SIRS Epidemics on Complex Networks: Concurrence of Exact Markov Chain and Approximated Models

We study the SIRS (Susceptible-Infected-Recovered-Susceptible) spreading processes over complex networks, by considering its exact $3^n$-state Markov chain model. The Markov chain model exhibits an interesting connection with its $2n$-state nonlinear "mean-field" approximation and the latter's corresponding linear approximation. We show that under the specific threshold where the disease-free state is a globally stable fixed point of both the linear and nonlinear models, the exact underlying Markov chain has an $O(\log n)$ mixing time, which means the epidemic dies out quickly. In fact, the epidemic eradication condition coincides for all the three models. Furthermore, when the threshold condition is violated, which indicates that the linear model is not stable, we show that there exists a unique second fixed point for the nonlinear model, which corresponds to the endemic state. We also investigate the effect of adding immunization to the SIRS epidemics by introducing two different models, depending on the efficacy of the vaccine. Our results indicate that immunization improves the threshold of epidemic eradication. Furthermore, the common threshold for fast-mixing of the Markov chain and global stability of the disease-free fixed point improves by the same factor for the vaccination-dominant model.Comment: A short version of this paper has been submitted to CDC 201

Large-Scale Intelligent Systems: From Network Dynamics to Optimization Algorithms

The expansion of large-scale technological systems such as electrical grids, transportation networks, health care systems, telecommunication networks, the Internet (of things), and other societal networks has created numerous challenges and opportunities at the same time. These systems are often not yet as robust, efficient, sustainable, or smart as we would want them to be. Fueled by the massive amounts of data generated by all these systems, and with the recent advances in making sense out of data, there is a strong desire to make them more intelligent. However, developing large-scale intelligent systems is a multifaceted problem, involving several major challenges. First, large-scale systems typically exhibit complex dynamics due to the large number of entities interacting over a network. Second, because the system is composed of many interacting entities, that make decentralized (and often self-interested) decisions, one has to properly design incentives and markets for such systems. Third, the massive computational needs caused by the scale of the system necessitate performing computations in a distributed fashion, which in turn requires devising new algorithms. Finally, one has to create algorithms that can learn from the copious amounts of data and generalize well. This thesis makes several contributions related to each of these four challenges. Analyzing and understanding the network dynamics exhibited in societal systems is crucial for developing systems that are robust and efficient. In Part I of this thesis, we study one of the most important families of network dynamics, namely, that of epidemics, or spreading processes. Studying such processes is relevant for understanding and controlling the spread of, e.g., contagious diseases among people, ideas or fake news in online social networks, computer viruses in computer networks, or cascading failures in societal networks. We establish several results on the exact Markov chain model and the nonlinear "mean-field" approximations for various kinds of epidemics (i.e., SIS, SIRS, SEIRS, SIV, SEIV, and their variants). Designing incentives and markets for large-scale systems is critical for their efficient operation and ensuring an alignment between the agents' decentralized decisions and the global goals of the system. To that end, in Part II of this thesis, we study these issues in markets with non-convex costs as well as networked markets, which are of vital importance for, e.g., the smart grid. We propose novel pricing schemes for such markets, which satisfy all the desired market properties. We also reveal issues in the current incentives for distributed energy resources, such as renewables, and design optimization algorithms for efficient management of aggregators of such resources. With the growing amounts of data generated by large-scale systems, and the fact that the data may already be dispersed across many units, it is becoming increasingly necessary to run computational tasks in a distributed fashion. Part III concerns developing algorithms for distributed computation. We propose a novel consensus-based algorithm for the task of solving large-scale systems of linear equations, which is one of the most fundamental problems in linear algebra, and a key step at the heart of many algorithms in scientific computing, machine learning, and beyond. In addition, in order to deal with the issue of heterogeneous delays in distributed computation, caused by slow machines, we develop a new coded computation technique. In both cases, the proposed methods offer significant speed-ups relative to the existing approaches. Over the past decade, deep learning methods have become the most successful learning algorithms in a wide variety of tasks. However, the reasons behind their success (as well as their failures in some respects) are largely unexplained. It is widely believed that the success of deep learning is not just due to the deep architecture of the models, but also due to the behavior of the optimization algorithms, such as stochastic gradient descent (SGD), used for training them. In Part IV of this thesis, we characterize several properties, such as minimax optimality and implicit regularization, of SGD, and more generally, of the family of stochastic mirror descent (SMD). While SGD performs an implicit regularization, this regularization can be effectively controlled using SMD with a proper choice of mirror, which in turn can improve the generalization error.</p

Improving Distributed Gradient Descent Using Reed-Solomon Codes

Today's massively-sized datasets have made it necessary to often perform computations on them in a distributed manner. In principle, a computational task is divided into subtasks which are distributed over a cluster operated by a taskmaster. One issue faced in practice is the delay incurred due to the presence of slow machines, known as \emph{stragglers}. Several schemes, including those based on replication, have been proposed in the literature to mitigate the effects of stragglers and more recently, those inspired by coding theory have begun to gain traction. In this work, we consider a distributed gradient descent setting suitable for a wide class of machine learning problems. We adapt the framework of Tandon et al. (arXiv:1612.03301) and present a deterministic scheme that, for a prescribed per-machine computational effort, recovers the gradient from the least number of machines $f$ theoretically permissible, via an $O(f^2)$ decoding algorithm. We also provide a theoretical delay model which can be used to minimize the expected waiting time per computation by optimally choosing the parameters of the scheme. Finally, we supplement our theoretical findings with numerical results that demonstrate the efficacy of the method and its advantages over competing schemes

Distributed Solution of Large-Scale Linear Systems via Accelerated Projection-Based Consensus

Solving a large-scale system of linear equations is a key step at the heart of many algorithms in machine learning, scientific computing, and beyond. When the problem dimension is large, computational and/or memory constraints make it desirable, or even necessary, to perform the task in a distributed fashion. In this paper, we consider a common scenario in which a taskmaster intends to solve a large-scale system of linear equations by distributing subsets of the equations among a number of computing machines/cores. We propose an accelerated distributed consensus algorithm, in which at each iteration every machine updates its solution by adding a scaled version of the projection of an error signal onto the nullspace of its system of equations, and where the taskmaster conducts an averaging over the solutions with momentum. The convergence behavior of the proposed algorithm is analyzed in detail and analytically shown to compare favorably with the convergence rate of alternative distributed methods, namely distributed gradient descent, distributed versions of Nesterov's accelerated gradient descent and heavy-ball method, the block Cimmino method, and ADMM. On randomly chosen linear systems, as well as on real-world data sets, the proposed method offers significant speed-up relative to all the aforementioned methods. Finally, our analysis suggests a novel variation of the distributed heavy-ball method, which employs a particular distributed preconditioning, and which achieves the same theoretical convergence rate as the proposed consensus-based method

Opportunities for Price Manipulation by Aggregators in Electricity Markets

Aggregators are playing an increasingly crucial role in the integration of renewable generation in power systems. However, the intermittent nature of renewable generation makes market interactions of aggregators difficult to monitor and regulate, raising concerns about potential market manipulation by aggregators. In this paper, we study this issue by quantifying the profit an aggregator can obtain through strategic curtailment of generation in an electricity market. We show that, while the problem of maximizing the benefit from curtailment is hard in general, efficient algorithms exist when the topology of the network is radial (acyclic). Further, we highlight that significant increases in profit are possible via strategic curtailment in practical settings

Opportunities for Price Manipulation by Aggregators in Electricity Markets

Aggregators are playing an increasingly crucial role for integrating renewable generation into power systems. However, the intermittent nature of renewable generation makes market interactions of aggregators difficult to monitor and regulate, raising concerns about potential market manipulations. In this paper, we address this issue by quantifying the profit an aggregator can obtain through strategic curtailment of generation in an electricity market. We show that, while the problem of maximizing the benefit from curtailment is hard in general, efficient algorithms exist when the topology of the network is radial (acyclic). Further, we highlight that significant increases in profit can be obtained through strategic curtailment in practical settings

Opportunities for Price Manipulation by Aggregators in Electricity Markets

Aggregators are playing an increasingly crucial role for integrating renewable generation into power systems. However, the intermittent nature of renewable generation makes market interactions of aggregators difficult to monitor and regulate, raising concerns about potential market manipulations. In this paper, we address this issue by quantifying the profit an aggregator can obtain through strategic curtailment of generation in an electricity market. We show that, while the problem of maximizing the benefit from curtailment is hard in general, efficient algorithms exist when the topology of the network is radial (acyclic). Further, we highlight that significant increases in profit can be obtained through strategic curtailment in practical settings

Opportunities for Price Manipulation by Aggregators in Electricity Markets

Aggregators of distributed generation are playing an increasingly crucial role in the integration of renewable energy in power systems. However, the intermittent nature of renewable generation makes market interactions of aggregators difficult to monitor and regulate, raising concerns about potential market manipulation by aggregators. In this paper, we study this issue by quantifying the profit an aggregator can obtain through strategic curtailment of generation in an electricity market. We show that, while the problem of maximizing the benefit from curtailment is hard in general, efficient algorithms exist when the topology of the network is radial (acyclic). Further, we highlight that significant increases in profit are possible via strategic curtailment in practical settings

Distributed Solution of Large-Scale Linear Systems via Accelerated Projection-Based Consensus

Solving a large-scale system of linear equations is a key step at the heart of many algorithms in scientific computing, machine learning, and beyond. When the problem dimension is large, computational and/or memory constraints make it desirable, or even necessary, to perform the task in a distributed fashion. In this paper, we consider a common scenario in which a taskmaster intends to solve a large-scale system of linear equations by distributing subsets of the equations among a number of computing machines/cores. We propose a new algorithm called Accelerated Projection-based Consensus , in which at each iteration every machine updates its solution by adding a scaled version of the projection of an error signal onto the nullspace of its system of equations, and the taskmaster conducts an averaging over the solutions with momentum. The convergence behavior of the proposed algorithm is analyzed in detail and analytically shown to compare favorably with the convergence rate of alternative distributed methods, namely distributed gradient descent, distributed versions of Nesterov's accelerated gradient descent and heavy-ball method, the block Cimmino method, and Alternating Direction Method of Multipliers. On randomly chosen linear systems, as well as on real-world data sets, the proposed method offers significant speed-up relative to all the aforementioned methods. Finally, our analysis suggests a novel variation of the distributed heavy-ball method, which employs a particular distributed preconditioning and achieves the same theoretical convergence rate as that in the proposed consensus-based method