arXiv subjects

Scott D. Pauls

Publications and source records attributed to Scott D. Pauls.

16 recordsLinked to original sources

arXiv2022

Spatial Games of Fake News

To curb the spread of fake news on social media platforms, recent studies have considered an online crowdsourcing fact-checking approach as one possible intervention method to reduce misinformation. However, it remains unclear under what conditions crowdsourcing fact-checking efforts deter the spread of misinformation. To address this issue, we model such distributed fact-checking as `peer policing' that will reduce the perceived payoff to share or disseminate false information (fake news) and also reward the spread of trustworthy information (real news). By simulating our model on synthetic square lattices and small-world networks, we show that the presence of social network structure enables fake news spreaders to be self-organized into echo chambers, thereby providing a boost to the efficacy of fake news and thus its resistance to fact-checking efforts. Additionally, to study our model in a more realistic setting, we utilize a Twitter network dataset and study the effectiveness of deliberately choosing specific individuals to be fact-checkers. We find that targeted fact-checking efforts can be highly effective, seeing the same level of success with as little as a fifth of the number of fact-checkers, but it depends on the structure of the network in question. In the limit of weak selection, we obtain closed-form analytical conditions for critical threshold of crowdsourced fact-checking in terms of the payoff values in our fact-checker/fake news game. Our work has practical implications for developing model-based mitigation strategies for controlling the spread of misinformation that interferes with the political discourse.

physics.soc-ph↗

arXiv2021

The dual problems of coordination and anti-coordination on random bipartite graphs

In some scenarios ("anti-coordination games"), individuals are better off choosing different actions than their neighbors while in other scenarios ("coordination games"), it is beneficial for individuals to choose the same strategy as their neighbors. Despite having different incentives and resulting population dynamics, it is largely unknown which collective outcome, anti-coordination or coordination, is easier to achieve. To address this issue, we focus on the distributed graph coloring problem on bipartite graphs. We show that with only two strategies, anti-coordination games (2-colorings) and coordination games (uniform colorings) are dual problems that are equally difficult to solve. To prove this, we construct an isomorphism between the Markov chains arising from the corresponding anti-coordination and coordination games under certain specific individual stochastic decision-making rules. Our results provide novel insights into solving collective action problems on networks.

physics.soc-ph↗

arXiv2021

Random Choices can Facilitate the Solving of Collective Network Coloring Problems by Artificial Agents

Global coordination is required to solve a wide variety of challenging collective action problems from network colorings to the tragedy of the commons. Recent empirical study shows that the presence of a few noisy autonomous agents can greatly improve collective performance of humans in solving networked color coordination games. To provide further analytical insights into the role of behavioral randomness, here we study myopic artificial agents attempt to solve similar network coloring problems using decision update rules that are only based on local information but allow random choices at various stages of their heuristic reasonings. We consider that agents are distributed over a random bipartite network which is guaranteed to be solvable with two colors. Using agent-based simulations and theoretical analysis, we show that the resulting efficacy of resolving color conflicts is dependent on the specific implementation of random behavior of agents, including the fraction of noisy agents and at which decision stage noise is introduced. Moreover, behavioral randomness can be finely tuned to the specific underlying population structure such as network size and average network degree in order to produce advantageous results in finding collective coloring solutions. Our work demonstrates that distributed greedy optimization algorithms exploiting local information should be deployed in combination with occasional exploration via random choices in order to overcome local minima and achieve global coordination.

physics.soc-ph↗

arXiv2020

Evolutionary Kuramoto Dynamics

Common models of synchronizable oscillatory systems consist of a collection of coupled oscillators governed by a collection of differential equations. The ubiquitous Kuramoto models rely on an {\em a priori} fixed connectivity pattern facilitates mutual communication and influence between oscillators. In biological synchronizable systems, like the mammalian suprachaismatic nucleus, enabling communication comes at a cost -- the organism expends energy creating and maintaining the system -- linking their development to evolutionary selection. Here, we introduce and analyze a new evolutionary game theoretic framework modeling the behavior and evolution of systems of coupled oscillators. Each oscillator in our model is characterized by a pair of dynamic behavioral traits: an oscillatory phase and whether they connect and communicate to other oscillators or not. Evolution of the system occurs along these dimensions, allowing oscillators to change their phases and/or their communication strategies. We measure success of mutations by comparing the benefit of phase synchronization to the organism balanced against the cost of creating and maintaining connections between the oscillators. Despite such a simple setup, this system exhibits a wealth of nontrivial behaviors, mimicking different classical games -- the Prisoner's Dilemma, the snowdrift game, and coordination games -- as the landscape of the oscillators changes over time. Despite such complexity, we find a surprisingly simple characterization of synchronization through connectivity and communication: if the benefit of synchronization $B(0)$ is greater than twice the cost $c$, $B(0) > 2c$, the organism will evolve towards complete communication and phase synchronization. Taken together, our model demonstrates possible evolutionary constraints on both the existence of a synchronized oscillatory system and its overall connectivity.

q-bio.PE↗

arXiv2017

A new framework for dynamical models on multiplex networks

Many complex systems have natural representations as multi-layer networks. While these formulations retain more information than standard single-layer network models, there is not yet a fully developed theory for computing network metrics and statistics on these objects. We introduce a family of models of multiplex processes motivated by dynamical applications and investigate the properties of their spectra both theoretically and computationally. We study special cases of multiplex diffusion and Markov dynamics, using the spectral results to compute their rates of convergence. We use our framework to define a version of multiplex eigenvector centrality, which generalizes some existing notions in the literature. Last, we compare our operator to structurally-derived models on synthetic and real-world networks, helping delineate the contexts in which the different frameworks are appropriate.

cs.SI↗

arXiv2017

Spectral Clustering Methods for Multiplex Networks

Multiplex networks offer an important tool for the study of complex systems and extending techniques originally designed for single--layer networks is an important area of study. One of the most important methods for analyzing networks is clustering the nodes into communities that represent common connectivity patterns. In this paper we extend spectral clustering to multiplex structures and discuss some of the difficulties that arise in attempting to define a natural generalization. In order to analyze our approach, we describe three simple, synthetic multiplex networks and compare the performance of different multiplex models. Our results suggest that a dynamically motivated model is more successful than a structurally motivated model in discovering the appropriate communities.

cs.SI↗

arXiv2011

A measure of centrality based on the spectrum of the Laplacian

We introduce a family of new centralities, the k-spectral centralities. k-Spectral centrality is a measurement of importance with respect to the deformation of the graph Laplacian associated with the graph. Due to this connection, k-spectral centralities have various interpretations in terms of spectrally determined information. We explore this centrality in the context of several examples. While for sparse unweighted networks 1-spectral centrality behaves similarly to other standard centralities, for dense weighted networks they show different properties. In summary, the k-spectral centralities provide a novel and useful measurement of relevance (for single network elements as well as whole subnetworks) distinct from other known measures.

physics.data-an↗

arXiv2008

Topological structures in the equities market network

We present a new method for articulating scale-dependent topological descriptions of the network structure inherent in many complex systems. The technique is based on "Partition Decoupled Null Models,'' a new class of null models that incorporate the interaction of clustered partitions into a random model and generalize the Gaussian ensemble. As an application we analyze a correlation matrix derived from four years of close prices of equities in the NYSE and NASDAQ. In this example we expose (1) a natural structure composed of two interacting partitions of the market that both agrees with and generalizes standard notions of scale (eg., sector and industry) and (2) structure in the first partition that is a topological manifestation of a well-known pattern of capital flow called "sector rotation.'' Our approach gives rise to a natural form of multiresolution analysis of the underlying time series that naturally decomposes the basic data in terms of the effects of the different scales at which it clusters. The equities market is a prototypical complex system and we expect that our approach will be of use in understanding a broad class of complex systems in which correlation structures are resident.

q-fin.PR↗

arXiv2007

Variation of Perimeter Measure in sub-Riemannian geometry

We derive a formula for the first variation of horizontal perimeter measure for $C^2$ hypersurfaces of completely general sub-Riemannian manifolds, allowing for the existence of characteristic points. For $C^2$ hypersurfaces in vertically rigid sub-Riemannian manifolds we also produce a second variation formula for variations supported away from the characteristic locus. This variation formula is used to show the bubble sets in \hn{2} are stable under volume preserving variations.

math.DG↗

arXiv2006

H-minimal graphs of low regularity in the Heisenberg group

In this paper we investigate H-minimal graphs of lower regularity. We show that noncharactersitic C^1 H-minimal graphs whose components of the unit horizontal Gauss map are in W^{1,1} are ruled surfaces with C^2 seed curves. In a different direction, we investigate ways in which patches of C^1 H-minimal graphs can be glued together to form continuous piecewise C^1 H-minimal surfaces. We apply these description of H-minimal graphs to the question of the existence of smooth solutions to the Dirichlet problem with smooth data. We find a necessary condition for the existence of smooth solutions and produce examples where the conditions are satisfied and where they fail. In particular we illustrate the failure of the smoothness of the data to force smoothness of the solution to the Dirichlet problem by producing a class of smooth curves whoses H-minimal spanning graphs cannot be C^2.

math.DG↗

arXiv2006

Minimal surfaces in the Heisenberg group

We investigate the minimal surface problem in the three dimensional Heisenberg group, H, equipped with its standard Carnot-Caratheodory metric. Using a particular surface measure, we characterize minimal surfaces in terms of a sub-elliptic partial differential equation and prove an existence result for the Plateau problem in this setting. Further, we provide a link between our minimal surfaces and Riemannian constant mean curvature surfaces in H equipped with different Riemannian metrics approximating the Carnot-Caratheodory metric. We generate a large library of examples of minimal surfaces and use these to show that the solution to the Dirichlet problem need not be unique. Moreover, we show that the minimal surfaces we construct are in fact X-minimal surfaces in the sense of Garofalo and Nhieu.

math.DG↗

arXiv2005

C^1 Hypersurfaces of the Heisenberg Group are N-Rectifiable

We show that C^1 hypersurfaces in the Heisenberg group are countably N-rectifiable. As a corollary, this shows that all C^1_H graphs over the xy-plane are countable N-rectifiable, showing the equivalence of this notion of rectifiability with that of Franchi, Serra Cassano and Serapioni for such surfaces.

math.DG↗

arXiv2005

Minimal surfaces in the roto-translation group with applications to a neuro-biological image completion model

We investigate solutions to the minimal surface problem with Dirichlet boundary conditions in the roto-translation group equipped with a subRiemannian metric. By work of G. Citti and A. Sarti, such solutions are amodal completions of occluded visual data when using a model of the first layer of the visual cortex. Using a characterization of smooth minimal surfaces as ruled surfaces, we give a method to compute a minimal spanning surface given fixed boundary data presuming such a surface exists. Moreover, we describe a number of obstructions to existence and uniqueness but also show that under suitable conditions, smooth minimal spanning surfaces with good properties exist. Not only does this provide an explicit realization of the disocclusion process for the neurobiological model, but it also has application to contructing disocclusion algorithms in digital image processing.

math.DG↗

arXiv2005

Constant mean curvature surfaces in sub-Riemannian geometry

We investigate the minimal and isoperimetric surface problems in a large class of sub-Riemannian manifolds, the so-called Vertically Rigid spaces. We construct an adapted connection for such spaces and, using the variational tools of Bryant, Griffiths and Grossman, derive succinct forms of the Euler-Lagrange equations for critical points for the associated variational problems. Using the Euler-Lagrange equations, we show that minimal and isoperimetric surfaces satisfy a constant horizontal mean curvature conditions away from characteristic points. Moreover, we use the formalism to construct a horizontal second fundamental form, $II_0$, for vertically rigid spaces and, as a first application, use $II_0$ to show that minimal surfaces cannot have points of horizontal positive curvature and, that minimal surfaces in Carnot groups cannot be locally horizontally geometrically convex. We note that the convexity condition is distinct from others currently in the literature.

math.DG↗

arXiv2005

The Bernstein Problem in the Heisenberg Group

We establish the following theorem of Bernstein type for the first Heisenberg group: Let S be a C^2 connected H-minimal surface which is a graph over some plane P, then S is either a non-characteristic vertical plane, or its generalized seed curve satisfies a type of constant curvature condition.

math.DG↗

arXiv2004

A notion of rectifiability modeled on Carnot groups

We introduce a notion of rectifiability modeled on Carnot groups. Precisely, we say that a subset E of a Carnot group M and N is a subgroup of M, we say E is N-rectifiable if it is the Lipschitz image of a positive measure subset of N. First, we discuss the implications of N-rectifiability, where N is a Carnot group (not merely a subgroup of a Carnot group), which include N-approximability and the existence of approximate tangent cones isometric to N almost everywhere in E. Second, we prove that, under a stronger condition concerning the existence of approximate tangent cones isomorphic to N almost everywhere in a set E, that E is N-rectifiable. Third, we investigate the rectifiability properties of level sets of C^1_N functions, where N is a Carnot group. We show that for almost every real number t and almost every noncharacteristic point x in a level set of f, there exists a subgroup T_x of H and r >0 so that f^{-1}(t) intersected with B_H(x,r) is T_x-approximable at x and an approximate tangent cone isomorphic to T_x at x.

math.CA↗