Complex Networks: New Models and Distributed Algorithms

University of Pennsylvania ScholarlyCommons Publicly Accessible Penn Dissertations Fall 2009 Complex Networks: New Models and Distributed Algorithms Alireza Tahbaz Salehi University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/edissertations Part of the Controls and Control Theory Commons Recommended Citation Tahbaz Salehi, Alireza, "Complex Networks: New Models and Distributed Algorithms" (2009). Publicly Accessible Penn Dissertations. 84. https://repository.upenn.edu/edissertations/84 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/edissertations/84 For more information, please contact [email protected]. Complex Networks: New Models and Distributed Algorithms Abstract Over the past few years, a consensus has emerged among scientists and engineers that net-centric technology can provide unprecedented levels of performance, robustness, and efficiency. Success stories such as the Internet, distributed sensor networks, and multi-agent networks of mobile robots are only a few examples that support this view. The important role played by complex networks has been widely observed in various physical, natural, and social systems. Given the complexity of many of these systems, it is important to understand the fundamental rules that govern them and introduce appropriate models that capture such principles, while abstracting away the redundant details. The main goal of this thesis is to contribute to the emerging field of network" science'' in two ways. The first part of the thesis focuses on the question of information aggregation over complex networks. The problem under study is the asymptotic behavior of agents in a network when they are willing to share information with their neighbors. We start by focusing on conditions under which all agents in the network will asymptotically agree on some quantity of interest, what is known as the consensus problem. We present conditions that guarantee asymptotic agreement when inter-agent communication links change randomly over time. We then propose a distributed (non-Bayesian) algorithm that enables agents to not only agree, but also learn the true underlying state of the world. We prove that our proposed learning rule results in successful information aggregation, in the sense that all agents asymptotically learn the truth as if they were completely informed of all signals and updated their beliefs rationally. Moreover, the simplicity of our local update rule guarantees that agents eventually achieve full learning, while at the same time, avoiding highly complex computations that are essential for full Bayesian learning over networks. The second part of this thesis focuses on presenting a new modeling paradigm that greatly expands the tool set for mathematical modeling of networks, beyond graphs. The approach taken is based on using simplicial complexes, which are objects of study in algebraic topology, as generalizations of graphs to higher dimensions. We show how simplicial complexes serve as more faithful models of the network and are able to capture many of its global topological properties. Furthermore, we develop distributed algorithms for computing various topological invariants of the network. These concepts and algorithms are further explored in the context of a specific application: coverage verification in coordinate-free sensor networks, where sensor nodes have no access to location, distance, or orientation information. We propose real-time, scalable, and decentralized schemes for detection of coverage holes, as well as computation of a minimal set of sensors required to monitor a given region of interest. The presented algorithms clarify the benefits of using simplicial complexes and simplicial homology, instead of applying tools from graph theory, in modeling and analyzing complex networks. Degree Type Dissertation Degree Name Doctor of Philosophy (PhD) Graduate Group Electrical & Systems Engineering First Advisor Ali Jadbabaie Keywords Complex networks, distributed algorithms, distributed control, multi-agent systems Subject Categories Controls and Control Theory This dissertation is available at ScholarlyCommons: https://repository.upenn.edu/edissertations/84 Acknowledgements This thesis could not have been written without the support of my advisor, Professor Ali Jadbabaie. Ali taught me how to think independently while providing me with guidance on all aspects of research. His encouragement during these years will always be of immense value to me. I am also greatly indebted to Professor Robert Ghrist for the many helpful conversations that we had over the past years. The second half of this thesis would have been impossible without his guidance. I am thankful to the other members of my committee: Professors George Pappas, Daniel Koditchek, and Asuman Ozdaglar. Their valuable suggestions and considerable advice have greatly improved the quality of this thesis. I would also like to thank Professor Alvaro Sandroni for his contributions in Chapter 3. An important role was played by my many friends at Penn. Special thanks to Elena, Nima, Agung, Michael, Farhad, Gil, Antonio, Andreas, and many others. Without them, my years in Philadelphia would not have been as enjoyable. Last but not least, my deepest thanks to my parents, Farzaneh and Ali, and my sister, Ghazaleh, who gave me their unconditional love and support. I owe them much more than I would ever be able to express. ii ABSTRACT COMPLEX NETWORKS: NEW MODELS AND DISTRIBUTED ALGORITHMS Alireza Tahbaz Salehi Supervisor: Ali Jadbabaie Over the past few years, a consensus has emerged among scientists and engineers that net-centric technology can provide unprecedented levels of performance, robustness, and efficiency. Success stories such as the Internet, distributed sensor networks, and multi-agent networks of mobile robots are only a few examples that support this view. The important role played by complex networks has been widely observed in various physical, natural, and social systems. Given the complexity of many of these systems, it is important to understand the fundamental rules that govern them and introduce appropriate models that capture such principles, while abstracting away the redundant details. The main goal of this thesis is to contribute to the emerging field of “network science” in two ways. The first part of the thesis focuses on the question of information aggregation over complex networks. The problem under study is the asymptotic behavior of agents in a network when they are willing to share information with their neighbors. We start by focusing on conditions under which all agents in the network will asymptotically agree on some quantity of interest, what is known as the consensus problem. We present conditions that guarantee asymptotic agreement when inter-agent communication links change randomly over time. We then propose a distributed (non-Bayesian) algorithm that enables agents to not only agree, but also learn the true underlying state of the world. We prove that our proposed learning rule results in successful information aggregation, in the sense that all agents asymptotically learn the truth as if they were completely informed of all signals and updated their beliefs rationally. Moreover, the simplicity of our local update rule guarantees that agents eventually achieve full learning, while at the same time, avoiding highly iii complex computations that are essential for full Bayesian learning over networks. The second part of this thesis focuses on presenting a new modeling paradigm that greatly expands the tool set for mathematical modeling of networks, beyond graphs. The approach taken is based on using simplicial complexes, which are objects of study in algebraic topology, as generalizations of graphs to higher dimensions. We show how simplicial complexes serve as more faithful models of the network and are able to capture many of its global topological properties. Furthermore, we develop distributed algorithms for computing various topological invariants of the network. These concepts and algorithms are further explored in the context of a specific application: coverage verification in coordinate-free sensor networks, where sensor nodes have no access to location, distance, or orientation information. We propose real-time, scalable, and decentralized schemes for detection of coverage holes, as well as computation of a minimal set of sensors required to monitor a given region of interest. The presented algorithms clarify the benefits of using simplicial complexes and simplicial homology, instead of applying tools from graph theory, in modeling and analyzing complex networks. iv Contents Acknowledgements ii Abstract iii Contents v List of Figures viii 1 Overview 1 1.1 Information Aggregation over Complex Networks . 3 1.2 New Models of Networks: Beyond Graphs . 7 I Information Aggregation Over Complex Networks 9 2 Network Consensus Algorithms 10 2.1 Distributed Consensus Algorithms . 11 2.1.1 Ergodicity . 13 2.1.2 Coefficients of Ergodicity . 15 2.1.3 Consensus and Joint Connectivity . 18 2.2 Consensus over Random Networks . 21 2.2.1 Independent and Identically Distributed Random Networks . 22 v 2.2.2 Ergodic Stationary Random Networks . 32 2.3 Related Literature . 37 3 Social Learning 39 3.1 Summary of Results and Related Literature . 40 3.1.1 Summary of Results . 41 3.1.2 Related Literature . 44 3.2 A Model of Non-Bayesian Social Learning . 46 3.2.1 Agents and Observations . 46 3.2.2 Social

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