Sequential Importance Sampling Algorithms for Estimating the All-Terminal Reliability Polynomial of Sparse Graphs

The all-terminal reliability polynomial of a graph counts its connected subgraphs of various sizes. Algorithms based on sequential importance sampling (SIS) have been proposed to estimate a graph\u27s reliability polynomial. We show upper bounds on the relative error of three sequential importance sampling algorithms. We use these to create a hybrid algorithm, which selects the best SIS algorithm for a particular graph G and particular coefficient of the polynomial. This hybrid algorithm is particularly effective when G has low degree. For graphs of average degree < 11, it is the fastest known algorithm; for graphs of average degree <= 45 it is the fastest known polynomial-space algorithm. For example, when a graph has average degree 3, this algorithm estimates to error epsilon in time O(1.26^n * epsilon^{-2}). Although the algorithm may take exponential time, in practice it can have good performance even on medium-scale graphs. We provide experimental results that show quite practical performance on graphs with hundreds of vertices and thousands of edges. By contrast, alternative algorithms are either not rigorous or are completely impractical for such large graphs

Latency and Lifetime Enhancements in Industrial Wireless Sensor Networks : A Q-Learning Approach for Graph Routing

Industrial wireless sensor networks usually have a centralized management approach, where a device known as network manager is responsible for the overall configuration, definition of routes, and allocation of communication resources. Graph routing is used to increase the reliability of communication through path redundancy. Some of the state-of-the-art graph-routing algorithms use weighted cost equations to define preferences on how the routes are constructed. The characteristics and requirements of these networks complicate to find a proper set of weight values to enhance network performance. Reinforcement learning can be useful to adjust these weights according to the current operating conditions of the network. In this article, we present the Q-learning reliable routing with a weighting agent approach, where an agent adjusts the weights of a state-of-the-art graph-routing algorithm. The states of the agent represent sets of weights, and the actions change the weights during network operation. Rewards are given to the agent when the average network latency decreases or the expected network lifetime increases. Simulations were conducted on a WirelessHART simulator considering industrial monitoring applications with random topologies. Results show, in most cases, a reduction of the average network latency while the expected network lifetime and the communication reliability are at least as good as what is obtained by the state-of-the-art graph-routing algorithms

The number and degree distribution of spanning trees in the Tower of Hanoi graph

The number of spanning trees of a graph is an important invariant related to topological and dynamic properties of the graph, such as its reliability, communication aspects, synchronization, and so on. However, the practical enumeration of spanning trees and the study of their properties remain a challenge, particularly for large networks. In this paper, we study the number and degree distribution of the spanning trees in the Hanoi graph. We first establish recursion relations between the number of spanning trees and other spanning subgraphs of the Hanoi graph, from which we find an exact analytical expression for the number of spanning trees of the n-disc Hanoi graph. This result allows the calculation of the spanning tree entropy which is then compared with those for other graphs with the same average degree. Then, we introduce a vertex labeling which allows to find, for each vertex of the graph, its degree distribution among all possible spanning trees.Postprint (author's final draft

Novel insights into the impact of graph structure on SLAM

© 2014 IEEE. SLAM can be viewed as an estimation problem over graphs. It is well known that the topology of each dataset affects the quality of the corresponding optimal estimate. In this paper we present a formal analysis of the impact of graph structure on the reliability of the maximum likelihood estimator. In particular, we show that the number of spanning trees in the graph is closely related to the D-optimality criterion in experimental design. We also reveal that in a special class of linear-Gaussian estimation problems over graphs, the algebraic connectivity is related to the E-optimality design criterion. Furthermore, we explain how the average node degree of the graph is related to the ratio between the minimum of negative log-likelihood achievable and its value at the ground truth. These novel insights give us a deeper understanding of the SLAM problem. Finally we discuss two important applications of our analysis in active measurement selection and graph pruning. The results obtained from simulations and experiments on real data confirm our theoretical findings

Effect of Unequal Clustering Algorithms in WirelessHART networks

The use of Graph Routing in Wireless Highway Addressable Remote Transducer (WirelessHART) networks offers the benefit of increased reliability of communications because of path redundancy and multi-hop network paths. Nonetheless, Graph Routing in a WirelessHART network creates a hotspot challenge resulting from unbalanced energy consumption. This paper proposes the use of unequal clustering algorithms based on Graph Routing in WirelessHART networks to help with balancing energy consumption, maximizing reliability, and reducing the number of hops in the network. Graph Routing is compared with pre-set and probabilistic unequal clustering algorithms in terms of energy consumption, packet delivery ratio, throughput and average end-to-end delay. A simulation test reveals that Graph Routing has improved energy consumption, throughput and reduced average end-to-end delay when conducted using probabilistic unequal clustering algorithms. However, there is no significant change in the packet delivery ratio, as most packets reach their destination successfully anyway

Obtaining and Using Cumulative Bounds of Network Reliability

In this chapter, we study the task of obtaining and using the exact cumulative bounds of various network reliability indices. A network is modeled by a non-directed random graph with reliable nodes and unreliable edges that fail independently. The approach based on cumulative updating of the network reliability bounds was introduced by Won and Karray in 2010. Using this method, we can find out whether the network is reliable enough with respect to a given threshold. The cumulative updating continues until either the lower reliability bound becomes greater than the threshold or the threshold becomes greater than the upper reliability bound. In the first case, we decide that a network is reliable enough; in the second case, we decide that a network is unreliable. We show how to speed up cumulative bounds obtaining by using partial sums and how to update bounds when applying different methods of reduction and decomposition. Various reliability indices are considered: k-terminal probabilistic connectivity, diameter constrained reliability, average pairwise connectivity, and the expected size of a subnetwork that contains a special node. Expected values can be used for unambiguous decision-making about network reliability, development of evolutionary algorithms for network topology optimization, and obtaining approximate reliability values