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We consider generic families of gradient-like dynamical systems with a parameter space $P$ which is a 2-dimensional simply connected domain. We prove that if over the boundary of $P$ there is a S or Z shaped bifurcation graph containing two opposing fold bifurcation points while over the rest of the boundary there are no other bifurcation points then there is an odd number of cusps in the interior of $P$.

Modern computational systems have an unprecedented ability to detect, leverage and influence human attention. Prior work identified user engagement and dwell time as two key metrics of attention in digital environments, but these metrics have yet to be integrated into a unified model that can advance the theory andpractice of digital attention. We draw on work from cognitive science, digital advertising, and AI to propose a two-stage model of attention for social media environments that disentangles engagement and dwell. In an online experiment, we show that attention operates differently in these two stages and find clear evidence of dissociation: when dwelling on posts (Stage 1), users attend more to sensational than credible content, but when deciding whether to engage with content (Stage 2), users attend more to credible than sensational content. These findings have implications for the design and development of computational systems that measure and model human attention, such as newsfeed algorithms on social media., Comment: All Things Attention NeurIPS Workshop

Unlike traditional media, social media typically provides quantified metrics of how many users have engaged with each piece of content. Some have argued that the presence of these cues promotes the spread of misinformation. Here we investigate the causal effect of social cues on users' engagement with social media posts. We conducted an experiment with N=628 Americans on a custom-built newsfeed interface where we systematically varied the presence and strength of social cues. We find that when cues are shown, indicating that a larger number of others have engaged with a post, users were more likely to share and like that post. Furthermore, relative to a control without social cues, the presence of social cues increased the sharing of true relative to false news. The presence of social cues also makes it more difficult to precisely predict how popular any given post would be. Together, our results suggest that -- instead of distracting users or causing them to share low-quality news -- social cues may, in certain circumstances, actually boost truth discernment and reduce the sharing of misinformation. Our work suggests that social cues play important roles in shaping users' attention and engagement on social media, and platforms should understand the effects of different cues before making changes to what cues are displayed and how.

Likelihood-based inference in stochastic non-linear dynamical systems, such as those found in chemical reaction networks and biological clock systems, is inherently complex and has largely been limited to small and unrealistically simple systems. Recent advances in analytically tractable approximations to the underlying conditional probability distributions enable long-term dynamics to be accurately modelled, and make the large number of model evaluations required for exact Bayesian inference much more feasible. We propose a new methodology for inference in stochastic non-linear dynamical systems exhibiting oscillatory behaviour and show the parameters in these models can be realistically estimated from simulated data. Preliminary analyses based on the Fisher Information Matrix of the model can guide the implementation of Bayesian inference. We show that this parameter sensitivity analysis can predict which parameters are practically identifiable. Several Markov chain Monte Carlo algorithms are compared, with our results suggesting a parallel tempering algorithm consistently gives the best approach for these systems, which are shown to frequently exhibit multi-modal posterior distributions.

Social media platforms are increasingly deploying complex interventions to help users detect false news. Labeling false news using techniques that combine crowd-sourcing with artificial intelligence (AI) offers a promising way to inform users about potentially low-quality information without censoring content, but also can be hard for users to understand. In this study, we examine how users respond in their sharing intentions to information they are provided about a hypothetical human-AI hybrid system. We ask i) if these warnings increase discernment in social media sharing intentions and ii) if explaining how the labeling system works can boost the effectiveness of the warnings. To do so, we conduct a study ($N=1473$ Americans) in which participants indicated their likelihood of sharing content. Participants were randomly assigned to a control, a treatment where false content was labeled, or a treatment where the warning labels came with an explanation of how they were generated. We find clear evidence that both treatments increase sharing discernment, and directional evidence that explanations increase the warnings' effectiveness. Interestingly, we do not find that the explanations increase self-reported trust in the warning labels, although we do find some evidence that participants found the warnings with the explanations to be more informative. Together, these results have important implications for designing and deploying transparent misinformation warning labels, and AI-mediated systems more broadly., Comment: Forthcoming in ICWSM 2022

A great deal of empirical research has examined who falls for misinformation and why. Here, we introduce a formal game-theoretic model of engagement with news stories that captures the strategic interplay between (mis)information consumers and producers. A key insight from the model is that observed patterns of engagement do not necessarily reflect the preferences of consumers. This is because producers seeking to promote misinformation can use strategies that lead moderately inattentive readers to engage more with false stories than true ones -- even when readers prefer more accurate over less accurate information. We then empirically test people's preferences for accuracy in the news. In three studies, we find that people strongly prefer to click and share news they perceive as more accurate -- both in a general population sample, and in a sample of users recruited through Twitter who had actually shared links to misinformation sites online. Despite this preference for accurate news -- and consistent with the predictions of our model -- we find markedly different engagement patterns for articles from misinformation versus mainstream news sites. Using 1,000 headlines from 20 misinformation and 20 mainstream news sites, we compare Facebook engagement data with 20,000 accuracy ratings collected in a survey experiment. Engagement with a headline is negatively correlated with perceived accuracy for misinformation sites, but positively correlated with perceived accuracy for mainstream sites. Taken together, these theoretical and empirical results suggest that consumer preferences cannot be straightforwardly inferred from empirical patterns of engagement.

Embryonic development leads to the reproducible and ordered appearance of complexity from egg to adult. The successive differentiation of different cell types, that elaborates this complexity, result from the activity of gene networks and was likened by Waddington to a flow through a landscape in which valleys represent alternative fates. Geometric methods allow the formal representation of such landscapes and codify the types of behaviors that result from systems of differential equations. Results from Smale and coworkers imply that systems encompassing gene network models can be represented as potential gradients with a Riemann metric, justifying the Waddington metaphor. Here, we extend this representation to include parameter dependence and enumerate all 3-way cellular decisions realisable by tuning at most two parameters, which can be generalized to include spatial coordinates in a tissue. All diagrams of cell states vs model parameters are thereby enumerated. We unify a number of standard models for spatial pattern formation by expressing them in potential form. Turing systems appear non-potential yet in suitable variables the dynamics are low dimensional, potential, and a time independent embedding recovers the biological variables. Lateral inhibition is described by a saddle point with many unstable directions. A model for the patterning of the Drosophila eye appears as relaxation in a bistable potential. Geometric reasoning provides intuitive dynamic models for development that are well adapted to fit time-lapse data.

When users on social media share content without considering its veracity, they may unwittingly be spreading misinformation. In this work, we investigate the design of lightweight interventions that nudge users to assess the accuracy of information as they share it. Such assessment may deter users from posting misinformation in the first place, and their assessments may also provide useful guidance to friends aiming to assess those posts themselves. In support of lightweight assessment, we first develop a taxonomy of the reasons why people believe a news claim is or is not true; this taxonomy yields a checklist that can be used at posting time. We conduct evaluations to demonstrate that the checklist is an accurate and comprehensive encapsulation of people's free-response rationales. In a second experiment, we study the effects of three behavioral nudges -- 1) checkboxes indicating whether headings are accurate, 2) tagging reasons (from our taxonomy) that a post is accurate via a checklist and 3) providing free-text rationales for why a headline is or is not accurate -- on people's intention of sharing the headline on social media. From an experiment with 1668 participants, we find that both providing accuracy assessment and rationale reduce the sharing of false content. They also reduce the sharing of true content, but to a lesser degree that yields an overall decrease in the fraction of shared content that is false. Our findings have implications for designing social media and news sharing platforms that draw from richer signals of content credibility contributed by users. In addition, our validated taxonomy can be used by platforms and researchers as a way to gather rationales in an easier fashion than free-response., Comment: In CSCW'21

Previous experiments have found mixed results on whether honesty is intuitive or requires deliberation. Here we add to this literature by building on prior work of Capraro (2017). We report a large study (N=1,389) manipulating time pressure vs time delay in a deception game. We find that, in this setting, people are more honest under time pressure, and that this result is not driven by confounds present in earlier work., Comment: Journal of Behavioral and Experimental Economics, In press

Extensive cooperation among unrelated individuals is unique to humans, who often sacrifice personal benefits for the common good and work together to achieve what they are unable to execute alone. The evolutionary success of our species is indeed due, to a large degree, to our unparalleled other-regarding abilities. Yet, a comprehensive understanding of human cooperation remains a formidable challenge. Recent research in social science indicates that it is important to focus on the collective behavior that emerges as the result of the interactions among individuals, groups, and even societies. Non-equilibrium statistical physics, in particular Monte Carlo methods and the theory of collective behavior of interacting particles near phase transition points, has proven to be very valuable for understanding counterintuitive evolutionary outcomes. By studying models of human cooperation as classical spin models, a physicist can draw on familiar settings from statistical physics. However, unlike pairwise interactions among particles that typically govern solid-state physics systems, interactions among humans often involve group interactions, and they also involve a larger number of possible states even for the most simplified description of reality. The complexity of solutions therefore often surpasses that observed in physical systems. Here we review experimental and theoretical research that advances our understanding of human cooperation, focusing on spatial pattern formation, on the spatiotemporal dynamics of observed solutions, and on self-organization that may either promote or hinder socially favorable states., Comment: 48 two-column pages, 35 figures; Review accepted for publication in Physics Reports

We integrate dual-process theories of human cognition with evolutionary game theory to study the evolution of automatic and controlled decision-making processes. We introduce a model where agents who make decisions using either automatic or controlled processing compete with each other for survival. Agents using automatic processing act quickly and so are more likely to acquire resources, but agents using controlled processing are better planners and so make more effective use of the resources they have. Using the replicator equation, we characterize the conditions under which automatic or controlled agents dominate, when coexistence is possible, and when bistability occurs. We then extend the replicator equation to consider feedback between the state of the population and the environment. Under conditions where having a greater proportion of controlled agents either enriches the environment or enhances the competitive advantage of automatic agents, we find that limit cycles can occur, leading to persistent oscillations in the population dynamics. Critically, however, these limit cycles only emerge when feedback occurs on a sufficiently long time scale. Our results shed light on the connection between evolution and human cognition, and demonstrate necessary conditions for the rise and fall of rationality., Comment: 9 pages, 7 figures

We explore a replicator-mutator model of the repeated Prisoner's Dilemma involving three strategies: always cooperate (ALLC), always defect (ALLD), and tit-for-tat (TFT). The dynamics resulting from single unidirectional mutations are considered, with detailed results presented for the mutations TFT $\rightarrow$ ALLC and ALLD $\rightarrow$ ALLC. For certain combinations of parameters, given by the mutation rate $\mu$ and the complexity cost $c$ of playing tit-for-tat, we find that the population settles into limit cycle oscillations, with the relative abundance of ALLC, ALLD, and TFT cycling periodically. Surprisingly, these oscillations can occur for unidirectional mutations between any two strategies. In each case, the limit cycles are created and destroyed by supercritical Hopf and homoclinic bifurcations, organized by a Bogdanov-Takens bifurcation. Our results suggest that stable oscillations are a robust aspect of a world of ALLC, ALLD, and costly TFT; the existence of cycles does not depend on the details of assumptions of how mutation is implemented., Comment: 18 pages, 9 figures