Designing routing schemes is a multidimensional and complex task that depends on the objective function, the computational model (centralized vs. distributed), and the amount of uncertainty (online vs. offline). Nevertheless, there are quite a few well-studied general techniques, for a large variety of network problems. In contrast, in our view, practical techniques for designing robust routing schemes are scarce; while fault-tolerance has been studied from a number of angles, existing approaches are concerned with dealing with faults after the fact by rerouting, self-healing, or similar techniques. We argue that this comes at a high burden for the designer, as in such a system any algorithm must account for the effects of faults on communication. With the goal of initiating efforts towards addressing this issue, we showcase simple and generic transformations that can be used as a blackbox to increase resilience against (independently distributed) faults. Given a network and a routing scheme, we determine a reinforced network and corresponding routing scheme that faithfully preserves the specification and behavior of the original scheme. We show that reasonably small constant overheads in terms of size of the new network compared to the old are sufficient for substantially relaxing the reliability requirements on individual components. The main message in this paper is that the task of designing a robust routing scheme can be decoupled into (i) designing a routing scheme that meets the specification in a fault-free environment, (ii) ensuring that nodes correspond to fault-containment regions, i.e., fail (approximately) independently, and (iii) applying our transformation to obtain a reinforced network and a robust routing scheme that is fault-tolerant.
The literature also considered random fault models. In the network reliability problem, the goal is to compute the probability that the (connected) input network becomes disconnected under random independent edge failures. The reliability of a network is the probability that the network remains connected after this random process. Karger @cite_15 gave a fully polynomial randomized approximation scheme for the network reliability problem. Chechik et. al @cite_17 studied a variant of the task, in which the goal is to compute a sparse sub-network that approximates the reliability of the input network. We, on the other hand, construct a reinforced network that increases the reliability of the input network; note also that our requirements are much stricter than merely preserving connectivity. For a more detailed survey on Fault-Tolerant Logical Network Structures, see @cite_4 .
@cite_15 @cite_4 @cite_17
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The classic all-terminal network reliability problem posits a graph, each of whose edges fails independently with some given probability. The goal is to determine the probability that the network becomes disconnected due to edge failures. This problem has obvious applications in the design of communication networks. Since the problem is @math -complete and thus believed hard to solve exactly, a great deal of research has been devoted to estimating the failure probability. In this paper, we give a fully polynomial randomized approximation scheme that, given any n-vertex graph with specified failure probabilities, computes in time polynomial in n and @math an estimate for the failure probability that is accurate to within a relative error of @math with high probability. We also give a deterministic polynomial approximation scheme for the case of small failure probabilities. Some extensions to evaluating probabilities of @math -connectivity, strong connectivity in directed Eulerian graphs and @math -way disconnection, and to evaluating the Tutte polynomial are also described. This version of the paper corrects several errata that appeared in the previous journal publication [D. R. Karger, SIAM J. Comput., 29 (1999), pp. 492--514]. We consider network reliability in layered networks where the lower layer experiences random link failures. In layered networks, each failure at the lower layer may lead to multiple failures at the upper layer. We generalize the classical polynomial expression for network reliability to the multilayer setting. Using random sampling techniques, we develop polynomial-time approximation algorithms for the failure polynomial. Our approach gives an approximate expression for reliability as a function of the link failure probability, eliminating the need to resample for different values of the failure probability. Furthermore, it gives insight on how the routings of the logical topology on the physical topology impact network reliability. We show that maximizing the min cut of the (layered) network maximizes reliability in the low-failure-probability regime. Based on this observation, we develop algorithms for routing the logical topology to maximize reliability. We consider network reliability in layered networks where the lower layer experiences random link failures. In layered networks, each failure at the lower layer may lead to multiple failures at the upper layer. We generalize the classical polynomial expression for network reliability to the multi-layer setting. Using random sampling techniques, we develop polynomial time approximation algorithms for the failure polynomial. Our approach gives an approximate expression for reliability as a function of the link failure probability, eliminating the need to resample for different values of the failure probability. Furthermore, it gives insight on how the routings of the logical topology on the physical topology impact network reliability. We show that maximizing the min cut of the (layered) network maximizes reliability in the low failure probability regime. Based on this observation, we develop algorithms for routing the logical topology to maximize reliability. We consider the network reliability problem in secure wireless sensor networks that employ a heterogeneous random key predistribution scheme. This scheme is introduced recently, as a generalization of the Eschenauer-Gligor scheme, to account for the cases when the network comprises sensors with varying level of resources or connectivity requirements; e.g., regular nodes vs. cluster heads. The scheme induces the inhomogeneous random key graph, denoted G(n;μ,K, P ), where each of the n nodes are independently assigned to one of r classes according to a probability distribution μ = (μ1, . . . , μr) and then a class-i node is assigned Ki keys uniformly at random from a pool of size P ; two nodes that share a key are then connected by an edge. We analyze the reliability of G(n;μ,K, P ) against random link failures. Namely, we consider G(n;μ,K, P, α) formed by deleting each edge of G(n;μ,K, P ) independently with probability 1 − α, and study the probability that the resulting graph i) has no isolated node; and ii) is connected. We present scaling conditions on K , P , and α such that both events take place with probability zero or one, respectively, as the number of nodes gets large. We also present numerical results to support these zero-one laws in the finite-node regime. Given a connected graph G and a failure probability p(e) for each edge e in G, the reliability of G is the probability that G remains connected when each edge e is removed independently with probability p(e). In this paper it is shown that every n-vertex graph contains a sparse backbone, i.e., a spanning subgraph with O(nlogn) edges whose reliability is at least (1-n^-^@W^(^1^)) times that of G. Moreover, for any pair of vertices s, t in G, the (s,t)-reliability of the backbone, namely, the probability that s and t remain connected, is also at least (1-n^-^@W^(^1^)) times that of G. Our proof is based on a polynomial time randomized algorithm for constructing the backbone. In addition, it is shown that the constructed backbone has nearly the same Tutte polynomial as the original graph (in the quarter-plane x>=1, y>1), and hence the graph and its backbone share many additional features encoded by the Tutte polynomial. To be considered for an IEEE Jack Keil Wolf ISIT Student Paper Award. We study the secure and reliable connectivity of wireless sensor networks. Security is assumed to be ensured by the random pairwise key predistribution scheme of Chan, Perrig, and Song, and unreliable wireless links are represented by independent on off channels. Modeling the network by an intersection of a random K-out graph and an Erdős-Renyi graph, we present scaling conditions (on the number of nodes, the scheme parameter K, and the probability of a wireless channel being on) such that the resulting graph contains no nodes with degree less than k with high probability, when the number of nodes gets large. Results are given in the form of zero-one laws and are shown to improve the previous results by Yagan and Makowski on the absence of isolated nodes (i.e., absence of nodes with degree zero). Via simulations, the established zero-one laws are shown to hold also for the property of k-connectivity; i.e., the property that graph remains connected despite the deletion of any k − 1 nodes or edges. We consider computational aspects of a game theoretic approach to network reliability. Consider a network where failure of one node may disrupt communication between two other nodes. We model this network as a simple coalitional game, called the vertex Connectivity Game (CG). In this game, each agent owns a vertex, and controls all the edges going to and from that vertex. A coalition of agents wins if it fully connects a certain subset of vertices in the graph, called the primary vertices. We show that power indices, which express an agent's ability to affect the outcome of the vertex connectivity game, can be used to identify significant possible points of failure in the communication network, and can thus be used to increase network reliability. We show that in general graphs, calculating the Banzhaf power index is #P-complete, but suggest a polynomial algorithm for calculating this index in trees. We also show a polynomial algorithm for computing the core of a CG, which allows a stable division of payments to coalition agents. We consider secure and reliable connectivity in wireless sensor networks that utilize the heterogeneous random key predistribution scheme. We model the unreliability of wireless links by an on off channel model that induces an Erdős-Renyi graph, while the heterogeneous scheme induces an inhomogeneous random key graph. The overall network can thus be modeled by the intersection of both graphs. We present conditions (in the form of zero-one laws) on how to scale the parameters of the intersection model, so that with high probability: i) all of its nodes are connected to at least @math other nodes, i.e., the minimum node degree of the graph is no less than @math , and ii) the graph is @math -connected, i.e., the graph remains connected even if any @math nodes leave the network. These results are shown to complement and generalize several previous results in the literature. We also present numerical results to support our findings in the finite-node regime. Finally, we demonstrate via simulations that our results are also useful when the on off channel model is replaced with the more realistic disk communication model . Consider a network consisting of two subnetworks (communities) connected by some external edges. Given the network topology, the community detection problem can be cast as a graph partitioning problem that aims to identify the external edges as the graph cut that separates these two subnetworks. In this paper, we consider a general model where two arbitrarily connected subnetworks are connected by random external edges. Using random matrix theory and concentration inequalities, we show that when one performs community detection via spectral clustering there exists an abrupt phase transition as a function of the random external edge connection probability. Specifically, the community detection performance transitions from almost perfect detectability to low detectability near some critical value of the random external edge connection probability. We derive upper and lower bounds on the critical value and show that the bounds are equal to each other when two subnetwork sizes are identical. Using simulated and experimental data we show how these bounds can be empirically estimated to validate the detection reliability of any discovered communities. Several enumeration and reliability problems are shown to be # P-complete, and hence, at least as hard as NP-complete problems. Included are important problems in network reliability analysis, namely, computing the probability that a graph is connected and counting the number of minimum cardinality @math -cuts or directed network cuts. Also shown to be # P-complete are counting vertex covers in a bipartite graph, counting antichains in a partial order, and approximating the probability that a graph is connected and the probability that a pair of vertices is connected. Networks-on-Chip constitute the interconnection architecture of future, massively parallel multiprocessors that assemble hundreds to thousands of processing cores on a single chip. Their integration is enabled by ongoing miniaturization of chip manufacturing technologies following Moore's Law. It comes with the downside of the circuit elements' increased susceptibility to failure. Research on fault-tolerant Networks-on-Chip tries to mitigate partial failure and its effect on network performance and reliability by exploiting various forms of redundancy at the suitable network layers. The article at hand reviews the failure mechanisms, fault models, diagnosis techniques, and fault-tolerance methods in on-chip networks, and surveys and summarizes the research of the last ten years. It is structured along three communication layers: the data link, the network, and the transport layers. The most important results are summarized and open research problems and challenges are highlighted to guide future research on this topic. Practitioners agree that unreliable links, which sometimes deliver messages and sometime do not, are an important characteristic of wireless networks. In contrast, most theoretical models of radio networks fix a static set of links and assume that these links are reliable. This gap between theory and practice motivates us to investigate how unreliable links affect theoretical bounds on broadcast in radio networks. To that end we consider a model that includes two types of links: reliable links, which always deliver messages, and unreliable links, which sometimes fail to deliver messages. We assume that the reliable links induce a connected graph, and that unreliable links are controlled by a worst-case adversary. In the new model we show an Ω(n log n) lower bound on deterministic broadcast in undirected graphs, even when all processes are initially awake and have collision detection, and an Ω(n) lower bound on randomized broadcast in undirected networks of constant diameter. This separates the new model from the classical, reliable model. On the positive side, we give two algorithms that tolerate unreliability: an O(n3 2 √log n)-time deterministic algorithm and a randomized algorithm which terminates in O(n log2 n) rounds with high probability. Traditionally, models of fault-tolerant distributed computing assume that failures are “uniform”: processes are equally probable to fail and a failure of one process does not affect reliability of the others. In real systems, however, processes may not be equally reliable. Moreover, failures may be correlated because of software or hardware features shared by subsets of processes. In this paper, we survey recent results addressing the question of what can and what cannot be computed in systems with non-identical and non-independent failures The undisrupted operation of logical structures and services is a crucial requirement in modern day communication networks. As the vertices and edges of the network may occasionally fail or malfunction, it is desirable to make those structures robust against failures. Fault-Tolerant (FT) Logical Network Structures are low cost high resilient structures, constructed on top of a given network, that satisfy certain desirable performance requirements concerning, e.g., connectivity, distance or capacity. In this survey, we review some recent results for designing FT network structures. We classify the existing construction algorithms into four classes and discuss the settings in which a given approach might become useful. We hope that this would pave the way towards the development of a more generalized theory for the design of fault-tolerant networks. In this paper, we study crucial elements of a complex network, namely its nodes and connections, which play a key role in maintaining the network’s structure and function under unexpected structural perturbations of nodes and edges removal. Specifically, we want to identify vital nodes and edges whose failure (either random or intentional) will break the most number of connected triples (or triangles) in the network. This problem is extremely important, because connected triples form the foundation of strong connections in many real-world systems, such as mutual relationships in social networks, reliable data transmission in communication networks, and stable routing strategies in mobile networks. Disconnected triples, analog to broken mutual connections, can greatly affect the network’s structure and disrupt its normal function, which can further lead to the corruption of the entire system. The analysis of such crucial elements will shed light on key factors behind the resilience and robustness of many complex systems in practice. We formulate the analysis under multiple optimization problems and show their intractability. We next propose efficient approximation algorithms, namely, DAK-n and DAK-e , which guarantee an @math -approximate ratio (compared with the overall optimal solutions) while having the same time complexity as the best triangle counting and listing algorithm on power-law networks. This advantage makes our algorithms scale extremely well even for very large networks. In an application perspective, we perform comprehensive experiments on real social traces with millions of nodes and billions of edges. Empirical results indicate that our approaches achieve comparably better solution quality while are up to @math faster than the current state-of-the-art methods. We re-introduce the coded model of fault-tolerant computation in which the input and output of a computational device are treated as words in an error-correcting code. A computational device correctly computes a function in the coded model if its input and output, once decoded, are a valid input and output of the function. In the coded model, it is reasonable to hope to simulate all computational devices by devices whose size is greater by a constant factor but which are exponentially reliable even if each of their components can fail with some constant probability. We consider fine-grained parallel computations in which each processor has a constant probability of producing the wrong output at each time step. We show that any parallel computation that runs for time t on w processors can be performed reliably on a faulty machine in the coded model using wlog sup 0(1 )w processors and time tlog sup 0(1) w. The failure probability of the computation will be at most t spl middot exp(-w sup 1 4 ). The codes used to communicate with our fault-tolerant machines are generalized Reed-Solomon codes and can thus be encoded and decoded in O(nlog sup 0(1) n) sequential time and are independent of the machine they are used to communicate with. We also show how coded computation can be used to self-correct many linear functions in parallel with arbitrarily small overhead. In the unconditionally reliable message transmission (URMT) problem, two non-faulty players, the sender S and the receiver R are part of a synchronous network modeled as a directed graph. S has a message that he wishes to send to R; the challenge is to design a protocol such that after exchanging messages as per the protocol, the receiver R should correctly obtain S's message with arbitrarily small error probability Δ, in spite of the influence of a Byzantine adversary that may actively corrupt up to t nodes in the network (we denote such a URMT protocol as (t, (1 - Δ))-reliable). While it is known that (2t + 1) vertex disjoint directed paths from S to R are necessary and sufficient for (t, 1)-reliable URMT (that is with zero error probability), we prove that a strictly weaker condition, which we define and denote as (2t, t)-special-connectivity, together with just (t+1) vertex disjoint directed paths from S to R, is necessary and sufficient for (t, (1' - Δ))-reliable URMT with arbitrarily small (but non-zero) error probability, Δ. Thus, we demonstrate the power of randomization in the context of reliable message transmission. In fact, for any positive integer k > 0, we show that there always exists a digraph Gk such that (k, 1)-reliable URMT is impossible over Gk whereas there exists a (2k, (1 - Δ))-reliable URMT protocol, Δ > 0 in Gk. In a digraph G on which (t, (1 - Δ))-reliable URMT is possible, an edge is called critical if the deletion of that edge renders (t, (1 - Δ))-reliable URMT impossible. We give an example of a digraph G on n vertices such that G has Ω(n2) critical edges. This is quite baffling since no such graph exists for the case of perfect reliable message transmission (or equivalently (t, 1)-reliable URMT) or when the underlying graph is undirected. Such is the anomalous behavior of URMT protocols (when "randomness meet directedness") that it makes it extremely hard to design efficient protocols over arbitrary digraphs. However, if URMT is possible between every pair of vertices in the network, then we present efficient protocols for the same. A natural requirement of many distributed structures is fault-tolerance: after some failures, whatever remains from the structure should still be effective for whatever remains from the network. In this paper we examine spanners of general graphs that are tolerant to vertex failures, and significantly improve their dependence on the number of faults @math , for all stretch bounds. For stretch @math we design a simple transformation that converts every @math -spanner construction with at most @math edges into an @math -fault-tolerant @math -spanner construction with at most @math edges. Applying this to standard greedy spanner constructions gives @math -fault tolerant @math -spanners with @math edges. The previous construction by Chechik, Langberg, Peleg, and Roddity [STOC 2009] depends similarly on @math but exponentially on @math (approximately like @math ). For the case @math and unit-length edges, an @math -approximation algorithm is known from recent work of Dinitz and Krauthgamer [arXiv 2010], where several spanner results are obtained using a common approach of rounding a natural flow-based linear programming relaxation. Here we use a different (stronger) LP relaxation and improve the approximation ratio to @math , which is, notably, independent of the number of faults @math . We further strengthen this bound in terms of the maximum degree by using the Local Lemma. Finally, we show that most of our constructions are inherently local by designing equivalent distributed algorithms in the LOCAL model of distributed computation. In this paper we show how the permutation Monte Carlo method, originally developed for reliability networks, can be successfully adapted for stochastic flow networks, and in particular for estimation of the probability that the maximal flow in such a network is above some fixed level, called the threshold. A stochastic flow network is defined as one, where the edges are subject to random failures. A failed edge is assumed to be erased (broken) and, thus, not able to deliver any flow. We consider two models; one where the edges fail with the same failure probability and another where they fail with different failure probabilities. For each model we construct a different algorithm for estimation of the desired probability; in the former case it is based on the well known notion of the D-spectrum and in the later one—on the permutational Monte Carlo. We discuss the convergence properties of our estimators and present supportive numerical results. It may not be feasible for sensor networks monitoring nature and inaccessible geographical regions to include powered sinks with Internet connections. We consider the scenario where sinks are not present in large-scale sensor networks, and unreliable sensors have to collectively resort to storing sensed data over time on themselves. At a time of convenience, such cached data from a small subset of live sensors may be collected by a centralized (possibly mobile) collector. In this paper, we propose a decentralized algorithm using fountain codes to guarantee the persistence and reliability of cached data on unreliable sensors. With fountain codes, the collector is able to recover all data as long as a sufficient number of sensors are alive. We use random walks to disseminate data from a sensor to a random subset of sensors in the network. Our algorithms take advantage of the low decoding complexity of fountain codes, as well as the scalability of the dissemination process via random walks. We have proposed two algorithms based on random walks. Our theoretical analysis and simulation-based studies have shown that, the first algorithm maintains the same level of fault tolerance as the original centralized fountain code, while introducing lower overhead than naive random-walk based implementation in the dissemination process. Our second algorithm has lower level of fault tolerance than the original centralized fountain code, but consumes much lower dissemination cost.