Stacy Patterson

Coherence in Large-Scale Networks: Dimension-Dependent Limitations of Local Feedback

We consider distributed consensus and vehicular formation control problems. Specifically we address the question of whether local feedback is sufficient to maintain coherence in large-scale networks subject to stochastic disturbances. We define macroscopic performance measures which are global quantities that capture the notion of coherence; a notion of global order that quantifies how closely the formation resembles a solid object. We consider how these measures scale asymptotically with network size in the topologies of regular lattices in 1, 2, and higher dimensions, with vehicular platoons corresponding to the 1-D case. A common phenomenon appears where a higher spatial dimension implies a more favorable scaling of coherence measures, with a dimensions of 3 being necessary to achieve coherence in consensus and vehicular formations under certain conditions. In particular, we show that it is impossible to have large coherent 1-D vehicular platoons with only local feedback. We analyze these effects in terms of the underlying energetic modes of motion, showing that they take the form of large temporal and spatial scales resulting in an accordion-like motion of formations. A conclusion can be drawn that in low spatial dimensions, local feedback is unable to regulate large-scale disturbances, but it can in higher spatial dimensions. This phenomenon is distinct from, and unrelated to string instability issues which are commonly encountered in control problems for automated highways.

Convergence Rates of Distributed Average Consensus With Stochastic Link Failures

We consider a distributed average consensus algorithm over a network in which communication links fail with independent probability. In such stochastic networks, convergence is defined in terms of the variance of deviation from average. We first show how the problem can be recast as a linear system with multiplicative random inputs which model link failures. We then use our formulation to derive recursion equations for the second order statistics of the deviation from average in networks with and without additive noise. We give expressions for the convergence behavior in the asymptotic limits of small failure probability and large networks. We also present simulation-free methods for computing the second order statistics in each network model and use these methods to study the behavior of various network examples as a function of link failure probability.

Leader selection for optimal network coherence

We consider the problem of leader-based distributed coordination in networks where agents are subject to stochastic disturbances, but where certain designated leaders are immune to those disturbances. Specifically, we address the effect of leader selection on the coherence of the network, defined in terms of an H2 norm of the system. This quantity captures the level of agreement of the nodes in the face of the external disturbances. We show that network coherence depends on the eigenvalues of a principal submatrix of the Laplacian matrix, and we formulate an optimization problem to select the set of leaders that results in the highest coherence. As this optimization problem is combinatorial in nature, we also present several greedy algorithms for leader selection that rely on more easily computable bounds of the H2 norm and the eigenvalues of the system. Finally, we illustrate the effectiveness of these algorithms using several network examples.

Distributed average consensus with stochastic communication failures

We consider a distributed average consensus algorithm over a network in which communication links fail with independent probability. Convergence in such stochastic networks is defined in terms of the variance of deviation from average. We characterize the decay factor of the variance in terms of the eigenvalues of a Lyapunov-like matrix recursion. We give expressions for the decay factors in the asymptotic limits of small failure probability and large networks. We also present a simulation-free method for computing the decay factor for any particular graph instance and use this method to study the behavior of various network examples as a function of link failure probability.

Serializability, not serial: concurrency control and availability in multi-datacenter datastores

We present a framework for concurrency control and availability in multi-datacenter datastores. While we consider Googles Megastore as our motivating example, we define general abstractions for key components, making our solution extensible to any system that satisfies the abstraction properties. We first develop and analyze a transaction management and replication protocol based on a straightforward implementation of the Paxos algorithm. Our investigation reveals that this protocol acts as a concurrency prevention mechanism rather than a concurrency control mechanism. We then propose an enhanced protocol called Paxos with Combination and Promotion (Paxos-CP) that provides true transaction concurrency while requiring the same per instance message complexity as the basic Paxos protocol. Finally, we compare the performance of Paxos and Paxos-CP in a multi-datacenter experimental study, and we demonstrate that Paxos-CP results in significantly fewer aborted transactions than basic Paxos.

From static distributed systems to dynamic systems

A noteworthy advance in distributed computing is due to the recent development of peer-to-peer systems. These systems are essentially dynamic in the sense that no process can get a global knowledge on the system structure. They mainly allow processes to look up for data that can be dynamically added/suppressed in a permanently evolving set of nodes. Although protocols have been developed for such dynamic systems, to our knowledge, up to date no computation model for dynamic systems has been proposed. Nevertheless, there is a strong demand for the definition of such models as soon as one wants to develop provably correct protocols suited to dynamic systems. This paper proposes a model for (a class of) dynamic systems. That dynamic model is defined by (1) a parameter (an integer denoted a) and (2) two basic communication abstractions (query-response and persistent reliable broadcast). The new parameter is a threshold value introduced to capture the liveness part of the system (it is the counterpart of the minimal number of processes that do not crash in a static system). To show the relevance of the model, the paper adapts an eventual leader protocol designed for the static model, and proves that the resulting protocol is correct within the proposed dynamic model. In that sense, the paper has also a methodological flavor, as it shows that simple modifications to existing protocols can allow them to work in dynamic systems.

Distributed Compressed Sensing for Static and Time-Varying Networks

We consider the problem of in-network compressed sensing from distributed measurements. Every agent has a set of measurements of a signal x, and the objective is for the agents to recover x from their collective measurements using only communication with neighbors in the network. Our distributed approach to this problem is based on the centralized Iterative Hard Thresholding algorithm (IHT). We first present a distributed IHT algorithm for static networks that leverages standard tools from distributed computing to execute in-network computations with minimized bandwidth consumption. Next, we address distributed signal recovery in networks with time-varying topologies. The network dynamics necessarily introduce inaccuracies to our in-network computations. To accommodate these inaccuracies, we show how centralized IHT can be extended to include inexact computations while still providing the same recovery guarantees as the original IHT algorithm. We then leverage these new theoretical results to develop a distributed version of IHT for time-varying networks. Evaluations show that our distributed algorithms for both static and time-varying networks outperform previously proposed solutions in time and bandwidth by several orders of magnitude.

Effect of topological dimension on rigidity of vehicle formations: Fundamental limitations of local feedback

We consider the role of topological dimension in problems of network consensus and vehicular formations where only local feedback is available. In particular, we consider the simple network topologies of regular lattices in 1, 2 and higher dimensions. Performance measures for consensus and formation problems are proposed that measure the deviation from average and rigidity or tightness of formations respectively. A common phenomenon appears where in dimensions 1 and 2, consensus is impossible in the presence of any amount of additive stochastic perturbations, and in the limit of large formations. In dimensions 3 and higher, consensus is indeed possible. We show that microscopic error measures that involve only neighboring sites do not suffer from this effect. This phenomenon reflects the fact that in dimensions 1 and 2, local stabilizing feedbacks can not suppress long spatial wavelength ¿meandering¿ motions. These effects are significantly more pronounced in vehicular problems than in consensus, and yet they are unrelated to string stability issues.

Network coherence in fractal graphs

We study distributed consensus algorithms in fractal networks where agents are subject to external disturbances. We characterize the coherence of these networks in terms of an H2 norm of the system that captures how closely agents track the consensus value. We show that, in first-order systems, the coherence measure is closely related to the global mean first passage time of a simple random walk. We can therefore draw directly from the literature on random walks in fractal graphs to derive asymptotic expressions for the coherence in terms of the network size and dimension. We then show how techniques employed in the random walks setting can be extended to analyze the coherence of second-order consensus algorithms in fractal graphs with tree-like structures, and we present asymptotic results for these second-order systems.

Distributed sparse signal recovery for sensor networks

We propose a distributed algorithm for sparse signal recovery in sensor networks based on Iterative Hard Thresholding (IHT). Every agent has a set of measurements of a signal x, and the objective is for the agents to recover x from their collective measurements at a minimal communication cost and with low computational complexity. A naïve distributed implementation of IHT would require global communication of every agents full state in each iteration. We find that we can dramatically reduce this communication cost by leveraging solutions to the distributed top-K problem in the database literature. Evaluations show that our algorithm requires up to three orders of magnitude less total bandwidth than the best-known distributed basis pursuit method.