The Behavioral SI* Model, with Applications to the Swine Flu and COVID-19 Pandemics

The 1927 SIR contagion model is the dynamical system for an infection that passes at a constant rate in random pairwise meetings. Our Behavioral SI* Model assumes that everyone has access to a constant elasticity of avoidance technology. We then derive the passing rate in fully solvable Nash equilibrium of the game where everyone optimizes. The resulting dynamics are log-linear, and incidence is log-linear in prevalence, with slope less than one. The SI* models yields extreme predictions for major contagions, not realized. At breakout, the SI* models capture exponential growth. In our BSI* model, increasing avoidance behavior bends the curve, and induces herd immunity at lower prevalence but a later time. Our model is tractable, and better explains incidence data during the 2009 Swine Flu and the COVID-19 pandemic. In both cases, we statistically reject the SIR model. For Swine Flu, across states, the prevalence elasticity ranges from 0.8 to 0.9. We find a similar slope at breakout in the COVID-19 pandemic, and verify that its curve bending matches our BSI* formula. The BSI* model captures mandated social distancing or lockdowns in downward shifts of the line in log-prevalance - log-incidence space

Diffusion Dynamics in Interconnected Communities

In this dissertation, multi-community-based Susceptible-Infected-Recovered (SIR) and Susceptible-Infected-Susceptible (SIS) models of infection/innovation diffusion are introduced for heterogeneous social networks in which agents are viewed as belonging to one of a finite number of communities. Agents are assumed to have well-mixed interactions within and between communities. The communities are connected through a backbone graph which defines an overall network structure for the models. The models are used to determine conditions for outbreak of an initial infection. The role of the strengths of the connections between communities in the development of an outbreak as well as long-term behavior of the diffusion is also studied. Percolation theory is brought to bear on these questions as an independent approach separate from the main dynamic multi-community modeling approach of the dissertation. Results obtained using both approaches are compared and found to be in agreement in the limit of infinitely large populations in all communities. Based on the proposed models, three classes of marketing problems are formulated and studied: referral marketing, seeding marketing and dynamic marketing. It is found that referral marketing can be optimized relatively easily because the associated optimization problem can be formulated as a convex optimization. Also, both seeding marketing and dynamic marketing are shown to enjoy a useful property, namely ``continuous monotone submodularity." Based on this property, a greedy heuristic is proposed which yields solutions with approximation ratio no less than 1-1/e. Also, dynamic marketing for SIS models is reformulated into an equivalent convex optimization to obtain an optimal solution. Both cost minimization and trade-off of cost and profit are analyzed. Next, the proposed modeling framework is applied to study competition of multiple companies in marketing of similar products. Marketing of two classes of such products are considered, namely marketing of durable consumer goods (DCG) and fast-moving consumer goods (FMCG). It is shown that an epsilon-equilibrium exists in the DCG marketing game and a pure Nash equilibrium exists in the FMCG marketing game. The Price of Anarchy (PoA) in both marketing games is found to be bounded by 2. Also, it is shown that any two Nash equilibria for the FMCG marketing game agree almost everywhere, and a distributed algorithm converging to the Nash equilibrium is designed for the FMCG marketing game. Finally, a preliminary investigation is carried out to explore possible concepts of network centrality for diffusions. In a diffusion process, the centrality of a node should reflect the influence that the node has on the network over time. Among the preliminary observations in this work, it is found that when an infection does not break out, diffusion centrality is closely related to Katz centrality; when an infection does break out, diffusion centrality is closely related to eigenvector centrality

Influence Maximization in Social Networks: A Survey

Online social networks have become an important platform for people to communicate, share knowledge and disseminate information. Given the widespread usage of social media, individuals' ideas, preferences and behavior are often influenced by their peers or friends in the social networks that they participate in. Since the last decade, influence maximization (IM) problem has been extensively adopted to model the diffusion of innovations and ideas. The purpose of IM is to select a set of k seed nodes who can influence the most individuals in the network. In this survey, we present a systematical study over the researches and future directions with respect to IM problem. We review the information diffusion models and analyze a variety of algorithms for the classic IM algorithms. We propose a taxonomy for potential readers to understand the key techniques and challenges. We also organize the milestone works in time order such that the readers of this survey can experience the research roadmap in this field. Moreover, we also categorize other application-oriented IM studies and correspondingly study each of them. What's more, we list a series of open questions as the future directions for IM-related researches, where a potential reader of this survey can easily observe what should be done next in this field

A Convex Framework for Epidemic Control in Networks

With networks becoming pervasive, research attention on dynamics of epidemic models in networked populations has increased. While a number of well understood epidemic spreading models have been developed, little to no attention has been paid to epidemic control strategies; beyond heuristics usually based on network centrality measures. Since epidemic control resources are typically limited, the problem of optimally allocating resources to control an outbreak becomes of interest. Existing literature considered homogeneous networks, limited the discussion to undirected networks, and largely proposed network centrality-based resource allocation strategies. In this thesis, we consider the well-known Susceptible-Infected-Susceptible spreading model and study the problem of minimum cost resource allocation to control an epidemic outbreak in a networked population. First, we briefly present a heuristic that outperforms network centrality-based algorithms on a stylized version of the problem previously studied in the literature. We then solve the epidemic control problem via a convex optimization framework on weighted, directed networks comprising heterogeneous nodes. Based on our spreading model, we express the problem of controlling an epidemic outbreak in terms of spectral conditions involving the Perron-Frobenius eigenvalue. This enables formulation of the epidemic control problem as a Geometric Program (GP), for which we derive a convex characterization guaranteeing existence of an optimal solution. We consider two formulations of the epidemic control problem -- the first seeks an optimal vaccine and antidote allocation strategy given a constraint on the rate at which the epidemic comes under control. The second formulation seeks to find an optimal allocation strategy given a budget on the resources. The solution framework for both formulations also allows for control of an epidemic outbreak on networks that are not necessarily strongly connected. The thesis further proposes a fully distributed solution to the epidemic control problem via a Distributed Alternating Direction Method of Multipliers (ADMM) algorithm. Our distributed solution enables each node to locally compute its optimum allocation of vaccines and antidotes needed to collectively globally contain the spread of an outbreak, via local exchange of information with its neighbors. Contrasting previous literature, our problem is a constrained optimization problem associated with a directed network comprising non-identical agents. For the different problem formulations considered, illustrations that validate our solutions are presented. This thesis, in sum, proposes a paradigm shift from heuristics towards a convex framework for contagion control in networked populations

Localizing the Source of an Epidemic Using Few Observations

Localizing the source of an epidemic is a crucial task in many contexts, including the detection of malicious users in social networks and the identification of patient zeros of disease outbreaks. The difficulty of this task lies in the strict limitations on the data available: In most cases, when an epidemic spreads, only few individuals, who we will call sensors, provide information about their state. Furthermore, as the spread of an epidemic usually depends on a large number of variables, accounting for all the possible spreading patterns that could explain the available data can easily result in prohibitive computational costs. Therefore, in the field of source localization, there are two central research directions: The design of practical and reliable algorithms for localizing the source despite the limited data, and the optimization of data collection, i.e., the identification of the most informative sensors. In this dissertation we contribute to both these directions. We consider network epidemics starting from an unknown source. The only information available is provided by a set of sensor nodes that reveal if and when they become infected. We study how many sensors are needed to guarantee the identification of the source. A set of sensors that guarantees the identification of the source is called a double resolving set (DRS); the minimum size of a DRS is called the double metric dimension (DMD). Computing the DMD is, in general, hard, hence estimating it with bounds is desirable. We focus on G(N,p) random networks for which we derive tight bounds for the DMD. We show that the DMD is a non-monotonic function of the parameter p, hence there are critical parameter ranges in which source localization is particularly difficult. Again building on the relationship between source localization and DRSs, we move to optimizing the choice of a fixed number K of sensors. First, we look at the case of trees where the uniqueness of paths makes the problem simpler. For this case, we design polynomial time algorithms for selecting K sensors that optimize certain metrics of interest. Next, turning to general networks, we show that the optimal sensor set depends on the distribution of the time it takes for an infected node u to infect a non-infected neighbor v, which we call the transmission delay from u to v. We consider both a low- and a high-variance regime for the transmission delays. We design algorithms for sensor placement in both cases, and we show that they yield an improvement of up to 50% over state-of-the-art methods. Finally, we propose a framework for source localization where some sensors (called dynamic sensors) can be added while the epidemic spreads and the localization progresses. We design an algorithm for joint source localization and dynamic sensor placement; This algorithm can handle two regimes: offline localization, where we localize the source after the epidemic spread, and online localization, where we localize the source while the epidemic is ongoing. We conduct an empirical study of offline and online localization and show that, by using dynamic sensors, the number of sensors we need to localize the source is up to 10 times less with respect to a strategy where all sensors are deployed a priori. We also study the resistance of our methods to high-variance transmission delays and show that, even in this setting, using dynamic sensors, the source can be localized with less than 5% of the nodes being sensors

Analysis And Control Of Networked Systems Using Structural And Measure-Theoretic Approaches

Network control theory provides a plethora of tools to analyze the behavior of dynamical processes taking place in complex networked systems. The pattern of interconnections among components affects the global behavior of the overall system. However, the analysis of the global behavior of large scale complex networked systems offers several major challenges. First of all, analyzing or characterizing the features of large-scale networked systems generally requires full knowledge of the parameters describing the system\u27s dynamics. However, in many applications, an exact quantitative description of the parameters of the system may not be available due to measurement errors and/or modeling uncertainties. Secondly, retrieving the whole structure of many real networks is very challenging due to both computation and security constraints. Therefore, an exact analysis of the global behavior of many real-world networks is practically unfeasible. Finally, the dynamics describing the interactions between components are often stochastic, which leads to difficulty in analyzing individual behaviors in the network. In this thesis, we provide solutions to tackle all the aforementioned challenges. In the first part of the thesis, we adopt graph-theoretic approaches to address the problem caused by inexact modeling and imprecise measurements. More specifically, we leverage the connection between algebra and graph theory to analyze various properties in linear structural systems. Using these results, we then design efficient graph-theoretic algorithms to tackle topology design problems in structural systems. In the second part of the thesis, we utilize measure-theoretic techniques to characterize global properties of a network using local structural information in the form of closed walks or subgraph counts. These methods are based on recent results in real algebraic geometry that relates semidefinite programming to the multidimensional moment problem. We leverage this connection to analyze stochastic networked spreading processes and characterize safety in nonlinear dynamical systems