Average Consensus in the Presence of Delays and Dynamically Changing Directed Graph Topologies

Classical approaches for asymptotic convergence to the global average in a distributed fashion typically assume timely and reliable exchange of information between neighboring components of a given multi-component system. These assumptions are not necessarily valid in practical settings due to varying delays that might affect transmissions at different times, as well as possible changes in the underlying interconnection topology (e.g., due to component mobility). In this work, we propose protocols to overcome these limitations. We first consider a fixed interconnection topology (captured by a - possibly directed - graph) and propose a discrete-time protocol that can reach asymptotic average consensus in a distributed fashion, despite the presence of arbitrary (but bounded) delays in the communication links. The protocol requires that each component has knowledge of the number of its outgoing links (i.e., the number of components to which it sends information). We subsequently extend the protocol to also handle changes in the underlying interconnection topology and describe a variety of rather loose conditions under which the modified protocol allows the components to reach asymptotic average consensus. The proposed algorithms are illustrated via examples.Comment: 37 page

ARQ-based Average Consensus over Unreliable Directed Network Topologies

In this paper, we address the discrete-time average consensus problem, where nodes exchange information over unreliable communication links. We enhance the Robustified Ratio Consensus algorithm by exploiting features of the Automatic Repeat ReQuest (ARQ) protocol used for error control of data transmissions, in order to allow the nodes to reach asymptotic average consensus even when operating within unreliable directed networks. This strategy, apart from handling time-varying delays induced by retransmissions of erroneous packets, can also handle packet drops that occur when exceeding a predefined packet retransmission limit imposed by the ARQ protocol. Invoking the ARQ protocol allows nodes to: (a) exploit the incoming error-free acknowledgement feedback to initially acquire or later update their out-degree, (b) know whether a packet has arrived or not, and (c) determine a local upper-bound on the delays imposed by the retransmission limit. By augmenting the network's corresponding weight matrix, we show that nodes utilizing our proposed ARQ-based Ratio Consensus algorithm can reach asymptotic average consensus over unreliable networks, while maintaining low running sum values

Approximation of Markov Processes by Lower Dimensional Processes via Total Variation Metrics

The aim of this paper is to approximate a finite-state Markov process by another process with fewer states, called herein the approximating process. The approximation problem is formulated using two different methods. The first method, utilizes the total variation distance to discriminate the transition probabilities of a high dimensional Markov process and a reduced order Markov process. The approximation is obtained by optimizing a linear functional defined in terms of transition probabilities of the reduced order Markov process over a total variation distance constraint. The transition probabilities of the approximated Markov process are given by a water-filling solution. The second method, utilizes total variation distance to discriminate the invariant probability of a Markov process and that of the approximating process. The approximation is obtained via two alternative formulations: (a) maximizing a functional of the occupancy distribution of the Markov process, and (b) maximizing the entropy of the approximating process invariant probability. For both formulations, once the reduced invariant probability is obtained, which does not correspond to a Markov process, a further approximation by a Markov process is proposed which minimizes the Kullback-Leibler divergence. These approximations are given by water-filling solutions. Finally, the theoretical results of both methods are applied to specific examples to illustrate the methodology, and the water-filling behavior of the approximations.Comment: 38 pages, 17 figures, submitted to IEEE-TA

Mammalian Inscuteable Regulates Spindle Orientation and Cell Fate in the Developing Retina

During mammalian neurogenesis, progenitor cells can divide with the mitotic spindle oriented parallel or perpendicular to the surface of the neuroepithelium. Perpendicular divisions are more likely to be asymmetric and generate one progenitor and one neuronal precursor. Whether the orientation of the mitotic spindle actually determines their asymmetric outcome is unclear. Here, we characterize a mammalian homolog of Inscuteable (mInsc), a key regulator of spindle orientation in Drosophila. mInsc is expressed temporally and spatially in a manner that suggests a role in orienting the mitotic spindle in the developing nervous system. Using retroviral RNAi in rat retinal explants, we show that downregulation of mInsc inhibits vertical divisions. This results in enhanced proliferation, consistent with a higher frequency of symmetric divisions generating two proliferating cells. Our results suggest that the orientation of neural progenitor divisions is important for cell fate specification in the retina and determines their symmetric or asymmetric outcome

Mammalian Inscuteable Regulates Spindle Orientation and Cell Fate in the Developing Retina

During mammalian neurogenesis, progenitor cells can divide with the mitotic spindle oriented parallel or perpendicular to the surface of the neuroepithelium. Perpendicular divisions are more likely to be asymmetric and generate one progenitor and one neuronal precursor. Whether the orientation of the mitotic spindle actually determines their asymmetric outcome is unclear. Here, we characterize a mammalian homolog of Inscuteable (mInsc), a key regulator of spindle orientation in Drosophila. mInsc is expressed temporally and spatially in a manner that suggests a role in orienting the mitotic spindle in the developing nervous system. Using retroviral RNAi in rat retinal explants, we show that downregulation of mInsc inhibits vertical divisions. This results in enhanced proliferation, consistent with a higher frequency of symmetric divisions generating two proliferating cells. Our results suggest that the orientation of neural progenitor divisions is important for cell fate specification in the retina and determines their symmetric or asymmetric outcome