Embedded Edge Connectivity of Alternating Group Networks

According to Menger’s theorem, connectivity and edge connectivity are closely related to node-disjoint paths and edge-disjoint paths, respectively. Node- and edge-disjoint paths can keep the effective transmission of information and confidentiality. Therefore, node-disjoint paths and edge-disjoint paths are two important parameters to measure the reliability of a network. For a faulty node (resp. edge) set $S\subset V$ (resp. $S\subset E$), a connected graph $G=(V,E)$ is $S$-strongly Menger-node-connected (resp. Menger-edge-connected) if any two distinct nodes $x$ and $y$ in $G-S$ are connected by $\min \{\deg _{G-S}(x),\deg _{G-S}(y)\}$ internally node-disjoint (resp. edge-disjoint) paths in $G-S$, where $\deg _{G-S}(x)$ and $\deg _{G-S}(y)$ are the degrees of $x$ and $y$ in $G-S$, respectively. And most of the previous studies are based on networks that are triangle-free. In this paper, we consider the strongly Menger (edge) connectedness of a class of $r$-dimensional recursive networks RNCG $G_{r}$ with triangles. Moreover, we show that $G_{r}$ is $(rl-l-1)$-strongly Menger-node-connected. And then we show that $G_{r}$ is $[k+(r-1)l-2]$-strongly Menger-edge-connected of order 1 and $[2k+2(r-1)l-6]$-strongly Menger-edge-connected of order 2. Since the class of $r$-dimensional recursive networks RNCG $G_{r}$ includes not only data center networks DCell and generalized DCell but also interconnection network dragonfly, etc., all the results are appropriate for these networks.

Metric dimension and fault-tolerant metric dimension have potential applications in telecommunication, robot navigation and geographical routing protocols, among others. The computational complexity of these problems is known to be NP-complete. In this paper, we study the fault-tolerant metric dimension of various interconnection networks. By using the resolving sets in these networks, we locate fault-tolerant resolving sets in them. As a result, certain lower and upper bounds on the fault-tolerant metric dimension of those networks are obtained. We conclude the paper with some open problems.

Social Networks and Peer Effects at Works

A set of spanning trees in a graph G is called independent spanning trees (ISTs) if they are rooted at the same vertex r, and for each vertex v(= r) in G, the two paths from v to r in any two trees share no common vertex expect for v and r. ISTs can be applied in many research fields, such as fault-tolerant broadcasting and secure message distribution in reliable communication networks. Since Cayley graphs have been widely used to design interconnection networks, constructing ISTs on cayley graphs is worth studying. The alternating group network is a subclass of Cayley graphs, and the approach of constructing ISTs in alternating group networks is still unknown. In this paper, we propose a novel and simple top-down approach for constructing ISTs in alternating group networks. The main ideas of the algorithm are to use induction to develop small trees to big trees, to use a triangle breadth-first search (TBFS) process to create a backbone of an IST, and to use breadth-first search (BFS) process to connect the rest of nodes. Compared to other methods of different interconnection networks in the literature, the uniqueness of our method is that it does not need to determine the parent of one node by any rule, on the contrary others determine that by rules. The time complexity in n-dimensional alternating group network AN n is O(n 2 × n!), where n! is twice the number of nodes of AN n ; hence it is polynomial time. We implement the algorithm in PHP and run cases from AN 3 to AN 10. The results reveal that all spanning trees of AN 3 to AN 10 are independent and that our algorithm is accurate and efficient. INDEX TERMS Independent spanning trees, alternating group networks, triangle breadth-first search, interconnection networks, Cayley graph.

Any network can be modeled by a connected graph [Formula: see text], where [Formula: see text] is the set of nodes and [Formula: see text] is the set of direct communication links between the nodes. Many graph theoretical parameters are helpful in the study of the efficiency and reliability of an interconnection network model. A dominating set [Formula: see text] is a co-secure dominating set if, for each [Formula: see text], there exists [Formula: see text] such that [Formula: see text] is adjacent to [Formula: see text] and [Formula: see text] is a dominating set. The minimum cardinality of a co-secure dominating set in [Formula: see text] is called the co-secure domination number [Formula: see text] of [Formula: see text]. The study of a co-secure dominating set is essential in interconnection networks as it studies the security in it. Sierpiński graphs and generalized Sierpiński graphs are one among the well-known networks. Applications for Sierpiński graphs may be found in many fields, such as computer science, chemistry, psychology, probability, and dynamic systems. Sierpiński graphs are explored in fractal theory. Generalized Sierpiński graphs may be used to represent polymer networks and [Formula: see text] recursive networks. In this paper, we have obtained an upper bound for the co-secure domination number of the generalized Sierpiński graph of paths, and we have proved that this bound is sharp.

Deformable image registration (DIR) is a crucial technique in medical image analysis and is particularly important for 4D-CT-guided lung radiotherapy, where accurate spatiotemporal alignment and deformation plausibility are required for downstream tasks such as dose accumulation. However, many existing learning-based methods are limited in modeling global spatiotemporal dependencies and in preserving fine anatomical structures under large respiratory motion. METHODS: To address these issues, this paper proposes a wavelet transform-based regularized hybrid recursive spatiotemporal fusion registration network (W-STFNet). The proposed method incorporates SwinLSTM to effectively capture global spatiotemporal dependencies. To achieve better semantic integration of features across scales and time steps, a multi-scale spatiotemporal attention fusion (MSTAF) module is proposed, which improves the network's robustness and stability. Additionally, we design a novel frequency-domain loss function based on Discrete Wavelet Transform (DWT), which optimizes fine-grained structural matching by aligning high-frequency sub-bands, effectively improving the accuracy of high-frequency detail registration. The method is optimized in an unsupervised, patient-specific one-shot setting without anatomical annotations or multi-patient pretraining. RESULTS: Experiments on two public 4D-CT datasets (DIR-Lab and POPI-model) show that W-STFNet achieves competitive registration accuracy and stable performance across cases with varying deformation amplitudes. On DIR-Lab, W-STFNet attains a mean TRE of , and on POPI-model a mean TRE of , substantially reducing the initial misalignment. A two-sided paired Wilcoxon signed-rank test further supports that W-STFNet differs significantly from several learning-based baselines under the reported settings, although the absolute differences should be interpreted with respect to image resolution and annotation uncertainty. CONCLUSION: W-STFNet provides an annotation-free, patient-specific one-shot registration framework that achieves robust and competitive performance for 4D-CT lung DIR, particularly in handling image registration scenarios involving large deformations and complex temporal dynamics.

With the improvement of natural gas network interconnection, the network topology becomes increasingly complex. The significance of analyzing topology is gradually becoming prominent, and a systematic analysis method is required. This paper selects two typical natural gas pipeline networks: one in Europe, and the other in North China. Based on complex network theory and the nature of natural gas pipelines, topological models for the two typical networks were established and the evaluation indexes were developed based on four factors: network type, overall topological structure characteristics, path-related topological structure characteristics, and topological structure robustness. Using these indexes, the topological structure of the two typical networks is compared and analyzed quantitatively. The comparison shows that the European network topology has more redundancy, higher transmission efficiency, and greater robustness. The topology analysis method proposed in this paper is practical and suitable for the preliminary analysis of natural gas pipeline networks. The conclusions achieved by this method can assist operators in gaining an intuitive understanding of the overall characteristics, robustness, and key features of pipeline network topology, which is useful in the implementation of hierarchical prevention and control. It also serves as a solid theoretical foundation and guidance for network expansion, interconnection construction, and precise hydraulic simulation calculation in the next stage.

Multistage interconnection network provide communication among processors, memory modules and other devices in parallel computer systems. These networks are designed to provide fast and efficient communication with a reasonable cost. The number of stages, interconnection topology, and network configuration differentiate the reliability of each network. Reliability is used to measure the system capability to transform information from input to output devices and its depends on the reliability of its component. Evaluation of reliability of interconnection network has been attempted by researchers in the past. This paper is an depth study of reliability evaluation in multistage interconnection network. The measurement includes terminal reliability of regular network known as shuffle exchange network and gamma network.KeywordsMultistage Interconnection Networks (MINs)Terminal ReliabilityShuffle Exchange Network (SEN)Gamma NetworkSwitching ElementsThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The generalized k -connectivity κ k ( G ) and k -edge-connectivity λ k ( G ) of a graph G are a natural generalization of traditional connectivity κ( G ) and edge connectivity λ( G ) , respectively, which for κ( G )= κ 2 ( G ) and λ( G )= λ 2 ( G ) . They are important parameters which can often be used to measure the reliability and fault tolerance of interconnection networks. CRNs is a new family of composite networks based on the complete graph, which contain common networks and have the same structural properties as alternating group network, and may also include some unknown networks. In this paper, we investigate the generalized 3-connectivity and 3-edge-connectivity of CRNs, and show that κ 3 ( G l,m )= λ 3 ( G l,m )=m−2 .

In-depth studies of the topological properties of information and telecommunication networks contribute to the understanding of their functional capabilities, including stability. The study of the stability of complex networks to failures in operation when their components fail is based on modeling by sequentially removing nodes or edges of the network (percolation). The paper presents a comparative analysis of sequential and stochastic variants of percolation of network nodes and statistical estimates of the complex two-criterion network stability coefficient. During the study, methods for calculating the average path length based on graph theory were used. In the statistical analysis of the network stability, we applied the analysis of variance and pairwise comparisons according to the Tukey criterion, based on the provisions of the theory of mathematical statistics. The simulation is performed using the Barabashi–Albert and Erdős–Rényi random graph models. The difference between the method of stochastic percolation and sequential percolation is shown. The performed statistical analysis proved the influence of the factor changing the structure of networks on their stability due to stochastic percolation. The dynamics of network stability reduction under stochastic percolation for different types of networks is shown. It is revealed that in some cases, for example, in networks with high density, the stochastic percolation method is the most preferable one. The study shows the possible options for assessing the stability of networks without a priori knowledge about the type of connections between nodes and with a priori knowledge about the type of connections between nodes. In the former case, knowing the number of network nodes, one can calculate the limit values of stability, in the same way as if the nodes were deleted accidentally. The latter option can be used to calculate the stability of networks that are subject to random node failures, for example, when diagnosing technical systems.

The reliability of an interconnected power network is critical for maintaining a stable and efficient power supply. This research focuses on the detection, classification and mitigation of faults on the Ibom power interconnected network comprising of three synchronous generators, assuming each generator generate a 40 MW power, with a total power capacity of 120 MW power after synchronization. The network transmits it power through four transmission stations, first Afaha-Ube sub-transmission station, which has a load offtake of 60 MW, Itu sub-transmission station with 20 MW, Eket sub-transmission station with 25 MW, and Ekim sub-transmission station with 15 MW. All the sub-transmission station received 132 kV and step it down to 33kV. The system was first analyzed under normal operating conditions before various types of faults were introduced at the Afaha-Ube Substation bus, these included single line-to-ground (A-G), double line-to-ground (AB-G), and three-phase-to-ground (ABC-G), faults. To enhance fault detection, classification and system protection, current signals only were obtained through discrete wavelet transform (DWT). These signals were fed as input to train the ANFIS to properly detect and classified each fault type The ANFIS model was also trained to analyze fault current patterns and trigger a trip signal to the circuit breaker for fault isolation. Although faults persisted from 0.8 to 1.6 seconds, ANFIS was optimized to identify the fault and send a trip signal at 1.2 seconds, effectively mitigating damage and improving network stability. The result obtained shows that ANFIS successfully trip the Afaha-Ube bus station at exactly 1.2 seconds.

Multistage Interconnection Networks (MINs) are a cost-effective idea for connection between different nodes in multi-processor systems. In the meantime, understanding the impact of increasing the number of stages (i.e. a famous method for improving fault tolerance) on the reliability of these networks is essential. Last studies on this challenging issue, revealed that adding one switching stage to MINs is preferable to two in terms of reliability. However, there were some non-negligible gaps in previous researches: (i) The probability of failure for terminal nodes was not considered in the calculation of reliability. (ii) Numerical results were only based on the small size of the network, i.e. 8 × 8. (iii) Reliability analyses were time-independent. (iv) The lack of some important parameters such as the mean time to failure and availability was noticeable. Therefore, this paper will take a more comprehensive analysis of reliability to meet the above specifications.

Reliability evaluation of half hypercube networks

Under the current condition of the VLSI technique, 8/spl times/8 switches used in MINs have the best performance/cost ratios. Based on Delta network, Baseline network and current VLSI technique development level, this paper puts forward the idea of a new family of MINs using 8/spl times/8 switches which is called Super Recursive Baseline Interconnection Network (SRB), expounds its network topological Properties and routing techniques, makes performance analysis in this network and proves that it has superior qualities in network pass rates, bandwidths and performance/cost ratios, etc. Finally, this paper states that SRB is a family of Multistage Interconnection Network simple in routing, superior in performance/cost ratio, and easy in expanding.

The authors propose a new class of interconnection networks called recursive hierarchical fully-connected (RHFC) networks for high-performance parallel processing. The degree of a partially linked RHFC network is equal to that of its nucleus graph plus a fixed integer, leading to the scalability of such networks. The diameters of suitably constructed RHFC networks are asymptotically optimal within a small constant factor with respect to their node degrees. They then present the construction of recursive hierarchical swapped (RES) networks and multi-dimensional hypernets, both of which are subclasses of RHFC networks and can emulate a hypercube efficiently. They conclude that RHFC networks are highly modularized, make the use of fixed-degree building blocks possible for any practically realizable system, and lead to the construction of high-performance interconnection networks with reasonable cost.