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12
The Sleekest Link Algorithm
, 2003
"... How does Google decide which web sites are important? It uses an ingenious algorithm that exploits the structure of the web and is resistant to hacking. Here, we describe this PageRank algorithm, illustrate it by example, and show how it can be interpreted as a Jacobi iteration and a teleporting ran ..."
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How does Google decide which web sites are important? It uses an ingenious algorithm that exploits the structure of the web and is resistant to hacking. Here, we describe this PageRank algorithm, illustrate it by example, and show how it can be interpreted as a Jacobi iteration and a teleporting random walk. We also ask the algorithm to rank the undergraduate mathematics classes offered at the University of Strathclyde. PageRank draws upon ideas from linear algebra, graph theory and stochastic processes, and it throws up researchlevel challenges in scientific computing. It thus forms an exciting and modern application area that could brighten up many a mathematics class syllabus.
Effects of topology in networked systems: Stochastic methods and small worlds
 in Proc. 47th IEEE Conf. Decision Control, Cancun
"... Abstract—The topology of a networked control system has critical consequences for its performance. We provide first substantial examples on the effects of topology. Then we proceed to develop a rigorous evaluation of topology effects using stochastic methods inspired from statistical physics and Mar ..."
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Abstract—The topology of a networked control system has critical consequences for its performance. We provide first substantial examples on the effects of topology. Then we proceed to develop a rigorous evaluation of topology effects using stochastic methods inspired from statistical physics and Markov chains. This analysis leads us to proofs on faster convergence of distributed algorithms in networked systems for certain topologies and especially small world topologies, which are are given an ‘efficiency ’ characterization. Finally, these results lead to the development of selforganization of such systems in hierarchies that provably improve performance and response. I.
The Kemeny Constant For Finite Homogeneous Ergodic Markov Chains
"... This paper is dedicated to the memory of Professor David Gottlieb ..."
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This paper is dedicated to the memory of Professor David Gottlieb
CONSENSUS PROBLEMS AND THE EFFECTS OF GRAPH TOPOLOGY IN COLLABORATIVE CONTROL
, 2009
"... In this dissertation, several aspects of design for networked systems are addressed. The main focus is on combining approaches from system theory and graph theory to characterize graph topologies that result in efficient decision making and control. In this framework, modelling and design of sparse ..."
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In this dissertation, several aspects of design for networked systems are addressed. The main focus is on combining approaches from system theory and graph theory to characterize graph topologies that result in efficient decision making and control. In this framework, modelling and design of sparse graphs that are robust to failures and provide high connectivity are considered. A decentralized approach to path generation in a collaborative system is modelled using potential functions. Taking inspiration from natural swarms, various behaviors of the system such as target following, moving in cohesion and obstacle avoidance are addressed by appropriate encoding of the corresponding costs in the potential function and using gradient descent for minimizing the energy function. Different emergent behaviors emerge as a result of varying the weights attributed with different components of the potential function. Consensus problems are addressed as a unifying theme in many collaborative control problems and their robustness and convergence properties are studied. Implications of the continuous convergence property of consensus problems on their reachability and robustness are studied. The effects of link and agent faults on consen
Efficient and Robust Communication Topologies for Distributed Decision Making in Networked Systems
"... Abstract — Distributed decision making in networked systems depends critically on the timely availability of critical fresh information. Performance of networked systems, from the perspective of achieving goals and objectives in a timely and efficient manner is constrained by their collaboration and ..."
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Abstract — Distributed decision making in networked systems depends critically on the timely availability of critical fresh information. Performance of networked systems, from the perspective of achieving goals and objectives in a timely and efficient manner is constrained by their collaboration and communication structures (they are not necessarily the same) and their interplay with the networked system’s dynamics. Thus autonomous agents are critically influenced by their understanding of the network communication topology. We describe efficient communication topologies for distributed decision making and relate them to small world graphs and more generally to expander graphs. In most cases achieving the system objectives requires many agent to agent communications. A reasonable measure for system robustness to communication topology change is the number of spanning trees in the graph abstraction of the communication system. Solutions to this problem have also applications in trust and the relationship of trust to control. We address the problem of network formation with robustness and connectivity constraints. We show that the general combinatorial problem can be relaxed to a convex optimization problem. We solve the special case of adding a shortcut to a ring and provide insights for derivation of heuristics for the general case. We also analyze the small world effect in the context of abrupt increases in the number of spanning trees as a result of adding a few shortcuts to a base lattice in the WattsStrogatz framework. Finally we describe generalizations to expander graphs. I.
Dynamic Selforganization and Clustering in Distributed Networked Systems for Performance Improvement
, 2009
"... Abstract — We consider two closely related dynamic selforganization problems in networked control systems. Both are forms of dynamic clustering of nodes. The structure of networked control systems is often abstracted using graph theory. In this abstraction, the nodes of the graph represent the agent ..."
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Abstract — We consider two closely related dynamic selforganization problems in networked control systems. Both are forms of dynamic clustering of nodes. The structure of networked control systems is often abstracted using graph theory. In this abstraction, the nodes of the graph represent the agents and the edges between them represent the relation(s) or the possibility of communication between the corresponding agents. The topology of the communication network supporting a networked control system has critical consequences for its performance. The first problem we address is the development of a distributed selforganization algorithm, resulting into a dynamic two level hierarchy of leader and regular agents, which substantially improves the convergence speed of distributed algorithms utilized by the networked control system. For the second problem, we consider the collaborative control of a group of autonomous mobile agents (e.g. vehicles, robots) supported by a mobile wireless network, consisting of many ground and a few aerial nodes. The agents collaborate to achieve a common goal or objective, like to move in a particular area and cover it, while avoiding obstacles and collisions. Building upon our earlier work on deterministic, randomized and hybrid distributed coordination algorithms we consider the communication needs of the agents, and in particular the connectivity of their communication network as they move. We develop distributed algorithms that automatically select some agents and move them appropriately so as to maintain certain degree of desired connectivity among the moving agents. We characterize the tradeoff between the gain from maintaining a certain degree of connectivity vs. the combined cost of communications and the associated dynamic repositioning of agents. We also describe classes of efficient communication topologies and in particular their similarity to dynamic small world topologies and extensions. I.
Event Triggered Distributed Collaborative Control
 in Proc. European Control Conf. (ECC
, 2009
"... Abstract — We consider the collaborative control of a group of autonomous mobile agents. Building upon our earlier work we consider the communication needs and connectivity of the agents ’ network as they move. We develop algorithms that automatically sense the possibility of connectivity loss among ..."
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Abstract — We consider the collaborative control of a group of autonomous mobile agents. Building upon our earlier work we consider the communication needs and connectivity of the agents ’ network as they move. We develop algorithms that automatically sense the possibility of connectivity loss among the agents. We also consider the automatic detection of path disconnection when more than one path need to be maintained between pairs of agents. Using local probing schemes we formulate such problems as eventtriggered control problems. We develop distributed algorithms that automatically select some agents and move them appropriately so as to maintain certain degree of desired connectivity among the moving agents. We characterize the tradeoff between the gain from maintaining a certain degree of connectivity vs. the combined cost of communications and the associated dynamic repositioning of agents. The results illustrate the efficiency achieved by eventtriggered control in such problems. We also describe the resulting communication topologies and in particular their similarity to dynamic small world topologies. I.
MARKOV CHAIN SMALLWORLD MODEL WITH ASYMMETRIC TRANSITION PROBABILITIES ∗
"... Abstract. In this paper, a Markov chain smallworld model of D.J. Higham is broadened by incorporating asymmetric transition probabilities.Asymptotic results regarding the transient behavior of the extended model, as measured by its maximum mean first passage time, are established under the assumpti ..."
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Abstract. In this paper, a Markov chain smallworld model of D.J. Higham is broadened by incorporating asymmetric transition probabilities.Asymptotic results regarding the transient behavior of the extended model, as measured by its maximum mean first passage time, are established under the assumption that the size of the Markov chain is large.These results are consistent with the outcomes as obtained numerically from the model. The focus of this study is the effect of a varying degree of asymmetry on the transient behavior which the extended model exhibits.Being a quite interesting consequence, it turns out that such behavior is largely influenced by the strength of asymmetry.This discovery may find applications in realworld networks where unbalanced interaction is present. Key words. Asymmetry, Smallworld, Ring network, Markov chain, Mean first passage time.
ELA MARKOV CHAIN SMALLWORLD MODEL WITH ASYMMETRIC TRANSITION PROBABILITIES∗
, 2008
"... Markov chain smallworld model with asymmetric transition probabilities ..."
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TuA12.4 Average Consensus over Small World Networks: A Probabilistic Framework
"... Abstract — It has been observed that adding a few long range edges to certain graph topologies can significantly increase the rate of convergence for consensus algorithms. A notable example is the class of ringstructured WattsStrogatz small world graphs. Building on probabilistic methods for analy ..."
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Abstract — It has been observed that adding a few long range edges to certain graph topologies can significantly increase the rate of convergence for consensus algorithms. A notable example is the class of ringstructured WattsStrogatz small world graphs. Building on probabilistic methods for analyzing ‘smallworld phenomena’, developed in our earlier work, we provide here a probabilistic framework for analyzing this effect. We investigate what graph characteristics lead to such a significant improvement and develop bounds to analyze consensus problems on randomly varying graphs. I.