Underlying Communication Graph Research Articles

Almost-Everywhere Secure Computation with Edge Corruptions

We consider secure multi-party computation (MPC) in a setting where the adversary can separately corrupt not only the parties (nodes) but also the communication channels (edges), and can furthermore choose selectively and adaptively which edges or nodes to corrupt. Note that if an adversary corrupts an edge, even if the two nodes that share that edge are honest, the adversary can control the link and thus deliver wrong messages to both players. We consider this question in the information-theoretic setting, and require security against a computationally unbounded adversary. In a fully connected network the above question is simple (and we also provide an answer that is optimal up to a constant factor). What makes the problem more challenging is to consider the case of sparse networks. Partially connected networks are far more realistic than fully connected networks, which led Garay and Ostrovsky [Eurocrypt'08] to formulate the notion of (unconditional) almost everywhere (a.e.) secure computation in the node-corruption model, i.e., a model in which not all pairs of nodes are connected by secure channels and the adversary can corrupt some of the nodes (but not the edges). In such a setting, MPC among all honest nodes cannot be guaranteed due to the possible poor connectivity of some honest nodes with other honest nodes, and hence some of them must be "given up" and left out of the computation. The number of such nodes is a function of the underlying communication graph and the adversarial set of nodes. In this work we introduce the notion of almost-everywhere secure computation with edge corruptions, which is exactly the same problem as described above, except that we additionally allow the adversary to completely control some of the communication channels between two correct nodes--i.e., to "corrupt" edges in the network. While it is easy to see that an a.e. secure computation protocol for the original node-corruption model is also an a.e. secure computation protocol tolerating edge corruptions (albeit for a reduced fraction of edge corruptions with respect to the bound for node corruptions), no polynomial-time protocol is known in the case where a constant fraction of the edges can be corrupted (i.e., the maximum that can be tolerated) and the degree of the network is sublinear. We make progress on this front, by constructing graphs of degree O(n∈) (for arbitrary constant 0<∈<1) on which we can run a.e. secure computation protocols tolerating a constant fraction of adversarial edges. The number of given-up nodes in our construction is μn (for some constant 0<μ<1 that depends on the fraction of corrupted edges), which is also asymptotically optimal.

Conditions for Detectability in Distributed Consensus-Based Observer Networks

The paper discusses fundamental detectability properties associated with the problem of distributed state estimation using networked observers. The main result of the paper establishes connections between detectability of the plant through measurements, observability of the node filters through interconnections, and algebraic properties of the underlying communication graph, to ensure the interconnected filtering error dynamics are stabilizable via output injection.

Broadcast Gossip Algorithms for Consensus on Strongly Connected Digraphs

We study a general framework for broadcast gossip algorithms which use companion variables to solve the average consensus problem. Each node maintains an initial state and a companion variable. Iterative updates are performed asynchronously whereby one random node broadcasts its current state and companion variables and all other nodes receiving the broadcast update their state and companion variables. We provide conditions under which this scheme is guaranteed to converge to a consensus solution, where all nodes have the same limiting values, on any strongly connected directed graph. Under stronger conditions, which are reasonable when the underlying communication graph is undirected, we guarantee that the consensus value is equal to the average, both in expectation and in the mean-squared sense. Our analysis uses tools from non-negative matrix theory and perturbation theory. The perturbation results rely on a parameter being sufficiently small. We characterize the allowable upper bound as well as the optimal setting for the perturbation parameter as a function of the network topology, and this allows us to characterize the worst-case rate of convergence. Simulations illustrate that, in comparison to existing broadcast gossip algorithms, the approaches proposed in this paper have the advantage that they simultaneously can be guaranteed to converge to the average consensus and they converge in a small number of broadcasts.

LDAC: A localized and decentralized algorithm for efficiently countering wormholes in mobile wireless networks

Several protocols have been proposed to date to defend against wormhole attacks in wireless networks by adopting synchronized clocks, positioning devices, or directional antennas. These requirements and assumptions limit their applicability especially in the case of mobile networks where the degree of association and de-association between the nodes is relatively high. In this work, we present a novel lightweight countermeasure for the wormhole attack, called LDAC (Localized-Decentralized Algorithm for Countering wormholes). It is completely localized and works by looking for simple evidence that no attack is taking place, using only connectivity information, as implied by the underlying communication graph. LDAC is not confined to static networks but extends naturally to dynamic and even mobile ones. Rigorous arguments that prove the correctness of the algorithm are coupled with detailed performance evaluation along with an implementation on real sensor devices that demonstrates its efficiency in terms of memory requirements and processing overhead.

Dynamic Coalitional TU Games: Distributed Bargaining Among Players' Neighbors

We consider a sequence of transferable utility (TU) games where, at each time, the characteristic function is a random vector with realizations restricted to some set of values. The first part of the paper contributes to the definition of a robust (coalitional) TU game and the development of a distributed bargaining protocol. We prove the convergence with probability 1 of the bargaining process to a random allocation that lies in the core of the robust game under some mild conditions on the underlying communication graphs. The second part of the paper addresses the more general case where the robust game may have empty core. In this case, with the dynamic game we associate a dynamic average game by averaging over time the sequence of characteristic functions. Then, we consider an accordingly modified bargaining protocol. Assuming that the sequence of characteristic functions is ergodic and the core of the average game has a nonempty relative interior, we show that the modified bargaining protocol converges with probability 1 to a random allocation that lies in the core of the average game.

Virtual neighbor based connectivity preserving of multi-agent systems with bounded control inputs in the presence of unreliable communication links

In this work the connectivity preserving problem of multiple agents with limited communication ranges in the context of rendezvous tasks has been investigated. The communication among agents is considered to be unreliable and the control inputs of agents are required to be bounded. Based on the explicitly defined virtual neighbors, the proximity graph connectivity of the agent group has been preserved. Moreover, by using the constraint function approach, the control inputs are guaranteed to be bounded, which can be easily implemented in practice. It is proved that the proposed control law can not only ensure union connectivity of the underlying communication graph, but also drive the agents to rendezvous. The effectiveness of the control law is illustrated by numerical examples.

The Role of Persistent Graphs in the Agreement Seeking of Social Networks

This paper investigates the role persistent relations play for a social network to reach a global belief agreement under discrete-time or continuous-time evolution. Each directed arc in the underlying communication graph is assumed to be associated with a time-dependent weight function, which describes the strength of the information flow from one node to another. An arc is said to be persistent if its weight function has infinite L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> or l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> norm for continuous or discrete belief evolutions, respectively. The graph that consists of all persistent arcs is called the persistent graph of the underlying network. Three necessary and sufficient conditions on agreement or ε-agreement are established. We prove that the persistent graph fully determines the convergence to a common opinion in a social network. It is shown how the convergence rate explicitly depends on the diameter of the persistent graph. For a social networking service like Facebook, our results indicate how permanent friendships need to be and what network topology they should form for the network to be an efficient platform for opinion diffusion.

Efficient Decentralized Algorithms for the Distributed Trigger Counting Problem

Consider a distributed system with n processors, in which each processor receives some triggers from an external source. The distributed trigger counting (DTC) problem is to raise an alert and report to a user when the number of triggers received by the system reaches w, where w is a user-specified input. The problem has applications in monitoring, global snapshots, synchronizers and other distributed settings. In this paper, we first present a randomized and decentralized algorithm for the DTC problem with message complexity O(nlognlogw); furthermore, with high probability, no processor receives more than O(lognlogw) messages. Building on the ideas of this algorithm, we develop an improved algorithm having message complexity O(nlogw); furthermore, with high probability, no processor receives more than O(log w) messages. However, neither of the two algorithms provide any bound on the messages sent per processors. These algorithm assume complete connectivity between the processors. Next, we present a third algorithm having message complexity O(nlognlogw), wherein no processor exchanges more than O(lognlogw) messages with high probability; however, there is a negligible failure probability in raising the alert on receiving w triggers. This algorithm only requires that a constant degree tree be embeddable in the underlying communication graph.

Fast Information Spreading in Graphs with Large Weak Conductance

Gathering data from nodes in a network is at the heart of many distributed applications, most notably while performing a global task. We consider information spreading among $n$ nodes of a network, where each node $v$ has a message $m(v)$ which must be received by all other nodes. The time required for information spreading has been previously upper-bounded with an inverse relationship to the conductance of the underlying communication graph. This implies high running time bounds for graphs with small conductance. The main contribution of this paper is an information spreading algorithm which overcomes communication bottlenecks and thus achieves fast information spreading for a wide class of graphs, despite their small conductance. As a key tool in our study we use the recently defined concept of weak conductance, a generalization of classic graph conductance which measures how well-connected the components of a graph are. Our hybrid algorithm, which alternates between random and deterministic communication phases, exploits the connectivity within components by first applying partial information spreading, in which information is exchanged within well-connected components, and then sending messages across bottlenecks, thus spreading further throughout the network. This yields substantial improvements over the best known running times of algorithms for information spreading on any graph that has large weak conductance, from a polynomial to a polylogarithmic number of rounds.

A Token-Based Approach for Distributed Computation in Sensor Networks

We consider distributed algorithms for data aggregation and function computation in sensor networks. The algorithms perform pairwise computations along edges of an underlying communication graph. A token is associated with each sensor node, which acts as a transmission permit. Nodes with active tokens have transmission permits; they generate messages at a constant rate and send each message to a randomly selected neighbor. By using different strategies to control the transmission permits we can obtain tradeoffs between message and time complexity. Gossip corresponds to the case when all nodes have permits all the time. We study algorithms where permits are revoked after transmission and restored upon reception. Examples of such algorithms include Simple-Random Walk (SRW), Coalescent-Random-Walk (CRW), and Controlled Flooding (CFLD) and their hybrid variants. SRW has a single node permit, which is passed on in the network. CRW, initially has a permit for each node but these permits are revoked gradually. The final result for SRW and CRW resides at a single (or few) random node(s) making a direct comparison with GOSSIP difficult. A hybrid two-phase algorithm switching from CRW to CFLD at a suitable predetermined time can be employed to achieve consensus. We show that such hybrid variants achieve significant gains in both message and time complexity. The per-node message complexity for n-node graphs, such as 2-D mesh, torii, and random geometric graphs, scales as <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">polylog</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> )) and the corresponding time complexity scales as <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> ). The reduced per-node message complexity leads to reduced energy utilization in sensor networks.

Topological Boundary Detection in Wireless Sensor Networks

The awareness of boundaries in wireless sensor networks has many benefits. The identification of boundaries is especially challenging since typical wireless sensor networks consist of low-capability nodes that are unaware of their geographic location. In this paper, we propose a simple, efficient algorithm to detect nodes that are near the boundary of the sensor field as well as near the boundaries of holes. Our algorithm relies purely on the connectivity information of the underlying communication graph and does not require any information on the location of nodes. We introduce the 2-neighbor graph concept, and then make use of it to identify nodes near boundaries. The results of our experiment show that our algorithm carries out the task of topological boundary detection correctly and efficiently.

Network Coding: A Computational Perspective

In this work, we study the computational perspective of network coding, focusing on two issues. First, we address the computational complexity of finding a network code for acyclic multicast networks. Second, we address the issue of reducing the amount of computation performed by network nodes. In particular, we consider the problem of finding a network code with the minimum possible number of encoding nodes, i.e. nodes that generate new packets by performing algebraic operations on packets received over incoming links.We present a deterministic algorithm that finds a feasible network code for a multicast network over an underlying graph G(V,E) in time 0(\E\kh + \V\k2h2 + h4k3(k + h)), where k is the number of destinations and h is the number of packets. Our algorithm improves the best known running time for network code construction. In addition, our algorithm guarantees that the number of encoding nodes in the obtained network code is upper- bounded by 0(h3k2). Next, we address the problem of finding integral and fractional network codes with the minimum number of encoding nodes. We prove that in the majority of settings this problem is NP-hard. However, we show that if h = O(1),k = O(1), and the underlying communication graph is acyclic, then there exists an algorithm that solves this problem in polynomial time.

Bubbles: Adaptive Routing Scheme for High-Speed Dynamic Networks

This paper presents the first dynamic routing scheme for high-speed networks. The scheme is based on a hierarchical bubbles partition of the underlying communication graph. Dynamic routing schemes are ranked by their adaptability, i.e., the maximum number of sites to be updated upon a topology change. An advantage of our scheme is that it implies a small number of updates upon a topology change. In particular, for the case of a bounded degree network it is proved that our scheme is optimal in its adaptability by presenting a matching tight lower bound. Our bubble routing scheme is a combination of a distributed routing database, a routing strategy, and a routing database update. It is shown how to perform the routing database update on a dynamic network in a distributed manner.

On the complexity of deadlock detection in families of planar nets

We are interested in some properties of massively parallel computers which we model by finite automata connected together on a 2-dimensional grid. We wonder whether it is possible to anticipate a possible appearance of a deadlock in such nets. Thus, we look for efficient algorithms to predict whether deadlocks can appear in grids of bounded size. From the point of view of worst-case complexity, we prove that this problem is NP-complete whereas it is quadratic for linear structures. The method we use is a reduction from a tiling problem. We also prove that this problem, associated with a natural probability distribution on its instances, is RNP-complete (Random NP-complete) in the theory proposed by Levin and developed by Gurevich. Very few randomized problems are known to be RNP-complete. Under classical complexity hypotheses, this result proves that there does not exist any algorithm that solves this problem efficiently on the average case. We present other extentions of our results for different planar underlying communication graphs, and we present a Σ 2-complete problem for networks with inputs.

Annealed embeddings of communication patterns in an interconnection cached network

The communication needs of many parallel applications exhibit what we call switching locality. In such applications, each computation entity (process, thread, etc.) tends to restrict its communication to a small set of other entities. The physical location or proximity of these entities can be arbitrary, as long as the communication degree is small. The Interconnection Cached Network (ICN) is a reconfigurable network ideally suited for exploiting such locality. The use of fast small crossbar switches (Interconnection Caches) with a larger, but slower, reconfigurable network (optimized for connectivity) lets the ICN adapt to the communication requirements of individual applications, potentially achieving higher performance. Embedding communication patterns efficiently in an ICN, requires finding a bounded l-contraction of the underlying communication graph. The problem of identifying whether a graph has a bounded and l-contraction for a given integer l is known to be NP-complete for l>2. We describe a heuristic algorithm based on simulated annealing for this problem. We test the effectiveness of our approach by using it to embed graphs, representing regular communication patterns, for which the best solutions are deterministically known. The algorithm does not rely on any structural information of the communication pattern and is therefore applicable to irregular patterns as well. The results of applying our heuristics to embed such irregular graphs are also presented.

Parametric module allocation on partial k-trees

The problem of allocating modules to processors in a distributed system to minimize total costs when the underlying communication graph is a partial k-tree and all costs are linear functions of a real parameter t is considered. It is shown that if the number of processors is fixed, the sequence of optimum assignments that are obtained as t varies from zero to infinity can be constructed in polynomial time. As an auxiliary result, a linear time separator algorithm for k-trees is developed. The implications of the results for parametric versions of the weighted vertex cover, independent set, and 0-1 quadratic programming problems on partial k-trees are discussed. >