Pair Of Nodes Research Articles

Link prediction in complex networks based on resource transition capacity and local paths

Link prediction aims to identify missing links within static networks or estimate the probability of emerging links in dynamic networks, representing a critical and challenging research direction in complex network. Many similarity-based methods have been established from various viewpoints, however, there are relatively few methods that take into account both the path information between node pairs and the nodes along the path. To fill this gap, we propose two novel link prediction algorithms, namely LPRA and LPH. The core idea is that the similarity between node pairs is closely related to the local paths connecting the two nodes and the resource transition capability of the nodes along those paths. The algorithms utilize the local paths with adjustable lengths between node pairs and the topological information of the nodes on the path to calculate the similarity index, which incorporate the resource transition capabilities of all the nodes on the possible paths between node pairs. We conducted multiple groups of comparative experiments on 10 real-world networks to validate the effectiveness of the proposed algorithm. Experimental results demonstrate that LPRA and LPH outperform nine classical methods and five recently popular methods. Moreover, LPRA demonstrates a significant difference level (P-value [Formula: see text]) compared to most of the methods in the ANOVA of AUC accuracy, indicating a statistically significant performance advantage of LPRA.

Comparison of distributed and centralized quantum key management systems for meshed QKD networks

Recent developments in quantum key distribution (QKD) demonstrate the maturity of securing sensitive data against the emerging quantum computing threat. For QKD-secured long-haul and meshed optical transport networks (OTNs), quantum key management systems (QKMSs) are essential to overcome current distance limitations of available QKD devices. In this work, we present and compare two implementations of QKMSs, analyzing their scalability with an emulated QKD network (QKDN) utilizing recorded performance metrics from deployed QKD devices. First, we use a state-of-the-art Internet routing scheme, i.e., open shortest path first (OSPF), demonstrating that key management entities (KMEs) can solve the key routing problem utilizing distributed routing. Second, we apply software-defined networking (SDN) to implement centralized routing with a SDN controller. This paper compares distributed with centralized key routing regarding scalability, throughput, and latency. Both schemes facilitate up to six key relays between any pair of nodes in parallel with average key relay durations per hop below 300 ms given the Nobel-Germany topology and any-to-any demand matrix. With a network-wide joint key routing optimization in the SDN controller, up to 16.7% higher demands can be served compared to distributed key routing. Within the inherent compatibility of our study to network-function virtualization (NFV), we guideline future integration of QKMSs into deployed OTNs.

Spatiotemporal Interactive Modeling of Event-Based Dynamic Networks

Event-based dynamic networks exist in a wide range of areas, including traffic, biological, and social applications. Such a network consists of interaction event sequences over different locations, where each event may trigger or influence a series of subsequent events under certain intrinsic spatial structure because of their geographical and semantic proximities. Such influence patterns and triggering motivations reflect the nature and semantics of human/object behaviors in the network. Thus, modeling event-based dynamic networks properly is critically important. This article proposes a spatiotemporal interactive Hawkes process (SIHP) that describes how a series of events occurs and models the rate of interaction events between any pair of nodes on the network explicitly with the information from related historical events as well as geographical and semantic neighboring nodes. The proposed SIHP can not only learn the patterns of influence from historical interaction events on later ones, but can also understand the network dynamics by fully considering spatial structure knowledge. Specifically, we incorporate prior knowledge of spatial structure as a graph and design graph regularization in the SIHP, where model parameters are estimated by designing an alternating direction method of multiplier (ADMM) framework. Numerical experiments and a real case study on New York yellow taxi data validate the effectiveness of the proposed method.

Kriging data with measurement error: A review and a generalized approach

Filtered kriging with parametric error (FKPE) is a mathematically sound method that generalizes and addresses the oversimplifications of previous kriging algorithms designed to filter error. The proposed approach is developed for handling grade-dependent (heteroscedastic), non-stationary, and spatially correlated sampling errors. The covariance between each pair of measurements or nodes is estimated from their model of errors and the spatial continuity of the underlying process. This research is driven by the fact that sampling and analytical errors are inherent in the samples used in the mining industry. In recent decades, data from quality control programs monitoring these errors have become widely available. FKPE handles more types of errors than other kriging algorithms to filter error and, for any number of subsets, it avoids jointly modelling the models of co-regionalization required by co-kriging methods. The precision gains of FKPE over other methods depend on how well the error model is fit to the data. Therefore, a detailed analysis of the error model and how to estimate its components is discussed. In a five-data toy example with high independent and grade-dependent error, kriging methods with oversimplified error models overweighted a high-grade datum by 50% compared to FKPE estimates. The performance of FKPE and other methods is illustrated by two synthetic examples.

Subgraph Topology and Dynamic Graph Topology Enhanced Graph Learning and Pairwise Feature Context Relationship Integration for Predicting Disease-Related miRNAs.

As an increasing number of microRNAs (miRNAs) have become biomarkers of various human diseases, prediction of the candidate disease-related miRNAs is helpful for facilitating the early diagnosis of diseases. Most of the recent prediction models concentrated on learning of the features from the heterogeneous graph composed of miRNAs and diseases. However, they failed to fully exploit the subgraph structures consisting of multiple miRNA and disease nodes, and they also did not completely integrate the context relationships among the pairwise features. We proposed a prediction model, SFPred, to integrate and encode the local topologies from neighborhood subgraphs, the dynamically evolved heterogeneous graph topology, and the context among pairwise features. First, the importance of an miRNA (disease) node to another node is formulated according to the subgraphs composed of their neighbors. Second, the features of each miRNA (disease) node continuously change when the graph encoding gradually deepens for the miRNA-disease heterogeneous network. A strategy based on multi-layer perceptron (MLP) is designed to estimate the edge weights according to the changed node features and form the dynamic graph topology. Third, considering the context relationships among the features of a pair of miRNA and disease nodes, a context relationship sensitive transformer is constructed to integrate these relationships. Finally, since the previous encoding layer of the transformer contains more detailed features of the pairwise, we present a multiperspective residual strategy to supplement the detailed features to the following encoding layer from the channel perspective and the feature one, respectively. The extensive experiments confirmed that SFPred outperforms eight state-of-the-art methods for the prediction of miRNA-disease associations, and the ablation experiments validate the effectiveness of the proposed innovations. The recall rates for the top-ranked candidate miRNAs related to the diseases and the case studies on three diseases indicate SFPred's ability in screening the reliable candidates for subsequent biological experiments.

Topology Control of Low-Connection UAV Laser Network with Virtual Nodes

Space laser communication technology has advantages such as high bandwidth and low interference, but the limitation of the load capacity of an unmanned aerial vehicle (UAV) makes it difficult for UAV clusters to have sufficient communication links to build a stable communication network. Aiming to address the problem of the low upper limit of the node degree of the self-organized network after the introduction of space laser communication technology for UAVs, a virtual node topology control (VNTC) algorithm is proposed for constructing an efficient and stable communication network structure. The algorithm is based on the unit disc graph model and achieves a lower-node-degree upper bound while maintaining connectivity through node pairing and virtual graph construction. The theoretical analysis and experimental results of the algorithm show that the algorithm can effectively deal with the topology control problem in UAV laser networks. In contrast to the XTC algorithm (exceptional topology control), the VNTC algorithm maintains the connectivity and most of the planarizability of the network, and, at the same time, it performs better in reducing the node degree and number of edges, thus providing an effective solution for low-connectivity topology control.

Draw+: network-based computational drug repositioning with attention walking and noise filtering.

Drug repositioning, a strategy that repurposes already-approved drugs for novel therapeutic applications, provides a faster and more cost-effective alternative to traditional drug discovery. Network-based models have been adopted by many computational methodologies, especially those that use graph neural networks to predict drug-disease associations. However, these techniques frequently overlook the quality of the input network, which is a critical factor for achieving accurate predictions. We present a novel network-based framework for drug repositioning, named DRAW+, which incorporates noise filtering and feature extraction using graph neural networks and attention mechanisms. The proposed model first constructs a heterogeneous network that integrates the drug-disease association network with the similarity networks of drugs and diseases, which are upgraded through reduced-rank singular value decomposition. Next, a subgraph surrounding the targeted drug-disease node pair is extracted, allowing the model to focus on local structures. Graph neural networks are then applied to extract structural representation, followed by attention walking to capture key features of the subgraph. Finally, a multi-layer perceptron classifies the subgraph as positive or negative, which indicates the presence of the link between the target node pair. Experimental validation across three benchmark datasets showed that DRAW+ outperformed seven state-of-the-art methods, achieving the highest average AUROC and AUPRC, 0.963 and 0.564, respectively. Moreover, DRAW+ demonstrated its robustness by achieving the best performance across two additional datasets, further confirming its generalizability and effectiveness in diverse settings. The proposed network-based computational approach, DRAW+, demonstrates exceptional accuracy and robustness, confirming its effectiveness in drug repositioning tasks.

Voxel-Wise Brain Graphs From Diffusion MRI: Intrinsic Eigenspace Dimensionality and Application to Functional MRI.

Goal: Structural brain graphs are conventionally limited to defining nodes as gray matter regions from an atlas, with edges reflecting the density of axonal projections between pairs of nodes. Here we explicitly model the entire set of voxels within a brain mask as nodes of high-resolution, subject-specific graphs. Methods: We define the strength of local voxel-to-voxel connections using diffusion tensors and orientation distribution functions derived from diffusion MRI data. We study the graphs' Laplacian spectral properties on data from the Human Connectome Project. We then assess the extent of inter-subject variability of the Laplacian eigenmodes via a procrustes validation scheme. Finally, we demonstrate the extent to which functional MRI data are shaped by the underlying anatomical structure via graph signal processing. Results: The graph Laplacian eigenmodes manifest highly resolved spatial profiles, reflecting distributed patterns that correspond to major white matter pathways. We show that the intrinsic dimensionality of the eigenspace of such high-resolution graphs is only a mere fraction of the graph dimensions. By projecting task and resting-state data on low-frequency graph Laplacian eigenmodes, we show that brain activity can be well approximated by a small subset of low-frequency components. Conclusions: The proposed graphs open new avenues in studying the brain, be it, by exploring their organisational properties via graph or spectral graph theory, or by treating them as the scaffold on which brain function is observed at the individual level.

SPA: A poisoning attack framework for graph neural networks through searching and pairing

Graph Neural Networks (GNN) have played an important role in many fields, while GNNs also suffer from adversarial attacks that aim to malfunction the GNN model by changing the adjacency matrix (i.e. generating adversarial edges) or node features (i.e. generating adversarial features) in graph data. Although the gradient-based adversarial attack methods have achieved remarkable results in DNNs, optimizing discrete adversarial edges in graph data using continuous gradients may lead to sub-optimal solutions. In order to alleviate this situation, we propose a novel Searching and Pairing Attack (SPA) method to effectively generate adversarial edges by treating each adversarial edge as a combination of a pair of adversarial nodes. The proposed SPA method generates the adversarial edges through a Node Searching step and a Node Pairing step. The proposed Node Searching Ant Colony Optimization (NS-ACO) improves the attack effect by using the ability of heuristic algorithm to quickly find the approximate optimal solution, while in the Node Pairing (NP) step we propose a generative graph convolutional network with a novel Aggregate Cooperative (AC) layer to generate a set of nodes that meet the constraints, so as to obtain the perturbation set together with the Node Searching step. The proposed SPA method outperforms the state-of-the-art adversarial attack methods and achieves a misclassification rate of 32.5% in the poisoning attack on Cora dataset with a perturbation rate of 0.5%.

آلية إدارة مفتاح RSA المحسّنة لحماية حزم التحكم في شبكات تبديل الاندفاع البصري

Optical Burst Switch (OBS) network is a typical technology currently proposed, as an infrastructure to deal with most highly demanded communication services. OBS architecture consists of Data Burst (DB) as a payload and Burst Header Packet (BHP) as a control packet. BHP carries important information about path reservation and its corresponding DB. However, the information security (Privacy, Reliability, Confidentiality, Integrity, Availability, Authentication, and Authorization) of OBS network communication has been represented as the current issues, which affect the network performance in terms of data loss and data transmission delay. This paper focuses on security weaknesses of BHP transmission in OBS network against Data Burst Redirection (DBR) Attack. The current paper was conducted to develop a protection mechanism to ensure the confidentiality and authentication of BHP. In OBS, RSA public-key encryption algorithm has been enhanced and integrated. Moreover, the Self-Controlling key distribution technique has been implemented to ensure a high security level of key transmission between each pair of OBS nodes. Three different OBS environments have been designed and implemented. These environments were established on the bases of three different concepts; OBS Topology without Security Measures and without Security Attacks, OBS Topology under Security Attacks without Security Measures, and OBS Topology under Security Attacks with Security Measures. The obtained results are based on Burst Loss Ratio, Throughputs, and Average Delay Ratio. Such a result has successfully proved the trustworthiness and efficiency of Control Packet Protection Technique (CPPT-OBS) to prevent DBR attack. Keywords: OBS , DB, BHP, DBR, RSA, CPPT

Empirical networks are sparse: Enhancing multiedge models with zero-inflation.

Real-world networks are sparse. As we show in this article, even when a large number of interactions is observed, most node pairs remain disconnected. We demonstrate that classical multiedge network models, such as the , configuration models, and stochastic block models, fail to accurately capture this phenomenon. To mitigate this issue, zero-inflation must be integrated into these traditional models. Through zero-inflation, we incorporate a mechanism that accounts for the excess number of zeroes (disconnected pairs) observed in empirical data. By performing an analysis on all the datasets from the Sociopatterns repository, we illustrate how zero-inflated models more accurately reflect the sparsity and heavy-tailed edge count distributions observed in empirical data. Our findings underscore that failing to account for these ubiquitous properties in real-world networks inadvertently leads to biased models that do not accurately represent complex systems and their dynamics.

Graph Contrastive Pre-training for Anti-money Laundering

Anti-money laundering (AML) is vital to maintaining financial markets, social stability, and political authority. At present, many studies model the AML task as the graph and leverage graph neural network (GNN) for node/edge classification. Although these studies have achieved some achievements, they struggle with the issue of label scarcity in real-world scenarios. In this paper, we propose a graph contrastive pre-training framework for anti-money laundering (GCPAL), which mines supervised signals from the label-free transaction network to significantly reduce the dependence on annotations. Specifically, we construct three augmented views (i.e., two stochastic perturbed views and a KNN view). Perturbed views are beneficial to the model learning invariant information and improve the robustness against noise. KNN view provides implicit interactions to mitigate the link sparsity in the transaction network. Moreover, we extend the positive sample set using connected neighbors and node pairs with similar features to further enhance the expressiveness of the model. We evaluate the GCPAL on two datasets, and the extensive experimental results demonstrate that the GCPAL is consistently superior to other SOTA baselines, especially with scarce labels.

Establishment of multiple machine learning prognostic model for gene differences between primary tumors and lymph nodes in luminal breast cancer.

This study aimed to explore the correlation between primary tumors (PT) and paired metastatic lymph nodes (LN) and to develop a predictive model to provide evidence for forecasting patient prognoses. We obtained single-cell and bulk transcriptome data from the Gene Expression Omnibus database. Furthermore, mRNA transcriptomic data, encompassing 112 normal tissues and 1066 breast cancer samples, along with survival, clinical, and mutation information for breast cancer patients, were acquired from The Cancer Genome Atlas (TCGA). Employing a machine learning integration framework incorporating ten distinct algorithms, we developed and validated a prognostic model. We constructed a prognostic model named Lymph Node Metastasis-Related Scores (LMRS) using 26 differentially expressed genes trained on eight TCGA datasets. Across validation sets, the model demonstrated a high C-index, signifying its stability and effectiveness, outperforming 64 models from other studies. Notably, cytolytic activity and T cell co-stimulation were downregulated in the high LMRS group, alongside a downregulation of immune cells, including B cells, CD8 + T cells, iDCs, and TILs. Similarly, most immune checkpoints exhibited a decreasing trend with high LMRS expression. Finally, we selected the hub biomarkers PGK1 and HSP90 for pathological verification. Results indicated higher expression levels in PT and LN compared to normal and benign tumors, with higher expression levels in LN than in PT. This comprehensive analysis sheds light on gene expression differences between PT and LN in breast cancer, culminating in the development of a multiple-gene prognostic model with high clinical accuracy for prognosis prediction.

Domain-Adaptive Graph Attention-Supervised Network for Cross-Network Edge Classification.

Graph neural networks (GNNs) have shown great ability in modeling graphs; however, their performance would significantly degrade when there are noisy edges connecting nodes from different classes. To alleviate negative effect of noisy edges on neighborhood aggregation, some recent GNNs propose to predict the label agreement between node pairs within a single network. However, predicting the label agreement of edges across different networks has not been investigated yet. Our work makes the pioneering attempt to study a novel problem of cross-network homophilous and heterophilous edge classification (CNHHEC) and proposes a novel domain-adaptive graph attention-supervised network (DGASN) to effectively tackle the CNHHEC problem. First, DGASN adopts multihead graph attention network (GAT) as the GNN encoder, which jointly trains node embeddings and edge embeddings via the node classification and edge classification losses. As a result, label-discriminative embeddings can be obtained to distinguish homophilous edges from heterophilous edges. In addition, DGASN applies direct supervision on graph attention learning based on the observed edge labels from the source network, thus lowering the negative effects of heterophilous edges while enlarging the positive effects of homophilous edges during neighborhood aggregation. To facilitate knowledge transfer across networks, DGASN employs adversarial domain adaptation to mitigate domain divergence. Extensive experiments on real-world benchmark datasets demonstrate that the proposed DGASN achieves the state-of-the-art performance in CNHHEC.

Efficient Neural Collaborative Search for Pickup and Delivery Problems.

In this paper, we introduce Neural Collaborative Search (NCS), a novel learning-based framework for efficiently solving pickup and delivery problems (PDPs). NCS pioneers the collaboration between the latest prevalent neural construction and neural improvement models, establishing a collaborative framework where an improvement model iteratively refines solutions initiated by a construction model. Our NCS collaboratively trains the two models via reinforcement learning with an effective shared-critic mechanism. In addition, the construction model enhances the improvement model with high-quality initial solutions via curriculum learning, while the improvement model accelerates the convergence of the construction model through imitation learning. Besides the new framework design, we also propose the efficient Neural Neighborhood Search (N2S), an efficient improvement model employed within the NCS framework. N2S exploits a tailored Markov decision process formulation and two customized decoders for removing and then reinserting a pair of pickup-delivery nodes, thereby learning a ruin-repair search process for addressing the precedence constraints in PDPs efficiently. To balance the computation cost between encoders and decoders, N2S streamlines the existing encoder design through a light Synthesis Attention mechanism that allows the vanilla self-attention to synthesize various features regarding a route solution. Moreover, a diversity enhancement scheme is further leveraged to ameliorate the performance during the inference of N2S. Our NCS and N2S are both generic, and extensive experiments on two canonical PDP variants show that they can produce state-of-the-art results among existing neural methods. Remarkably, our NCS and N2S could surpass the well-known LKH3 solver especially on the more constrained PDP variant.

Identifying Associations between Small Nucleolar RNAs and Diseases via Graph Convolutional Network and Attention Mechanism.

Research has shown that small nucleolar RNAs (snoRNAs) play crucial roles in various biological processes, and understanding disease pathogenesis by studying their relationship with diseases is beneficial. Currently, known associations are insufficient, and conventional biological experiments are costly and time-consuming. Therefore, developing efficient computational methods is crucial for identifying potential snoRNA-disease associations. In this paper, a method to identify snoRNA-disease associations based on graph convolutional network and multi-view graph attention mechanism (GCASDA) is proposed. Firstly, the similarity matrices of snoRNAs and diseases are calculated based on biological entity-related information, and the weights of the edges between the snoRNA nodes and the disease nodes are supplemented by random forest. Then two homogeneous graphs and one heterogeneous graph are constructed. Subsequently, different types of embedded features are extracted from the graphs using specific graph convolutional network structure and integrated through a multi-view graph attention mechanism to obtain node embedded feature representations. Finally, for each pair of nodes, in addition to their global features, node interaction features are passed together to a multilayer perceptron neural network (MLP) to identify snoRNA-disease associations. Experimental results show that GCASDA achieves 0.9356 and 0.9294 in AUC and AUPR, respectively, and significantly outperformed other state-of-the-art methods on the basis of different evaluation metrics. Furthermore, the case study could further demonstrate the realistic feasibility of GCASDA.

Functional brain network organization supports cognitive function in patients with Alzheimer’s Disease‐related pathological burden and brain atrophy

AbstractBackgroundAlzheimer’s disease (AD) is defined by the accumulation of Aβ plaques and tau neurofibrillary tangles. The consequences of these pathologies include neurodegeneration and cognitive dysfunction. However, the process by which AD pathologies leads to cognitive impairment remains unclear. Some individuals can harbor pathological burden yet be cognitively unimpaired; these observations have led to ideas about cognitive reserve (Stern, Lancet, 2012).Dominant models of AD biomarkers do not include direct measures of brain function. To address this gap, we tested whether resting‐state brain system segregation explains discrepancies between AD‐related pathology, structural atrophy, and cognitive function. System segregation changes across the adult lifespan, is stratified in relation to life course exposures, and relates to cognitive function (Wig, Trends in Cognitive Sciences, 2017).MethodParticipants were recruited through the Alzheimer’s Disease Neuroimaging Initiative. Cognitive impairment was quantified using clinical dementia rating (CDR). The participant group included 184 healthy (CDR=0) adults, 101 very mild dementia (CDR=0.5), 24 mild dementia (CDR=1) and 2 moderate dementia (CDR=2) patients (age range: 55–96 years).Functional brain networks were generated using resting‐state fMRI time series. Network edges were defined as correlations between time series of each pair of nodes. Brain system segregation was calculated to quantify functional brain network organization for each participant (Chan et al., PNAS, 2014).ResultMultiple linear regression analyses were used to test whether Aβ and tau relate to brain system segregation, with age, gender, education, and head motion included as covariates. Neither Aβ nor tau burden was associated with brain system segregation (Aβ: p=0.45; tau: p=0.55).Among individuals exhibiting positive pathological burden (either Aβ, tau, or both) based on cut‐off values, individuals with higher system segregation were less cognitively impaired (Aβ+: p=0.007; tau+: p=0.021; Aβ and tau+: p=0.03).Finally, system segregation explained unique variance with respect to CDR status over and above Aβ, tau, and cortical thickness. Brain system segregation was negatively related to CDR status (p=0.003), while accounting for these other factors.ConclusionFunctional brain network organization supports observations of cognitive reserve, and explains unique variance with respect to AD‐related cognitive impairment independent of AD‐related pathology and neurodegeneration.

Low-Power Radiation-Hardened Static Random Access Memory with Enhanced Read Stability for Space Applications

In space environments, radiation particles affect the stored values of SRAM cells, and these effects, such as single-event upsets (SEUs) and single-event multiple-node upsets (SEMNUs), pose a threat to the reliability of systems used in the space industry. To mitigate the impacts of SEUs and SEMNUs, this paper proposes the Read Stability Improved and Low Power (RSLP16T) SRAM cell. It was confirmed that in SEU-induced simulations, all nodes of the RSLP16T could be restored with a charge amount of less than 100 fC. Additionally, it was verified that a similar level of restoration was possible for SEMNUs occurring in pair of storage nodes. The proposed cell achieves a high level of read stability due to a high pull-down cell ratio (current ratio, CR) at the storage nodes and the fact that only a pair of nodes is in contact with the bit lines during read operations. Because all node paths use a stacking structure for internal transistor configuration and a relatively higher number of cells are composed of PMOS, it consumes the least hold power. While these improvements come at the cost of slightly increased delay time and area, performance evaluation revealed that the equivalent quality metric (EQM) was the highest, indicating that the benefits outweigh the drawbacks. The proposed integrated circuit is implemented in the 90 nm CMOS process and operated on 1 V supply voltage.

Computing the Kirchhoff index of a family of phenylene chain networks

Abstract The Kirchhoff index is a fundamental topological metric that provides insights into the structural and electrical characteristics of networks. It is defined as the sum of resistance distances between all pairs of nodes, serving as a key factor in understanding the dynamics within networks. To investigate the impact of structural variations on the Kirchhoff index, we select a family of phenylene chain networks as our model and establish a methodology to explore the Kirchhoff index using the Laplacian spectrum. By analyzing the network structure, we introduce a parameter to control the number of iterations, providing a recursive relationship between the Laplacian matrix and its eigenvalues at intervals of generations. This approach enables the derivation of an analytical expression for both the sum of the reciprocals of all nonzero Laplacian eigenvalues and the Kirchhoff index.

Resistance distances in stretched Cantor product networks

In this paper, we consider resistance distances in stretched Cantor product networks, a family of non-self-similar networks. By constructing the networks in a iterated way, we give an approach to encode every node in their vertex set. And then we simplify the complex resistor networks by induction on the basic network pattern. Using classical results of circuit theory, we obtain the exact formulae of the resistance distances of some pairs of nodes in stretched Cantor product networks.