Reliability Of Interconnection Networks Research Articles

Characterization of matroidal connectivity of regular networks

High performance computing system, which takes an interconnection network as its infrastructure topology, has been utilized in scientific computing, big data analysis as well as artificial intelligence. With the rapid growth of infrastructure topology (interconnection network) in high performance computing system, the probability of network element malfunction increases dramatically. Generally, the reliability of interconnection network is measured by the traditional (edge) connectivity, which is limited by the minimum degree of the network. In order to overcome the defect, the matroidal connectivity has been newly proposed to extend the edge connectivity. For the regular network G, let Bk(G)={B1,B2,…,Bk} be a fixed edge partition ordered sequence of G and χk=(x1,x2,…,xk) be a sequence of k integers. For any D⊆E(G), if |D∩Bi|≤xi(i∈[k]) and G−D is disconnected, then D is named matroidal cut set of G. The matroidal connectivity of G, denoted by λ(G,Bk(G)), is the minimum size of matroidal cut set of G. In this work, we characterize the matroidal connectivity of a class of n-dimensional regular networks Gn. To be more specific, we show that λ(Gn,Bn(Gn))=(k−l+1)f(l+j), where f(n) is a function about n, and l,k,n are positive integers (l≥1), j=n−k. As empirical analysis, we directly determine the matroidal connectivity of some well-known interconnection networks, such as hypercube Qn, star graph STn and alternating group graph AGn.

Generalized Connectivity of the Mycielskian Graph under g-Extra Restriction

The g-extra connectivity is a very important index to evaluate the fault tolerance, reliability of interconnection networks. Let g be a non-negative integer, G be a connected graph with vertex set V and edge set E, a subset S⊆V is called a g-extra cut of G if the graph induced by the set G−S is disconnected and each component of G−S has at least g+1 vertices. The g-extra connectivity of G, denoted as κg(G), is the cardinality of the minimum g-extra cut of G. Mycielski introduced a graph transformation to discover chromatic numbers of triangle-free graphs that can be arbitrarily large. This transformation converts a graph G into a new compound graph called μ(G), also known as the Mycielskian graph of G. In this paper, we study the relationship on g-extra connectivity between the Mycielskian graph μ(G) and the graph G. In addition, we show that κ3(μ(G))=2κ1(G)+1 for κ1(G)≤min{4,⌊n2⌋}, and prove the bounds of κ2g+1(μ(G)) for g≥2.

Structure connectivity and substructure connectivity of split-star networks

The connectivity is an important metric to evaluate the fault tolerance and reliability of interconnection networks. The structure connectivity κ(G;H) and substructure connectivity κs(G;H) are the new proposed indicators to measure the robustness of a network when the network fails with different structures. The split-star network Sn2 is a famous interconnection network in parallel and distributed systems. In this paper, we will determine both κ(Sn2;H) and κs(Sn2;H) for H∈{Pl,Ck}, respectively, where 4≤l≤3n−5, 6≤k≤3n−5.

Reliability evaluation of half hypercube networks

With the increase of the number of processors in the interconnection network, the failure of processors is inevitable. When the processor fails, the information passing through the failed processor will be unreliable, which may cause fatal consequences. Therefore, the research on the reliability of interconnection networks is an important topic, and the reliability of networks can be characterized by connectivity and diagnosability. In this paper, we first establish that the h-extra connectivity of the n-dimensional half hypercube network HHn is κh(HHn)=−12h2+(⌈n/2⌉−12)h+⌈n/2⌉+1 for n≥7 and 0≤h≤⌈n/2⌉−4, and then determine that the h-extra diagnosability of the HHn under the PMC model (n≥7 and 0≤h≤⌈n/2⌉−4) and the MM* model (n≥7 and 0≤h≤⌈n/2⌉4−1) is th(HHn)=−12h2+(⌈n/2⌉+12)h+⌈n/2⌉+1. Finally, we obtain that HHn is [−12k2+(⌈n/2⌉−12)k+⌈n/2⌉+1]/k-diagnosable under the PMC model.

Intermittent Fault Diagnosis of Split-Star Networks and its Applications

With the rapid increase of the number of processors in multiprocessor systems and the fast expansion of interconnection networks, the reliability of interconnection network is facing severe challenges, where the fast recognition of fault processors is crucial. In practice, most of the processor failures are intermittent faults. In this article, we first determine the intermittent fault diagnosability <inline-formula><tex-math notation="LaTeX">$t_{I}^{PMC}(S_{n}^{2})$</tex-math></inline-formula> of <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula> -dimensional split-star network <inline-formula><tex-math notation="LaTeX">$S_{n}^{2}$</tex-math></inline-formula> under the PMC model. In addition, we propose a fast intermittent fault probabilistic diagnosis algorithm FIFPDPMC to identify the nodes with intermittent fault in the <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula> -dimensional split-star network <inline-formula><tex-math notation="LaTeX">$S_{n}^{2}$</tex-math></inline-formula> under the PMC model, and we calculated the time complexity of the algorithm FIFPDPMC. Then we implement the algorithm FIFPDPMC in the IoT-based wireless sensor network (IoTWSN) and a randomly generated network (RGN) under different number of nodes with intermittent fault, and we evaluate the performance and efficiency of the algorithm FIFPDPMC in terms of accuracy, precision, recall (TPR), F1, G-mean, FPR, TNR and FNR. Experimental results show that, as the number of stages of executing the algorithm FIFPDPMC increases, the number of nodes with intermittent fault being diagnosed by the algorithm FIFPDPMC increases, which implies that the algorithm FIFPDPMC has good performance and efficiency in both IoTWSN and RGN.

An O(log(N)) Algorithm View: Reliability Evaluation of Folded-crossed Hypercube in Terms of h-extra Edge-connectivity

An interconnection network can be modelled as a connected graph [Formula: see text]. The reliability of interconnection networks is critical for multiprocessor systems. Several conditional edge-connectivities have been introduced in the past for accurately reflecting various realistic network situations, with the [Formula: see text]-extra edge-connectivity being one such conditional edge-connectivity. The [Formula: see text]-extra edge-connectivity of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality of faulty edges whose deletion disconnects the graph [Formula: see text] with each resulting component containing at least [Formula: see text] processors. In general, for a connected graph [Formula: see text], determining whether the graph exists an [Formula: see text]-extra edge-cut is [Formula: see text]-hard. The folded-crossed hypercube [Formula: see text] is a variation of the crossed hypercube [Formula: see text] with [Formula: see text] processors. In this paper, after excavating the layer structure of folded-crossed hypercube, we investigate some recursive properties of [Formula: see text], based on some recursive properties, an effective [Formula: see text] algorithm of [Formula: see text]-extra edge-connectivity of folded-crossed hypercube is designed, which can determine the exact value and the [Formula: see text]-optimality of [Formula: see text] for each positive integer [Formula: see text]. Our results solve this problem thoroughly.

The &amp;lt;i&amp;gt;g&amp;lt;/i&amp;gt;-Good-Neighbor Connectivity of Some Cartesian Product Graphs

The g-good-neighbor connectivity of G is a generalization of the concept of connectivity, which is just for, and an important parameter in measuring the fault tolerance and reliability of interconnection network. Many well-known networks can be constructed by the Cartesian products of some simple graphs. In this paper, we determine the g-good-neighbor connectivity of some Cartesian product graphs. We give the exact value of g-good-neighbor connectivity of the Cartesian product of two complete graphs and for , mesh for , cylindrical grid and torus for .

The Generalized 3-Connectivity Of The Folded Hypercube FQn

The generalized $k$-connectivity of a graph $G$, denoted by $\kappa_k(G)$, is a generalization of the traditional connectivity. It is well known that the generalized $k$-connectivity is an important indicator for measuring the fault tolerance and reliability of interconnection networks. The $n$-dimensional folded hypercube $FQ_n$ is obtained from the $n$-dimensional hypercube $Q_n$ by adding an edge between any pair of vertices with complementary addresses. In this paper, we show that $\kappa_3(FQ_n)=n$ for $n\ge 2$, that is, for any three vertices in $FQ_n$, there exist $n$ internally disjoint trees connecting them.

The Component (Edge) Connectivity of Round Matching Composition Networks

The vertex (edge) connectivity has been regularly used to measure the fault tolerance and reliability of interconnection networks, while it has defects in the assumption that all neighbors of one node will fail concurrently. To overcome this deficiency, some new generalizations of traditional connectivity have been suggested to quantize the size or the number of the connected components of the survival graph. The [Formula: see text]-component (edge) connectivity, one generalization of vertex (edge) connectivity, has been proposed to characterize the vulnerability of multiprocessor systems based on the number of components of the survival graph. In this paper, we determine the [Formula: see text]-component (edge) connectivity of a family of networks, called the round matching composition networks [Formula: see text], which are a class of networks composed of [Formula: see text] ([Formula: see text]) clusters with the same order, linked by [Formula: see text] perfect matchings. By exploring the combinatorial properties and fault-tolerance of [Formula: see text], we establish the [Formula: see text]-component (edge) connectivity [Formula: see text] for [Formula: see text] and [Formula: see text], [Formula: see text] and [Formula: see text] for [Formula: see text].

Reliability of Divide-and-Swap Cube Based on r-Component Connectivity and Diagnosability

When performing big data processing and high-performance computing, large-scale multiprocessor systems have always been committed to ensuring high reliability and fault tolerance. Connectivity and diagnosability are two important metrics to evaluate the fault tolerance and reliability of interconnection networks. In this paper, we determine [Formula: see text]-component connectivity and diagnosability of divide-and-swap cube [Formula: see text]. In detail, we show that, for [Formula: see text] and [Formula: see text], [Formula: see text], [Formula: see text] under the PMC and MM[Formula: see text] models.

Cluster Connectivity And Super Cluster Connectivity Of DQcube

AbstractAs a fundamental metric, the connectivity to assess fault tolerance and reliability of interconnection networks has been extensively explored. However, classical connectivity is not very effective at evaluating large-scale networking systems. To overcome this deficiency, two new indices, cluster connectivity and super cluster connectivity, have been proposed to characterize the robustness of interconnection networks. This paper focuses on investigating $\mathcal{H} (\mathcal{H}^{*})$-cluster connectivity and super $\mathcal{H} (\mathcal{H}^{*})$-connectivity of composition graph $DQ_{n}$, based on disc-ring and hypercube, for $\mathcal{H}\in \{K_{1,r}\ |\ 0\leq r\leq n+1\}$, respectively. In detail, we show that $\kappa (DQ_{n}|K_{1,1} (K_{1,1}^{*}))=\kappa ^{\prime}(DQ_{n}|K_{1,1}(K_{1,1}^{*})) =n+1 (n\geq 3)$, $\kappa (DQ_{n}|K_{1,r}(K_{1,r}^{*})) =\left\lceil \frac{n}{2}\right\rceil +1 (2\leq r\leq 4)$ for $n\geq 3$, $\kappa ^{\prime}(DQ_{n}|K_{1}(K_{1}^{*})) =\kappa ^{\prime}(DQ_{n})=2n (n\geq 3)$, and for $2\leq r\leq 3$ and $k\geq 2$, $$\begin{align*} \kappa^{\prime}(DQ_{n}|K_{1,r}(K_{1,r}^{*}))=\left\{\begin{array}{@{}ll} n+1, &amp; if\ n=2k+1; \\ n, &amp; if\ n=2k. \end{array}\right. \end{align*}$$As by-products, we show that DQcube is super $K_{1,r}-\ (K_{1,r}^{*}-)$connected $(2\leq r\leq 3)$, and derive the 4-extra connectivity of DQcube $\kappa _{4}(DQ_{n}) = 5n-9 (n\geq 4)$.

The generalized 3-connectivity of two kinds of regular networks

The locally twisted cube LTQn and the generalized hypercube GQ(m1, m2, ⋯,mn) are two important kinds of regular graphs in the modelling interconnection networks. For measuring the fault tolerance and reliability of interconnection networks, the generalized k-connectivity was introduced, which is a generalization of the traditional connectivity. More precisely, let S⊆V(G) and κG(S) denote the maximum number r of trees T1,T2,⋯,Tr such that V(Ti)∩V(Tj)=S and E(Ti)∩E(Tj)=∅ for any integers 1≤i≠j≤r. For an integer k with 2≤k≤|V(G)|, the generalized k-connectivity of G, denoted by κk(G), is defined as κk(G)=min⁡{κG(S)|S⊆V(G)and|S|=k}. In this paper, we study the generalized 3-connectivity of the locally twisted cube LTQn and the generalized hypercube GQ(m1,m2,⋯,mn), and get two main results that κ3(LTQn)=n−1 and κ3(GQ(m1,m2,⋯,mn))=∑i=1n(mi−1)−1.

Structure and substructure connectivity of divide-and-swap cube

High fault tolerance and reliability of multiprocessor systems, modeled by interconnection network, are of great significance to assess the flexibility and effectiveness of the systems. Connectivity is an important metric to evaluate the fault tolerance and reliability of interconnection networks. As classical connectivity is not suitable for such large scale systems, a novel and generalized connectivity, structure connectivity and substructure connectivity, has been proposed to measure the robustness of networks and has witnessed rich achievements. The divide-and-swap cube DSCn is an interesting variant of hypercube that has nice hierarchical properties. In this paper, we mainly investigate H-structure-connectivity, denoted by κ(DSCn;H), and H-substructure-connectivity, denoted by κs(DSCn;H), for H∈{K1,K1,1,K1,m(2≤m≤d+1),C4}, respectively. In detail, we show that κ(DSCn;K1)=κs(DSCn;K1)=d+1 for n≥2, κ(DSCn;K1,1)=κs(DSCn;K1,1)=d+1 for n≥8, κ(DSCn;K1,m)=κs(DSCn;K1,m)=⌊d2⌋+1 with 2≤m≤d+1 for n≥4, κ(DSCn;C4)=3+2(d−2) for 4≤n≤8, ⌊d2⌋+1≤κ(DSCn;C4)≤d+1 for n≥16 and κs(DSCn;C4)=⌊d2⌋+1 for n≥4. Finally, we compare and analyze the ratios of structure (resp. substructure) connectivity to vertex degree of divide-and-swap cube with that of several well-known variants of hypercube.

Relation of Extra Edge Connectivity and Component Edge Connectivity for Regular Networks

Reliability of interconnection networks is important to design multiprocessor systems. The extra edge connectivity and component edge connectivity are two parameters for the reliability evaluation. The [Formula: see text]-extra edge connectivity [Formula: see text] is the cardinality of the minimum extra edge cut [Formula: see text] such that [Formula: see text] is not connected and each component of [Formula: see text] has at least [Formula: see text] vertices. The [Formula: see text]-component edge connectivity [Formula: see text] of a graph [Formula: see text] is the minimum edge number of a set [Formula: see text] such that [Formula: see text] is not connected and [Formula: see text] has at least [Formula: see text] components. In this paper, we find the relation of extra edge connectivity and component edge connectivity for regular networks. As an application, we determine the component edge connectivity of BC networks, [Formula: see text]-ary [Formula: see text]-cubes, enhanced hypercubes.

Impact of Raising Switching Stages on the Reliability of Interconnection Networks

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.

The Generalized Connectivity of Bubble-Sort Star Graphs

The connectivity plays an important role in measuring the fault tolerance and reliability of interconnection networks. The generalized [Formula: see text]-connectivity of a graph [Formula: see text], denoted by [Formula: see text], is an important indicator of a network’s ability for fault tolerance and reliability. The bubble-sort star graph, denoted by [Formula: see text], is a well known interconnection network. In this paper, we show that [Formula: see text] for [Formula: see text], that is, for any three vertices in [Formula: see text], there exist [Formula: see text] internally disjoint trees connecting them in [Formula: see text] for [Formula: see text], which attains the upper bound of [Formula: see text] given by Li et al. for [Formula: see text].

On g-extra conditional diagnosability of hierarchical cubic networks

Connectivity and diagnosability are important parameters in measuring the fault-tolerance and reliability of interconnection networks. Given a graph G and a non-negative integer g, the g-extra connectivity of G, denoted by κg(G), is the minimum cardinality of a set of vertices of G, if it exists, whose deletion disconnects G, and every remaining component has more than g vertices. The g-extra conditional diagnosability tg(G) of a graph G is the maximum value of t such that G is g-extra conditional t-diagnosable. The n-dimensional hierarchical cubic network HCNn was proposed by Ghose and Desai [12] as a hypercube-based topology while preserving its attractive features. For n≥5, we first determine κg(HCNn) for 0≤g≤n+1, then establish tg(HCNn) under the PMC model for 0≤g≤n+1, and tg(HCNn) under the MM⁎ model for 0≤g≤n−13, respectively.

Theg-Good-Neighbor Conditional Diagnosability of Locally Exchanged Twisted Cubes

Connectivity and diagnosability are important parameters in measuring the fault tolerance and reliability of interconnection networks. The $R^g$-vertex-connectivity of a connected graph $G$ is the minimum cardinality of a faulty set $X\subseteq V(G)$ such that $G-X$ is disconnected and every fault-free vertex has at least $g$ fault-free neighbors. The $g$-good-neighbor conditional diagnosability is defined as the maximum cardinality of a $g$-good-neighbor conditional faulty set that the system can guarantee to identify. The interconnection network considered here is the locally exchanged twisted cube $LeTQ(s,t)$. For $1\leq s\leq t$ and $0\leq g\leq s$, we first determine the $R^g$-vertex-connectivity of $LeTQ(s,t)$, then establish the $g$-good neighbor conditional diagnosability of $LeTQ(s,t)$ under the PMC model and MM$^*$ model, respectively.

The g-good neighbor conditional diagnosability of twisted hypercubes under the PMC and MM* model

Connectivity and diagnosability are important parameters in measuring the fault tolerance and reliability of interconnection networks. The g-good-neighbor conditional faulty set is a special faulty set that every fault-free vertex should have at least g fault-free neighbors. The Rg-vertex-connectivity of a connected graph G is the minimum cardinality of a g-good-neighbor conditional faulty set X⊆V(G) such that G−X is disconnected. The g-good-neighbor conditional diagnosability is a metric that can give the maximum cardinality of g-good-neighbor conditional faulty set that the system is guaranteed to identify. The twisted hypercube is a new variant of hypercubes with asymptotically optimal diameter introduced by X.D. Zhu. In this paper, we first determine the Rg-vertex-connectivity of twisted hypercubes, then establish the g-good neighbor conditional diagnosability of twisted hypercubes under the PMC model and MM* model, respectively.

The g-good-neighbor conditional diagnosability of star graphs under the PMC and MM* model

Diagnosability plays an important role in measuring the fault tolerance and reliability of interconnection networks. The g-good-neighbor conditional faulty set is a special faulty set that every fault-free vertex should have at least g fault-free neighbors. The g-good-neighbor conditional diagnosability is a metric that can give the maximum cardinality of g-good-neighbor conditional faulty set that the system is guaranteed to identify. This paper shows that the g-good-neighbor conditional diagnosabilities of star graph Sn under the PMC and MM* model are (n−g)(g+1)!−1 for n≥4 and 0≤g≤n−2. Our study shows the g-good-neighbor conditional diagnosability of star graph is several times larger than the classical diagnosability of star graph.