Reliability-Guided Trusted Repeater Node Selection in QKD-Enabled Metro Optical Networks Arup Kumar Marik∗ , Basabdatta Palit† , and Sadananda Behera∗ ∗ Dept. of Electronics and Communication Eng., National Institute of Technology Rourkela, Odisha, India
arXiv:2609.18797v1 [cs.NI] 16 Sep 2026
† Dept. of Information Technology, Indian Institute of Engineering Science and Technology, Shibpur, Howrah, India
Abstract—Quantum Key Distribution (QKD) in optical networks can provide information-theoretic security but is limited by its operability range, thereby requiring repeater nodes for extended coverage. Existing works typically assume that all repeater nodes and their associated key management systems are fully trusted, overlooking the risks posed by software exploits and insider threats. To address this, in this work, we propose a Bayesian fusion-based model to quantify the trustworthiness of each node. We have formulated a reliability-guided Trusted Repeater Node (TRN) selection framework for metro optical networks, where each node is assigned a trust score. We have subsequently mapped this trust score to link weights, which in turn has been used for reliable path computation for TRN selection, using the Dijkstra algorithm. We have also ranked the nodes using a composite score combining eigenvector and betweenness centrality, to capture their topographical relevance in the network. Simulation results on a reference topology demonstrate that the proposed method achieves approximately 8.93% higher path coverage than traditional centrality-based approaches like degree centrality, using the same number (around ten) of TRNs, thereby supporting reliable and resilient TRN selection for QKD-enabled optical networks. Index Terms—Quantum Key Distribution, Trusted Repeater Node, Node Reliability, Bayesian fusion model
I. I NTRODUCTION Quantum computing technologies have gained significant attention in recent years owing to their potential to compromise the security of conventional public-key cryptographic algorithms such as RSA [1], whose security relies on the computational hardness of the integer factorization problem. In particular, recent advances in quantum hardware, error correction, and algorithmic optimization have accelerated the practical realization of Shor’s algorithm [2], thereby raising concerns regarding the long-term resilience of classical cryptographic methods. To address these challenges, Quantum Key Distribution (QKD) has emerged as an informationtheoretically secure approach for cryptographic key distribution [3]. However, the practical deployment of QKD over optical fiber networks is constrained by transmission distance limitations, typically extending to only a few hundred kilometers [4], primarily due to signal attenuation and detector inefficiencies. To facilitate long-distance secure communication in the absence of quantum amplifiers, intermediate relay or repeater nodes are employed. These nodes receive, decrypt, and retransmit the secret keys to successive links, enabling the
distribution of quantum keys over longer distances. However, as these relays process the key using electronic systems, they present potential security risks. Hence, ensuring that such intermediary nodes remain secure and trustworthy is critical for preserving the overall confidentiality of the QKD network. Several research efforts have focused on optimizing the placement of Trusted Repeater Nodes (TRNs) in QKDenabled optical networks. Early studies primarily employed graph-theoretic heuristics such as the Minimum Spanning Tree (MST) and Single-Source Shortest Path Tree algorithms for TRN selection [5]. Although computationally simple, these methods often suffer from link congestion, limited resilience, and reduced secure key rates (SKR). A costefficient quantum network design using Steiner-tree-based heuristic algorithm was presented in [6], which minimizes TRN and dark-fiber deployment while outperforming MSTbased approaches in cost efficiency. However, it overlooks node reliability and physical security aspects that are essential for trustworthy network operation. A mixed-integer linear programming (MILP) approach was introduced in [7] to minimize fiber utilization; however, it assumed predetermined and fully trusted TRN locations, limiting its practical applicability. Authors in [8] proposed a span aggregation algorithm to reduce the number of TRNs in optical transport networks, achieving notable fiber cost savings across different topologies. However, this design introduces a trade-off between cost efficiency and SKR, and it does not account for node-level vulnerabilities arising from classical processing in TRNs. In [9], authors formulated the repeater allocation problem as an integer linear program (ILP) that minimizes the number of quantum repeaters while maintaining rate, fidelity, and robustness constraints. Although scalable to realistic network sizes, the ILP approach is computationally demanding and assumes limited end-node activity. A two-step ILP method proposed in [10] reduces computational complexity but continues to rely on static end-node assumptions. More recently, a trafficaware TRN placement strategy [11] based on the Hot-Link algorithm was proposed to adaptively allocate trusted nodes according to SKR demand and asymmetric traffic, improving utilization compared to static schemes. Most of these works have primarily focused on TRN placement in backbone optical networks and, in the existing literature [5]–[8], assume that these relay nodes are entirely
trustworthy [12], an assumption that is often impractical in realistic environments. The trustworthiness or reliability of nodes depends on their physical safeguards, firewall configurations, administrative oversight, and exposure to networkbased threats. Therefore, it is crucial to incorporate node reliability into the design and selection of TRNs to guarantee the end-to-end resilience and security of the QKD infrastructure. To address this gap, the authors in [13] proposed a reliability-aware TRN selection framework for metropolitan area networks by incorporating node reliability into pathweight computation. However, the node reliability values were assigned randomly (0.5–1.0) rather than derived from a reliability model. In addition, the framework relied on multiple tunable parameters for path weighting and TRN ranking, requiring retuning for different network topologies and reliability distributions. In contrast, the present work develops a systematic, simulation-driven reliability modelling framework based on Bayesian fusion method [14], providing a probabilistic approach for estimating node reliability. Rather than assigning reliability values arbitrarily, the proposed model quantifies node trustworthiness by aggregating multiple operational and security indicators into a unified reliability score. The estimated reliability values are then directly integrated into the TRN selection process, eliminating the need for manually tuned weighting parameters. Consequently, the proposed framework enables automated, topology-agnostic TRN selection for QKD-enabled optical networks and can be readily adapted to diverse network configurations without parameter retuning. The key contributions of this work can, therefore, be summarized as follows: 1) We have adopted a Bayesian fusion-based reliability modeling approach to estimate the reliability of nodes in QKD-enabled optical networks by combining multiple operational and security indicators into a unified probabilistic reliability score. 2) We have developed a reliability-aware TRN selection framework that incorporates the estimated node reliability into path-weight computation and combines it with centrality-based ranking to identify secure and reliable TRNs. We have validated the proposed framework on a 28node metro optical network and demonstrate its effectiveness over conventional centrality-based approaches through comprehensive simulation studies. Simulation results demonstrate the effectiveness of the proposed framework through comprehensive performance evaluation using reliability- and connectivity-based metrics. By jointly considering node reliability and structural centrality, the proposed method achieves approximately 8.93% higher path coverage than degree centrality while using the same number (around ten) of TRNs.
II. R ELIABILITY M ODELING U SING BAYESIAN F USION M ETHOD This section introduces the proposed Bayesian fusion-based reliability model designed to quantify the trustworthiness of nodes in Quantum Key Distribution (QKD)-enabled optical networks. We first describe the overall system model, followed by the proposed Bayesian fusion framework, and finally discuss the implementation of the framework. A. System Model The QKD network is represented as an undirected graph G(V, E), where V denotes the set of nodes and E the set of optical fiber links. Each link (u, v) ∈ E connects nodes u and v with a physical distance duv . The objective is to assign a reliability score Rv to each node v ∈ V , which will later be used for reliability-aware TRN selection to rank potential nodes for key relay operation and support secure path computation. In contrast to previous works, we have considered the node reliability to depend on multiple operational and securityrelated parameters, such as: • Patch Latency (PL): Average delay in applying critical updates; lower latency implies better maintenance. • Uptime (UP): Fraction of time the node remains operational; higher uptime denotes greater stability. • Hardware Security (HS): Presence of Trusted Platform Module, secure boot, or hardware encryption. • Physical Access (PA): Degree of physical protection, surveillance, and restricted access. • Firewall Score (FS): Strength of network-level defenses, including intrusion detection/prevention systems and access policies. These indicators collectively form the evidence set {Xvi } for each node v. We have assigned trust scores using a Bayesian fusion-based reliability model that systematically integrates these diverse reliability indicators into a unified probabilistic estimate for each node [14]. B. Bayesian Framework In order to implement Bayesian inferencing, we have modeled each node v as a binary random variable Hv ∈ {H1 , H0 }, where H1 denotes that the node is secure and H0 indicates that it is compromised. The prior probabilities P (H1 ) and P (H0 ) = 1 − P (H1 ) represent the initial belief about a node’s trustworthiness. Here, P (H1 ) is a tunable parameter. For each node, the set of measurable reliability indicators {Xvi }, normalized to the range [0, 1], serves as evidence, i.e., Xv = {Xv1 , Xv2 , Xv3 , Xv4 , Xv5 } = {PL, UP, HS, PA, FS}, (1) These reliability indicators, Xvi ∈ [0, 1], will vary randomly due to some external factors. So, we have modeled Xvi ’s to take random values from a beta distribution, which is suitable for representing bounded uncertainty [15]. Two beta
distributions are defined per indicator, one under the secure hypothesis and another under the compromised hypothesis: P (Xvi |H1 ) ∼ Beta(αi1 , βi1 ),
P (Xvi |H0 ) ∼ Beta(αi0 , βi0 ). (2) The beta distribution parameters were selected to represent realistic secure and compromised node behaviors while maintaining a partial overlap between the two hypotheses. This overlap allows the Bayesian inference process to distinguish trustworthy and vulnerable nodes without producing trivial classifications, thereby yielding meaningful posterior reliability estimates. A minimum reliability value of 0.5 implies that the node behaviour is highly uncertain making it only marginally reliable, while the maximum value 1 implies a fully secure node. In the next section, we explain how we have used the Bayesian inference model to compute the posterior reliability score Rv based on the observed reliability indicators. C. Posterior Reliability Estimation Given the observed set of indicators Xvi for a node v, the posterior probability representing the node’s reliability score is computed using Bayes’ theorem as: Rv = P (H1 |Xv1 , Xv2 , . . . , Xvn ) P (H1 )
n Q
P (Xvi |H1 )
(3)
i=1
= P (H1 )
n Q
P (Xvi |H1 ) + P (H0 )
i=1
n Q
. P (Xvi |H0 )
i=1
Here, the numerator represents the joint likelihood that node v is secure given the observed evidence. The resulting reliability score Rv lies within the interval [0.5, 1], where higher values indicate greater trustworthiness. D. Implementation and Computation The proposed Bayesian reliability framework was implemented in Python using the NumPy and SciPy libraries, and the node reliability was computed sequentially for each node as follows: 1) For simulation purposes, all reliability indicators Xvi ∈ X are assumed to be independent and identically distributed (i.i.d.) random variables generated from a Beta(5,3) distribution 2) Likelihoods P (Xvi |H1 ) and P (Xvi |H0 ), based on the previously computed Xvi values, are obtained using the parameterized beta distributions: Beta(4, 3) for the secure hypothesis (H1 ) and Beta(6, 7) for the compromised hypothesis (H0 ), capturing both reliability and uncertainty in the node behaviour. As mentioned earlier, the shape parameters of these beta distributions have been extensively tuned to ensure that the reliability scores lie between 0.5 and 1. 3) The joint likelihood is obtained by assuming conditional independence among the indicators. 4) The prior probability of a node being secured is uniformly set as P (H1 ) = 0.7 for all nodes. A prior
probability of 0.7 is used because metro optical networks are generally expected to operate securely, while still accounting for the possibility of occasional node failures or security compromises. Because the prior is identical for all nodes, the relative TRN ranking is governed by the node-specific evidence rather than the choice of prior. 5) Finally, Bayes’ theorem is applied to compute the posterior reliability Rv for each node. The computed posterior reliability scores Rv thus provide a quantitative measure of node trustworthiness and are subsequently integrated into the path-weight computation for reliability-aware TRN placement. III. R ELIABILITY-AWARE PATH W EIGHT M ODIFICATION In this section, we first explain how we have incorporated the reliability scores obtained from the proposed Bayesian fusion method into link weights. We then explain the proposed algorithm for ranking the TRNs using betweenness centrality and eigenvector centrality, based on modified Dijkstra’s method. A. Weight Modification with Reliability (Line 2, Algorithm 1) ′ Each link (u, v) is assigned a modified weight wuv based on the physical distance duv and the combined reliability of its endpoints (Ru · Rv ). The link weight is calculated as: ′ wuv =
duv . Ru · R v
(4)
Taking the inverse of (Ru · Rv ) penalizes links between less reliable nodes. The modified weights are used to generate a new graph G′ (V, E). We apply Dijkstra’s algorithm on this new graph to find the shortest path between all the source and destination pairs. Next, to capture the structural significance of each node based on the connections provided by Dijkstra’s algorithm, a composite trust score is constructed by combining betweenness and eigenvector centralities on the reliability-weighted graph, enabling the identification of topK nodes as optimal TRNs. Algorithm 1 presents the pseudocode of the proposed methodology. B. Node Ranking using Centrality Measures For each node v ∈ G′ (V, E) we compute the following centrality scores. 1) Betweenness Centrality (BC) quantifies the fraction of shortest paths in G′ (V, E) (computed using modified weights) that pass through node v, such that, X σst (v) BCv = , (5) σst s̸=v̸=t
where σst is the total number of shortest paths from source s to destination t, and σst (v) is the number of such paths that pass through v (excluding endpoints). 2) Eigenvector Centrality (EC) measures the influence of node v based on the scores of its neighbors. Let A = (av,z ) denote the adjacency matrix, where each element av,z corresponds to the inverse of the weight
Algorithm 1 Ranking of Potential TRNs in a Network Input: Network topology G(V, E), node reliabilities Rv , link distances duv Output: Ordered list of nodes for TRN 1: for all edge (u, v) ∈ E do 2: Compute modified weight: ′ wuv = (Rduuv ·Rv ) 3: Construct modified graph G′ (V, E). 4: for all node v ∈ V do 5: Compute BCv on G′ . 6: Compute ECv on G′ . 7: Compute TSv = BCv · ECv . 8: Rank all nodes in descending order of TSv . 9: return Top K nodes as potential TRNs.
of the link between vertices v and z in G′ . Specifically, ′ av,z = 1/wvz if vertex v is connected to vertex z, and av,z = 0 otherwise. The relative eigenvector centrality score, ECv , of vertex v is then defined as: 1 X av,z ECz (6) ECv = λ z∈N (v)
where N (v) is the set of neighbors of v, and λ is a constant. A high ECv implies connections to other influential nodes. A new composite score, which determines the importance of the node based on its topological location and its influence on neighbouring nodes, is defined as follows: TSv = BCv · ECv
(7)
Nodes with the highest TSv are considered ideal candidates for the TRN. This formulation ensures that a node receives a high score only when it is simultaneously central to shortestpath routing and well connected to other influential nodes. C. Ranking-Based TRN Selection Algorithm We then rank the nodes based on the reliability-aware composite total score (T Sv ) evaluated in the previous step. By weighting shortest-path computations with reliability and combining BC and EC, we identify nodes that are both structurally critical and trustworthy, supporting secure and efficient key distribution across metro optical networks. The computational complexity of Algorithm 1 is dominated by the betweenness centrality computation in weighted graphs, resulting in an overall complexity of O(|V ||E| + |V |2 log |V |). Hence, the algorithm is computationally feasible for metro-scale optical networks. IV. R ESULTS AND D ISCUSSIONS In this section, we have evaluated the proposed reliabilityaware TRN ranking framework on a reference metropolitan optical network topology [16], consisting of 28 nodes interconnected by 52 bidirectional fiber links.
Fig. 1. Topology of a reference metro optical network, where the link distances are in kilometers.
For each node v, the total score T Sv , which captures a node’s structural importance and influence, is computed as (7). We have carried out the total score computation over 1000 different random reliability values generated using the Bayesian fusion method described in Section II. For every instance, the corresponding link weights are updated according to the reliability-adjusted formulation (see line 2 of Algorithm 1), and both PC and RC metrics are computed for all nodes. The final average values of PC, RC, and the total score T Sv are obtained by averaging across all instances, following the procedure outlined in Algorithm 1 (see line 7). Nodes are subsequently ranked in descending order of their averaged T Sv values to identify the most suitable candidates for TRN selection. A. Node Ranking for TRN Selection Table I lists the top ten nodes identified as the most suitable candidates for TRN placement. The results indicate that nodes exhibiting both high connectivity and strong reliability tend to achieve higher total scores. Among them, Node 9 attains the maximum T Sv value of 0.062968, primarily due to its strategic network position (high BCv ) and strong influence from well-connected neighboring nodes (high ECv ). It maintains direct connections with Nodes 6, 7, 10, 11, and 12, all of which also appear within the top ten rankings, thereby enhancing its topological prominence (see Fig. 1 and Table I). Although Table I reports the top ten ranked nodes, the number of TRNs selected for key relay operation can be adapted based on specific network coverage or redundancy requirements. It is worth noting that only the selected TRNs are used for key relay operations along a path, while the remaining nodes serve as Optical Bypass (OB) nodes [17]. B. TRN Ranking Validation To evaluate the effectiveness of the proposed TRN ranking, we define two complementary performance metrics: Path
TABLE I S ET OF T OP 10 TRN N ODES O RDERED ACCORDING TO T HEIR T OTAL S CORE . Ranked Node 9 6 7 12 11
Total Score 0.062968 0.053521 0.031868 0.029856 0.027565
Ranked Node 10 16 13 17 8
Total Score 0.017679 0.015220 0.015009 0.014975 0.013445
Coverage (PC) and Reliability Contribution (RC). These metrics jointly capture the structural and reliability-based significance of nodes in the network. 1) Path Coverage (PC): PC quantifies the structural importance of nodes based on their participation in the network’s shortest paths, indicating how effectively the selected TRNs enhance key distribution reachability. Let Ps,t represent the shortest path between source s and destination t. For a given node v, the function δs,t (v) equals 1 if v lies on Ps,t (excluding s and t), and 0 otherwise. For node v, let ( X 1, v ∈ Ps,t \ {s, t}, Cv = δs,t (v), δs,t (v) = 0, otherwise. s,t∈V,s̸=t (8) The normalized value of the path coverage for node v is PCv = P
Cv
j∈V Cj
.
(9)
2) Reliability Contribution (RC): RC measures the reliability-weighted contribution of nodes to overall network robustness, capturing their influence on maintaining secure and resilient QKD operations. The reliability of a node v, denoted Rv , is obtained from the Bayesian reliability framework described earlier. Let Pv denote the set of shortest paths of all connections traversing node v. The reliability of a path Ps,t is Y Rs,t = Rv (10) v∈Ps,t \{s,t}
The overall reliability contribution of node v is X (v) Roverall = Rs,t ,
(11)
(s,t)∈Pv
and the normalized value of the reliability contribution is (v) R RCv = P overall . (i) i∈V Roverall
(12)
Figures 2 and 3 show the cumulative path coverage (CPC) and cumulative reliability contribution (CRC), respectively, achieved by the top-ranked TRNs as a function of the number of selected top-ranked TRNs. Both metrics follow a similar increasing trend, demonstrating that nodes with higher reliability contributions also occupy structurally critical positions within the network. The CPC curve rises sharply for the initial few nodes, covering approximately 75% of all shortest paths
Fig. 2. Cumulative path coverage (CPC) by TRNs according to total score.
Fig. 3. Cumulative Reliability Contribution (CRC) by TRNs according to total score.
with only the top 12 TRNs. Beyond this point, the marginal improvement decreases because the majority of shortest paths have already been covered by the highest-ranked nodes, and additional TRNs mainly introduce redundancy rather than new coverage. Similarly, the CRC curve exhibits a comparable trend. These results collectively confirm that a relatively small subset of highly ranked nodes, typically the top 7–8, achieves a balanced trade-off between coverage and reliability. Consequently, the combined analysis of CPC and CRC validates that the proposed reliability-aware ranking framework effectively identifies nodes that are both topologically central and operationally trustworthy, improving the reliability and robustness of TRN selection. C. Comparative Analysis To further validate the proposed ranking model, we compare its performance against a degree centrality-based heuristic using both CPC and CRC as evaluation metrics. Degree centrality is selected as the baseline because it is one of the simplest and most widely adopted graph-theoretic heuristics for identifying structurally important nodes, making it a suitable reference for evaluating the effectiveness of the proposed reliability-aware ranking framework. As shown in Fig. 4, the proposed approach consistently outperforms degree centrality across all values of K, with the most pronounced improvement observed for intermediate
reliability-aware shortest-path computation and graph centrality measures to rank candidate TRNs. Simulation results on a 28-node metro topology demonstrate up to 8.91% higher reliability contribution and 8.93% greater path coverage than degree centrality, highlighting the effectiveness of the proposed framework. Future work will investigate dynamic TRN selection under varying traffic demands. R EFERENCES
Fig. 4. Difference in CPC between proposed composite score and degree centrality-based TRN selection.
Fig. 5. Difference in CRC between proposed composite score and degree centrality-based TRN selection.
selections. For instance, at K = 10, the difference in CPC reaches a maximum of 8.93%, while for K = 8 and K = 12, the improvements are 7.64% and 4.01%, respectively. This demonstrates that the top-ranked nodes identified by our reliability-aware model not only occupy critical network positions but also contribute more significantly to overall reliability aggregation. A similar trend is also observed for CRC, where the proposed ranking achieves superior path coverage across nearly all K values. The gain remains most prominent throughout the range, highlighting the ability of the proposed scoring function to select structurally and reliabilityrelevant nodes that maximize network coverage. Overall, both metrics reveal that while degree centrality captures basic connectivity, it fails to account for node reliability and global path influence. The proposed composite score, integrating eigen vector and betweenness centralities with node reliability, results in TRN selections that are both structurally strategic and reliability-efficient, offering superior performance for mid-range TRN counts (approximately 7–12 TRNs). V. C ONCLUSIONS In this work, we proposed a reliability-aware TRN selection framework for QKD-enabled metro optical networks. A Bayesian fusion-based reliability model was combined with
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