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  1. Abstract

    Preferential attachment, homophily, and their consequences such as scale-free (i.e. power-law) degree distributions, the glass ceiling effect (the unseen, yet unbreakable barrier that keeps minorities and women from rising to the upper rungs of the corporate ladder, regardless of their qualifications or achievements) and perception bias are well-studied in undirected networks. However, such consequences and the factors that lead to their emergence in directed networks (e.g. author–citation graphs, Twitter) are yet to be coherently explained in an intuitive, theoretically tractable manner using a single dynamical model. To this end, we present a theoretical and numerical analysis of the novel Directed Mixed Preferential Attachment model in order to explain the emergence of scale-free degree distributions and the glass ceiling effect in directed networks with two groups (minority and majority). Specifically, we first derive closed-form expressions for the power-law exponents of the in-degree and out-degree distributions of each of the two groups and then compare the derived exponents with each other to obtain useful insights. These insights include answers to questions such as: when does the minority group have an out-degree (or in-degree) distribution with a heavier tail compared to the majority group? what factors cause the tail of the out-degree distribution of a group to be heavier than the tail of its own in-degree distribution? what effect does frequent addition of edges between existing nodes have on the in-degree and out-degree distributions of the majority and minority groups? Answers to these questions shed light on the interplay between structure (i.e. the in-degree and out-degree distributions of the two groups) and dynamics (characterized collectively by the homophily, preferential attachment, group sizes and growth dynamics) of various real-world directed networks. We also provide a novel definition of the glass ceiling faced by a group via the number of individuals with large out-degree (i.e. those with many followers) normalized by the number of individuals with large in-degree (i.e. those who follow many others) and then use it to characterize the conditions that cause the glass ceiling effect to emerge in a directed network. Our analytical results are supported by detailed numerical experiments. The DMPA model and its theoretical and numerical analysis provided in this article are useful for analysing various phenomena on directed networks in fields such as network science and computational social science.

     
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  2. Free, publicly-accessible full text available July 1, 2024
  3. Suppose L simultaneous independent stochastic systems generate observations, where the observations from each system depend on the underlying parameter of that system. The observations are unlabeled (anonymized), in the sense that an analyst does not know which observation came from which stochastic system. How can the analyst estimate the underlying parameters of the L systems? Since the anonymized observations at each time are an unordered set of L measurements (rather than a vector), classical stochastic gradient algorithms cannot be directly used. By using symmetric polynomials, we formulate a symmetric measurement equation that maps the observation set to a unique vector. We then construct an adaptive filtering algorithm that yields a statistically consistent estimate of the underlying parameters. 
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    Free, publicly-accessible full text available June 4, 2024
  4. This paper presents an inverse reinforcement learning (IRL) framework for Bayesian stopping time problems. By observing the actions of a Bayesian decision maker, we provide a necessary and sufficient condition to identify if these actions are consistent with optimizing a cost function. In a Bayesian (partially observed) setting, the inverse learner can at best identify optimality wrt the observed strategies. Our IRL algorithm identifies optimality and then constructs set-valued estimates of the cost function. To achieve this IRL objective, we use novel ideas from Bayesian revealed preferences stemming from microeconomics. We illustrate the proposed IRL scheme using two important examples of stopping time problems, namely, sequential hypothesis testing and Bayesian search. As a real-world example, we illustrate using a YouTube dataset comprising metadata from 190000 videos how the proposed IRL method predicts user engagement in online multimedia platforms with high accuracy. Finally, for finite datasets, we propose an IRL detection algorithm and give finite sample bounds on its error probabilities. 
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  5. In mathematical psychology, decision makers are modeled using the Lindbladian equations from quantum mechanics to capture important human-centric features such as order effects and violation of the sure thing principle. We consider human-machine interaction involving a quantum decision maker (human) and a controller (machine). Given a sequence of human decisions over time, how can the controller dynamically provide input messages to adapt these decisions so as to converge to a specific decision? We show via novel stochastic Lyapunov arguments how the Lindbladian dynamics of the quantum decision maker can be controlled to converge to a specific decision asymptotically. Our methodology yields a useful mathematical framework for human-sensor decision making. The stochastic Lyapunov results are also of independent interest as they generalize recent results in the literature. 
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  6. Inverse reinforcement learning (IRL) deals with estimating an agent’s utility function from its actions. In this paper, we consider how an agent can hide its strategy and mitigate an adversarial IRL attack; we call this inverse IRL (I-IRL). How should the decision maker choose its response to ensure a poor reconstruction of its strategy by an adversary performing IRL to estimate the agent’s strategy? This paper comprises four results: First, we present an adversarial IRL algorithm that estimates the agent’s strategy while controlling the agent’s utility function. Then, we propose an I-IRL result that mitigates the IRL algorithm used by the adversary. Our I-IRL results are based on revealed preference theory in microeconomics. The key idea is for the agent to deliberately choose sub-optimal responses so that its true strategy is sufficiently masked. Third, we give a sample complexity result for our main I-IRL result when the agent has noisy estimates of the adversary-specified utility function. Finally, we illustrate our I-IRL scheme in a radar problem where a meta-cognitive radar is trying to mitigate an adversarial target. 
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  7. This paper considers quickest detection scheme where the change in an underlying parameter influencing human decisions is to be detected by only observing the human decisions. Stemming from behavioral economics and mathematical psychology, we propose two generative models for the human decision maker. Namely, we consider an anticipatory decision making model and a quantum decision model. From a decision theoretic point of view, anticipatory models are time inconsistent, meaning that Bellman's principle of optimality does not hold. The appropriate formalism is thus the subgame Nash equilibrium. We show that the interaction between anticipatory agents and sequential quickest detection results in unusual (nonconvex) structure of the quickest change detection policy. In contrast the quantum decision model, despite its mathematical complexity, results in the typical convex quickest detection policy. The optimal quickest detection policy is shown to perform strictly worse than classical quickest detection for both models, via a Blackwell dominance argument. The model and structural results provided contribute to an understanding of the dynamics of human-sensor interfacing. 
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  8. This paper studies controlling segregation in social networks via exogenous incentives. We construct an edge formation game on a directed graph. A user (node) chooses the probability with which it forms an inter- or intra- community edge based on a utility function that reflects the tradeoff between homophily (preference to connect with individuals that belong to the same group) and the preference to obtain an exogenous incentive. Decisions made by the users to connect with each other determine the evolution of the social network. We explore an algorithmic recommendation mechanism where the exogenous incentive in the utility function is based on weak ties which incentivizes users to connect across communities and mitigates the segregation. This setting leads to a submodular game with a unique Nash equilibrium. In numerical simulations, we explore how the proposed model can be useful in controlling segregation and echo chambers in social networks under various settings. 
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