected to j itself. Non-edges are zeroed out. These matrices allow efficient construction of RA2 and its variants using only matrix operations. The time complexity is $\mathcal{O}\left(N^3\right)$, with the common neighbor matrix being the dominant operation, for dense graphs, and $\mathcal{O}\left(Nm\right)$ for sparse graphs, N being the number of nodes and m the number of links. On CPU, matrix multiplication is typically memory-bound and limited by sequential operations. GPUs, however, are optimized for matrix operations, leveraging thousands of parallel threads. This results in a substantial speedup when implementing the GPU version. Finally, we show in Appendix J that in controlled settings with nPSO networks the GPU advantage becomes apparent when networks exceed 1,000 nodes or 100,000 edges.

Figure 3. Runtime (in hours) is plotted against network size, measured by the number of edges, E, for dynamic dismantling. The annotated time indicates the runtime for the largest network. Evaluated on networks of up to 23,000 nodes and 507,000 edges ($n=1,475$).

Figure 3. Runtime (in hours) is plotted against network size, measured by the number of edges, E, for dynamic dismantling. The annotated time indicates the runtime for the largest network. Evaluated on networks of up to 23,000 nodes and 507,000 edges ($n=1,475$).

4.6. Leveraging CND Explainability to Engineer Network Robustness

A key advantage of our LGD-NA framework is its explainability. Indeed, we can directly explain why any of our network-automata-based and latent-geometry-driven measures prioritize specific nodes for dismantling. CND, our most performant network automata rule for dismantling, makes this explainability even more straightforward and shows that the vulnerability of a node is strongly related to the number of links its neighbors share with one another. The higher this number, the more common neighbors exist between the adjacent nodes of a vulnerable target node. This means that, to enhance the robustness of the network to the failure of a critical node, we should simply increase the number of links between its adjacent nodes. The strategy is as follows. First, identify the nodes with the highest dismantling scores according to a given measure. Here, we consider NBC, a global shortest-path count-based measure, and CND, a local topology common-neighbor-based network automata measure, because they use different rationales to estimate critical nodes and are the two best-performing measures in this study. Second, for these critical nodes, add new links between their adjacent nodes that are not already connected to each other. Robustness is defined as the ability of a system to continue functioning when subjected to perturbations (Artime et al. 2024). In this initial context, we define attack tolerance, quantified by the LCC AUC, as a robustness measure itself, representing the system’s structural integrity under dismantling attacks. We validate our reinforcement strategy in Table A10 and Figure A11. We clearly show that adding links between the adjacent nodes of the most critical nodes significantly increases the AUC—and therefore the robustness—by 36% to 95% for 1% of added links, and by 59% to 259% for 10% of added links. Remarkably, by reinforcing only the top 1% of nodes, we increase network robustness regardless of the dismantling method used—whether it is our CND or NBC.

4.7. Real-World Applications: Fault Tolerance, Security, and Communications

To demonstrate the practical utility of LGD-NA, we evaluate its performance on four distinct real-world systems using domain-specific functional metrics. First, we use the Drosophila Connectome (Shiu et al. 2024) (Fault Tolerance), where we utilize a Spiking Neural Network (SNN) model of the sugar-sensing circuit. The metric is the sensory neuron firing rate required to trigger the proboscis extension response. Second, the Terrorist Cell (Gutfraind & Genkin 2017) (Security & Communications), where we analyze the network responsible for the 2015 Paris and 2016 Brussels attacks. The metric is Commander Reach, defined as the percentage of operatives able to communicate with at least one of the three key commanders. Third, the Flight Map (Cardillo et al. 2013) (Fault Tolerance), where we measure Global Efficiency ($E_{glob}$). Fourth, a School Contact Network (Mastrandrea et al. 2015) (Security/Epidemics), where we simulate an epidemic using an SEIR model (Anderson & May 1991). The metric is the Final Outbreak Size. Our results in Figure 4 show that dismantling strategies effectively degrade the functional performance across all four systems. In the Drosophila Connectome and Terrorist Cell, we observe particularly sharp drops in performance metrics after removing only a small fraction of nodes ( 5%). We observe a more gradual deterioration in the global efficiency of the Flight Map and the viral spread within the School Contact network. This functional collapse is particularly significant for the two adversarial scenarios (Terrorist Cell and School Contact Network): it confirms that LGD-NA is effective for security and communication disruption, efficiently suppressing epidemic outbreaks and isolating hostile leadership with minimal intervention. We subsequently applied our strategy for engineering network robustness to these four scenarios, demonstrating its effectiveness. As shown in Table A11, the reinforced networks are significantly harder to dismantle, achieving robustness gains of up to 363%. This increased resilience is evident across both our original topological metric (LCC AUC) and the domain-specific functional metrics defined for each case. For the Drosophila Connectome, this analysis informs the resilient and redundant design of fault-tolerant neuromorphic circuits by mimicking its biological wiring (Ham et al. 2021; Suárez et al. 2021). In the Flight Map, it identifies specific hubs where reinforcement prevents systemic failure. Finally, for adversarial networks, our robustness analysis serves a diagnostic purpose when faced with incomplete data. Since social networks, and especially covert ones, often contain unobserved links (e.g., dormant ties or unreported contacts), calculating an empirical robustness ceiling allows us to estimate the margin of error required for successful security operations with partial observability.

5. Conclusion

The first limitation is that hardware constraints precluded testing on extremely large networks, though our results are validated across 1,475 real-world networks across a vast range of disciplines. Second, while practical runtimes could deviate from theoretical expectations, all methods were executed under identical hardware and optimization settings to ensure fair comparison. In addition, we acknowledge the dual-use potential of this research, as understanding network vulnerabilities is critical for both designing targeted attacks and engineering robust defensive strategies. To mitigate this, we proactively demonstrate a constructive application for enhancing network robustness and believe the societal benefit of openly publishing these defensive tools outweighs the risk of misuse. In summary, we introduced Latent Geometry-Driven Network Automata (LGD-NA), a framework that achieves state-of-the-art network dismantling using only local topological information. By applying simple network automata rules to estimate a network’s latent geometry, LGD-NA identifies critical nodes with significant speed advantages over global methods like Node Betweenness Centrality (NBC). Across 1,475 real-world networks and 32 complex systems domains, it consistently outperforms all other dismantling algorithms, including those based on machine learning (e.g., Graph Neural Networks) and spectral Laplacian-based ones. Notably, our minimalistic Common Neighbor Dissimilarity (CND) measure matches NBC’s efficacy while being orders of magnitude faster. Leveraging the explainability of CND, we introduce a novel strategy to engineer network robustness. Crucially, we demonstrate the practical utility of our framework across diverse domains, from informing the design of neuromorphic circuits and reinforcing transport hubs, to disrupting terrorist cells. This work establishes latent geometry as a powerful and efficient principle for both explaining vulnerabilities and engineering stronger networks.