Fault tolerant networks with small degree

We investigate the dynamics of a susceptible infected recovered (SIR) epidemic model on small networks with different topologies, as a stepping stone to determining how the structure of a contact network impacts the transmission of infection through a population. For an SIR model on a network of$N$nodes, there are$3^{N}$configurations that the network can be in. To simplify the analysis, we group the states together based on the number of nodes in each infection state and the symmetries of the network. We derive analytical expressions for the final epidemic size of an SIR model on small networks composed of three or four nodes with different topological structures. Differential equations which describe the transition of the network between states are also derived and solved numerically to confirm our analysis. A stochastic SIR model is numerically simulated on each of the small networks with the same initial conditions and infection parameters to confirm our results independently. We show that the structure of the network, degree of the initial infectious node, number of initial infectious nodes and the transmission rate all significantly impact the final epidemic size of an SIR model on small networks.

The Steiner tree problem is one of the fundamental optimization problems in distributed graph algorithms. Recently Saikia and Karmakar [27] proposed a deterministic distributed algorithm for the Steiner tree problem that constructs a Steiner tree in \(O(S + \sqrt{n} \log ^* n)\) rounds whose cost is optimal upto a factor of \(2(1 - 1/\ell )\), where n and S are the number of nodes and shortest path diameter [17] respectively of the given input graph and \(\ell \) is the number of terminal leaf nodes in the optimal Steiner tree. The message complexity of the algorithm is \(O(\varDelta (n - t) S + n^{3/2})\), which is equivalent to \(O(mS + n^{3/2})\), where \(\varDelta \) is the maximum degree of a vertex in the graph, t is the number of terminal nodes (we assume that \(t < n\)), and m is the number of edges in the given input graph. This algorithm has a better round complexity than the previous best algorithm for Steiner tree construction due to Lenzen and Patt-Shamir [21]. In this paper we present a deterministic distributed algorithm which constructs a Steiner tree in \(\tilde{O}(S + \sqrt{n})\) rounds and \(\tilde{O}(mS)\) messages and still achieves an approximation factor of \(2(1 - 1/\ell )\). Note here that \(\tilde{O}(\cdot )\) notation hides polylogarithmic factors in n. This algorithm improves the message complexity of Saikia and Karmakar’s algorithm by dropping the additive term of \(O(n^{3/2})\) at the expense of a logarithmic multiplicative factor in the round complexity. Furthermore, we show that for sufficiently small values of the shortest path diameter \((S=O(\log n))\), a \(2(1 - 1/\ell )\)-approximate Steiner tree can be computed in \(\tilde{O}(\sqrt{n})\) rounds and \(\tilde{O}(m)\) messages and these complexities almost coincide with the results of some of the singularly-optimal minimum spanning tree (MST) algorithms proposed in [9, 12, 23].

Recent researches about the fault and intrusion tolerance of the complex networks take the topology or functions of the networks as benchmark. This paper shows a novel framework of the fault and intrusion tolerance of complex networks in a controllability view, which takes the number of driver nodes as standard. The result, the SF networks have a better fault tolerance than the ER networks, is consistent with the result considered with available researches. But the SF networks and ER networks have poor intrusion tolerance. The reason is that it is more difficult to fully control the real SF networks than to fully control the real ER networks, so it is easier to thoroughly destroy the controllability of the real SF networks than the real ER networks.

The Fibonacci cube was proposed recently as an interconnection network. It has been shown that this new network topology possesses many interesting properties that are important in network design and applications. This paper addresses the following network simulation problem: Given a linear array, a ring or a two-dimensional mesh, how can be assign its nodes to the Fibonacci cube nodes so as to keep their adjacent nodes near each other in the Fibonacci cube. The authors first show a simple fact that there is a Hamiltonian path in any Fibonacci cube. They prove that any ring structure can be embedded into its corresponding optimum Fibonacci cube (the smallest Fibonacci cube with at least the number of nodes in the ring) with dilation 2, which is optimum for most cases. Then, they describe dilation 1 embeddings of a class of meshes into their corresponding optimum Fibonacci cubes. Finally, it is shown that an arbitrary mesh can be embedded into its corresponding optimum or near-optimum Fibonacci cube with dilation 2.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Early designs of mobile ad-hoc networks (MANETs) assumed they could be scaled to very large topologically flat networks where every node in the network has equal capability and access to the wireless channel. This assumption was called into question in 2000 by Gupta and Kumar [1] who established an asymptotic theoretical limit for node capacity as the number of nodes in a single channel MANET becomes large. Recent field experiments have confirmed that this theoretical limit may be difficult to overcome. To address MANET scalability alternative MANET designs have been developed in which a wireless channel are shared by only a subset of the entire collection of network nodes. Subnets are then linked by gateway nodes that operate on two or more channels and provide connectivity across the entire network. In these MANET designs gateway nodes transmit and receive on multiple subnets and can become bottlenecks in the network if there are too few gateways. On the other hand, if the network contains many gateways, then too much subnet bandwidth may be allocated to gateways and insufficient bandwidth allocated to single channel nodes. This paper examines the design of MANETs based on two subnets and determines the optimal number of gateway nodes which maximizes overall network performance for a uniform message traffic loading.

With the deep integration of communication technology and Internet of Things technology, the edge network structure is becoming increasingly dense and heterogeneous. At the same time, in the edge network environment, characteristics such as wide-area differentiated services, decentralized deployment of computing and network resources, and highly dynamic network environment lead to the deployment of redundant or insufficient edge cache nodes, which restricts the efficiency of network service caching and resource allocation. In response to the above problems, research on the joint optimization of service caching and resources in the decentralized edge network scenario is carried out. Therefore, we have conducted research on the collaborative caching of training data among multiple edge nodes and optimized the number of collaborative caching nodes. Firstly, we use a multi-queue model to model the collaborative caching process. This model can be used to simulate the in-network cache replacement process on collaborative caching nodes. In this way, we can describe the data flow and storage changes during the caching process more clearly. Secondly, considering the limitation of storage space of edge nodes and the demand for training data within a training epoch, we propose a stochastic gradient descent algorithm to obtain the optimal number of caching nodes. This algorithm entirely takes into account the resource constraints in practical applications and provides an effective way to optimize the number of caching nodes. Finally, the simulation results clearly show that the optimized number of caching nodes can significantly improve the adequacy rate and hit rate of the training data, with the adequacy rate reaching 84% and the hit rate reaching 100%.

We present an experimental study of parallel biconnected components algorithms employing several fundamental parallel primitives, e.g., prefix sum, list ranking, sorting, connectivity, spanning tree, and tree computations. Previous experimental studies of these primitives demonstrate reasonable parallel speedups. However, when these algorithms are used as subroutines to solve higher-level problems, there are two factors that hinder fast parallel implementations. One is parallel overhead, i.e., the large constant factors hidden in the asymptotic bounds; the other is the discrepancy among the data structures used in the primitives that brings non-negligible conversion cost. We present various optimization techniques and a new parallel algorithm that significantly improve the performance of finding biconnected components of a graph on symmetric multiprocessors (SMPs). Finding biconnected components has application in fault-tolerant network design, and is also used in graph planarity testing. Our parallel implementation achieves speedups up to 4 using 12 processors on a Sun E4500 for large, sparse graphs, and the source code is freely-available at our Web site.

The hierarchical dual-net (HDN) was introduced as a topology of interconnection networks for extremely large parallel computers. The HDN is constructed based on a symmetric product graph (base network). A k-level hierarchical dual-net, HDN(B,k,S), contains (2N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2k</sup> /(2 × Π <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i=1</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sup> s <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> ) nodes, where S = {G' <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> , G' <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> , . . . , G' <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sub> }, G' <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> is a super-node and s <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> = |G' <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> | is the number of nodes in the super-node at the level i for 1 ≤ i ≤ k, and N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> is the number of nodes in the base network B. The node degree of HDN(B, k, S) is d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> + k, where d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> is the node degree of the base network. The HDN is node and edge symmetric and can contain huge number of nodes with small node-degree and short diameter. Disjoint-path routing is a fundamental and critical issue for the performance of an interconnection network. In this paper, we propose an efficient algorithm for finding disjoint-paths on an HDN and give the performance simulation results.

We first introduce a flexible interconnection network, called the hierarchical dual-net (HDN), with low node degree and short diameter for constructing a large-scale supercomputer. The HDN is constructed based on a symmetric product graph (base network). A k-level hierarchical dual-net, HDN(B,k, S), contains (2N0)2k/(2Πi=1sik) nodes, where S = {si|1 ≤ i ≤ k} is the set of integers with each si representing the number of nodes in a super-node at the level i for 1 ≤ i ≤ k, and N0 is the number of nodes in the base network B. The node degree of HDN(B, k,S) is d0 + k, where d0 is the node degree of the base network. The benefit of the HDN is that we can select suitable si to control the growing speed of the number of nodes for constructing a supercomputer of the desired scale. Then we show that an HDN with the base network of p-ary q-cube is Hamiltonian and give an efficient algorithm for finding a Hamiltonian cycle in such hierarchical dual-nets.

We consider the problem of finding a local optimum for the Max-Cut problem with FLIP-neighborhood, in which exactly one node changes the partition. Schaffer and Yannakakis (SICOMP, 1991) showed PLS-completeness of this problem on graphs with unbounded degree. On the other side, Poljak (SICOMP, 1995) showed that in cubic graphs every FLIP local search takes O(n2) steps, where n is the number of nodes. Due to the huge gap between degree three and unbounded degree, Ackermann, Roglin, and Vocking (JACM, 2008) asked for the smallest d such that on graphs with maximum degree d the local MAX-CUT problem with FLIP-neighborhood is PLS-complete. In this paper, we prove that the computation of a local optimum on graphs with maximum degree five is PLS-complete. Thus, we solve the problem posed by Ackermann et al. almost completely by showing that d is either four or five (unless PLS ⊆ P). On the other side, we also prove that on graphs with degree O(log n) every FLIP local search has probably polynomial smoothed complexity. Roughly speaking, for any instance, in which the edge weights are perturbated by a (Gaussian) random noise with variance &sigma2, every FLIP local search terminates in time polynomial in n and σ-1, with probability 1-n-Ω(1). Putting both results together, we may conclude that although local MAX-CUT is likely to be hard on graphs with bounded degree, it can be solved in polynomial time for slightly perturbated instances with high probability.

Interconnection Networks are widely used in many applications such as parallel and massively parallel systems, industrial applications, VLSI circuits, electronic devices and telephone switches etc. The realization of data communication in parallel system is possible all the way through interconnection networks. The whole performance of the system depends on the type of interconnection networks used. Interconnection networks with embed ability by some other interconnection structure, high resilience, simple routing, constant node degree, and scalability are highly desirable. We had introduced Octagon-Cell Networks (OCN), in the past research, which is a new interconnection network. In this research, we introduce and prove the fundamental theorems for finding the optimal path length between any pair of source and destination nodes in Octagon-Cell Network of any depth. The basic objective of these theorems (excluding Horizontal and Vertical Moves) to calculate the optimal path length, is to check whether any intermediate node exists along the optimal path, for which its line number or serial number matches with the line number or the serial number of destination node respectively. We also introduce the theorems to find the total number of horizontal lines, which is the maximum line number of a node, as well the total number of vertical lines, which is the maximum serial number of a node in Octagon-Cell Network of any depth. The theorems are proved mathematically and are tested successfully on all possible cases.

Generalized [formula omitted]-sparse algorithms for constructing fault tolerant RBF networks

The increasingly large ensemble size of modern High-Performance Computing (HPC) systems has drastically increased the possibility of failures. Performance under failures and its optimization become timely important issues facing the HPC community. In this study, we propose an analytical model to predict the application performance. The model characterizes the impact of coordinated checkpointing and system failures on application performance, considering all the factors including workload, the number of nodes, failure arrival rate, recovery cost, and checkpointing interval and overhead. Based on the model, we gauge three parameters, the number of compute nodes, checkpointing interval, and the number of spare nodes to conduct a comprehensive study of performance optimization under failures. Performance scalability under failures is also studied to explore the performance improvement space for different parameters. Experimental results from both synthetic and actual system failure logs confirm that the proposed model and optimization methodologies are effective and feasible.

We explore the use of software-defined networking (SDN) technology in building a communication network for smart grid. With cyber-physical interdependence, such communication network may suffer from cross-network cascading failures. To prevent the failures, we perform virtual network function (VNF) orchestration jointly with power-disjoint routing. Our work is novel in proposing an efficient scheme to find power-disjoint routes at the same time of performing VNF orchestration. We formulate an optimization to maximize the ratio of power-disjoint route count to VNF orchestration cost. The optimization has a non-linear non-convex objective function. We propose a two-level hierarchical solution approach. At higher level, the scheme converts the problem into a fractional maximum flow circulation, which can be solved using simplex method to find the maximum number of power-disjoint routes. Given a higher level solution, the lower level aims to minimize the VNF orchestration cost while satisfying VNF chaining and placement requirements. This lower level hierarchy uses the Dijkstra’s algorithm in building a sequence of minimum spanning trees, each roots at the current VNF hosting node in a VNF chain. Extensive simulation results confirm that the proposed scheme can find the maximum number of power-disjoint routes and minimize the cost within a second, for a system with 120 communication nodes. The results show that the number of power-disjoint routes can be increased by increasing either the number of nodes or node degree, but only the node degree can keep the cost flat. Therefore, one should build a robust software-defined smart grid communication network by enhancing node connectivity.

We propose a new class of interconnection networks, called macro-star networks, which belong to the class of Cayley graphs and use the star graph as a basic building module. A macro-star network can have node degree that is considerably smaller than that of a star graph of the same size, and diameter that is sublogarithmic and asymptotically within a factor of 1.25 from a universal lower bound (given its node degree). We show that algorithms developed for star graphs can be emulated on suitably constructed macro-stars with asymptotically optimal slowdown. This enables us to obtain through emulation a variety of efficient algorithms for the macro-star network, thus proving its versatility. Basic communication tasks, such as the multimode broadcast and the total exchange, can be executed in macro-star networks in asymptotically optimal time under both the single-port and the all-port communication models. Moreover, no interconnection network with similar node degree can perform these communication tasks in time that is better by more than a constant factor than that required in a macro-star network. We show that macro-star networks can embed trees, meshes, hypercubes, as well as star, bubble-sort, and complete transposition graphs with constant dilation. We introduce several variants of the macro-star network that provide more flexibility in scaling up the number of nodes. We also discuss implementation issues and compare the new topology with the star graph and other popular topologies.