Search

Consider $n$ identical Kuramoto oscillators on a random graph. Specifically, consider \ER random graphs in which any two oscillators are bidirectionally coupled with unit strength, independently and at random, with probability $0\leq p\leq 1$. We say that a network is globally synchronizing if the oscillators converge to the all-in-phase synchronous state for almost all initial conditions. Is there a critical threshold for $p$ above which global synchrony is extremely likely but below which it is extremely rare? It is suspected that a critical threshold exists and is close to the so-called connectivity threshold, namely, $p\sim \log(n)/n$ for $n \gg 1$. Ling, Xu, and Bandeira made the first progress toward proving a result in this direction: they showed that if $p\gg \log(n)/n^{1/3}$, then \ER networks of Kuramoto oscillators are globally synchronizing with high probability as $n\rightarrow\infty$. Here we improve that result by showing that $p\gg \log^2(n)/n$ suffices. Our estimates are explicit: for example, we can say that there is more than a $99.9996\%$ chance that a random network with $n = 10^6$ and $p>0.01117$ is globally synchronizing., 9 pages, 2 figures

We characterize absorption-time distributions for birth-death Markov chains with an absorbing boundary. For "extinction-prone" chains (which drift on average toward the absorbing state) the asymptotic distribution is Gaussian, Gumbel, or belongs to a family of skewed distributions. The latter two cases arise when the dynamics slow down dramatically near the boundary. Several models of evolution, epidemics, and chemical reactions fall into these classes; in each case we establish new results for the absorption-time distribution. Applications to African sleeping sickness are discussed., Comment: 6 pages, 4 figures, 11 page supplemental material

Using a combination of theory, experiment, and simulation, we revisit the dynamics of two coupled metronomes on a moving platform. Our experiments show that the platform's motion is damped by a dry friction force of Coulomb type, not the viscous linear friction force that has often been assumed in the past. Prompted by this result, we develop a new mathematical model that builds on previously introduced models, but departs from them in its treatment of the friction on the platform. We analyze the model by a two-timescale analysis and derive the slow-flow equations that determine its long-term dynamics. The derivation of the slow flow is challenging, due to the stick-slip motion of the platform in some parameter regimes. Simulations of the slow flow reveal various kinds of long-term behavior including in-phase and antiphase synchronization of identical metronomes, phase locking and phase drift of non-identical metronomes, and metronome suppression and death. In these latter two states, one or both of the metronomes come to swing at such low amplitude that they no longer engage their escapement mechanisms. We find good agreement between our theory, simulations, and experiments, but stress that our exploration is far from exhaustive. Indeed, much still remains to be learned about the dynamics of coupled metronomes, despite their simplicity and familiarity., Comment: 25 pages, 19 figures

We study a system of $N$ interacting particles moving on the unit sphere in $d$-dimensional space. The particles are self-propelled and coupled all to all, and their motion is heavily overdamped. For $d=2$, the system reduces to the classic Kuramoto model of coupled oscillators; for $d=3$, it has been proposed to describe the orientation dynamics of swarms of drones or other entities moving about in three-dimensional space. Here we use group theory to explain the recent discovery that the model shows low-dimensional dynamics for all $N \ge 3$, and to clarify why it admits the analog of the Ott-Antonsen ansatz in the continuum limit $N \rightarrow \infty$. The underlying reason is that the system is intimately connected to the natural hyperbolic geometry on the unit ball $B^d$. In this geometry, the isometries form a Lie group consisting of higher-dimensional generalizations of the M\"obius transformations used in complex analysis. Once these connections are realized, the reduced dynamics and the generalized Ott-Antonsen ansatz follow immediately. This framework also reveals the seamless connection between the finite and infinite-$N$ cases. Finally, we show that special forms of coupling yield gradient dynamics with respect to the hyperbolic metric, and use that fact to obtain global stability results about convergence to the synchronized state.

The outcome of an election depends not only on which candidate is more popular, but also on how many of their voters actually turn out to vote. Here we consider a simple model in which voters abstain from voting if they think their vote would not matter. Specifically, they do not vote if they feel sure their preferred candidate will win anyway (a condition we call complacency), or if they feel sure their candidate will lose anyway (a condition we call dejectedness). The voters reach these decisions based on a myopic assessment of their local network, which they take as a proxy for the entire electorate: voters know which candidate their neighbors prefer and they assume-perhaps incorrectly-that those neighbors will turn out to vote, so they themselves cast a vote if and only if it would produce a tie or a win for their preferred candidate in their local neighborhood. We explore various network structures and distributions of voter preferences and find that certain structures and parameter regimes favor unrepresentative outcomes where a minority faction wins, especially when the locally preferred candidate is not representative of the electorate as a whole.

Significance Attaching sensors to crowd-sourced vehicles could provide a cheap and accurate way to monitor air pollution, road quality, and other aspects of a city’s health. But in order for so-called drive-by sensing to be practically useful, the sensor-equipped vehicle fleet needs to have large “sensing power”—that is, it needs to cover a large fraction of a city’s area during a given reference period. Here, we provide an analytic description of the sensing power of taxi fleets, which agrees with empirical data from nine major cities. Our results show taxis’ sensing power is unexpectedly large—in Manhattan; just 10 random taxis cover one-third of street segments daily, which certifies that drive-by sensing can be readily implemented in the real world., Sensors can measure air quality, traffic congestion, and other aspects of urban environments. The fine-grained diagnostic information they provide could help urban managers to monitor a city’s health. Recently, a “drive-by” paradigm has been proposed in which sensors are deployed on third-party vehicles, enabling wide coverage at low cost. Research on drive-by sensing has mostly focused on sensor engineering, but a key question remains unexplored: How many vehicles would be required to adequately scan a city? Here, we address this question by analyzing the sensing power of a taxi fleet. Taxis, being numerous in cities, are natural hosts for the sensors. Using a ball-in-bin model in tandem with a simple model of taxi movements, we analytically determine the fraction of a city’s street network sensed by a fleet of taxis during a day. Our results agree with taxi data obtained from nine major cities and reveal that a remarkably small number of taxis can scan a large number of streets. This finding appears to be universal, indicating its applicability to cities beyond those analyzed here. Moreover, because taxis’ motion combines randomness and regularity (passengers’ destinations being random, but the routes to them being deterministic), the spreading properties of taxi fleets are unusual; in stark contrast to random walks, the stationary densities of our taxi model obey Zipf’s law, consistent with empirical taxi data. Our results have direct utility for town councilors, smart-city designers, and other urban decision makers.

To explore basin geometry in high-dimensional dynamical systems, we consider a ring of identical Kuramoto oscillators. Many attractors coexist in this system; each is a twisted periodic orbit characterized by a winding number $q$, with basin size proportional to $e^{-kq^2}.$ We uncover the geometry behind this size distribution and find the basins are octopus-like, with nearly all their volume in the tentacles, not the head of the octopus (the ball-like region close to the attractor). We present a simple geometrical reason why basins with tentacles should be common in high-dimensional systems., Comment: published version

Seasonal influenza presents an ongoing challenge to public health. The rapid evolution of the flu virus necessitates annual vaccination campaigns, but the decision to get vaccinated or not in a given year is largely voluntary, at least in the USA, and many people decide against it. In some early attempts to model these yearly flu vaccine decisions, it was often assumed that individuals behave rationally, and do so with perfect information-assumptions that allowed the techniques of classical economics and game theory to be applied. However, these assumptions are not fully supported by the emerging empirical evidence about human decision-making behavior in this context. We develop a simple model of coupled disease spread and vaccination dynamics that instead incorporates experimental observations from social psychology to model annual vaccine decision-making more realistically. We investigate population-level effects of these new decision-making assumptions, with the goal of understanding whether the population can self-organize into a state of herd immunity, and if so, under what conditions. Our model agrees with the established results while also revealing more subtle population-level behavior, including biennial oscillations about the herd immunity threshold.

Seasonal influenza presents an ongoing challenge to public health. The rapid evolution of the flu virus necessitates annual vaccination campaigns, but the decision to get vaccinated or not in a given year is largely voluntary, at least in the United States, and many people decide against it. In early attempts to model these yearly flu vaccine decisions, it was often assumed that individuals behave rationally, and do so with perfect information -- assumptions that allowed the techniques of classical economics and game theory to be applied. However, the usual assumptions are contradicted by the emerging empirical evidence about human decision-making behavior in this context. We develop a simple model of coupled disease spread and vaccination dynamics that instead incorporates experimental observations from social psychology to model annual vaccine decision-making more realistically. We investigate population-level effects of these new decision-making assumptions, with the goal of understanding whether the population can self-organize into a state of herd immunity, and if so, under what conditions. Our model agrees with established results while also revealing more subtle population-level behavior, including biennial oscillations about the herd immunity threshold., 20 pages, 6 figures

Consider any network of $n$ identical Kuramoto oscillators in which each oscillator is coupled bidirectionally with unit strength to at least $\mu (n-1)$ other oscillators. There is a critical value of the connectivity, $\mu_c$, such that whenever $\mu>\mu_c$, the system is guaranteed to converge to the all-in-phase synchronous state for almost all initial conditions, but when $\mu 0.6838$. In this paper, we prove that $\mu_c\leq 0.75$ and explain why this is the best upper bound that one can obtain by a purely linear stability analysis., Comment: 6 pages, 1 figure

Information and communication technologies have opened the way to new solutions for urban mobility that provide better ways to match individuals with on-demand vehicles. However, a fundamental unsolved problem is how best to size and operate a fleet of vehicles, given a certain demand for personal mobility. Previous studies(1-5) either do not provide a scalable solution or require changes in human attitudes towards mobility. Here we provide a network-based solution to the following 'minimum fleet problem', given a collection of trips (specified by origin, destination and start time), of how to determine the minimum number of vehicles needed to serve all the trips without incurring any delay to the passengers. By introducing the notion of a 'vehicle-sharing network', we present an optimal computationally efficient solution to the problem, as well as a nearly optimal solution amenable to real-time implementation. We test both solutions on a dataset of 150 million taxi trips taken in the city of New York over one year(6). The real-time implementation of the method with near-optimal service levels allows a 30 per cent reduction in fleet size compared to current taxi operation. Although constraints on driver availability and the existence of abnormal trip demands may lead to a relatively larger optimal value for the fleet size than that predicted here, the fleet size remains robust for a wide range of variations in historical trip demand. These predicted reductions in fleet size follow directly from a reorganization of taxi dispatching that could be implemented with a simple urban app; they do not assume ride sharing(7-9), nor require changes to regulations, business models, or human attitudes towards mobility to become effective. Our results could become even more relevant in the years ahead as fleets of networked, self-driving cars become commonplace(10-14).

Droplet motion arises in many natural phenomena, ranging from the familiar gravity-driven slip and arrest of raindrops on windows to the directed transport of droplets for water harvesting by plants and animals under dry conditions. Deliberate transportation and manipulation of droplets are also important in many technological applications, including droplet-based microfluidic chemical reactors and for thermal management. Droplet motion usually requires gradients of surface energy or temperature or external vibration to overcome contact angle hysteresis. Here, we report a new phenomenon in which a drying droplet placed on a periodically compliant surface undergoes spontaneous, erratic motion in the absence of surface energy gradients and external stimuli such as vibration. By modeling the droplet as a mass-spring system on a substrate with periodically varying compliance, we show that the stability of equilibrium depends on the size of the droplet. Specifically, if the center of mass of the drop lies at a stable equilibrium point of the system, it will stay there until evaporation reduces its size and this fixed point becomes unstable; with any small perturbation, the droplet then moves to one of its neighboring fixed points.

In 1665, Huygens observed that two pendulum clocks hanging from the same board became synchronized in antiphase after hundreds of swings. On the other hand, modern experiments with metronomes placed on a movable platform show that they often tend to synchronize in phase, not antiphase. Here we study both in-phase and antiphase synchronization in a model of pendulum clocks and metronomes and analyze their long-term dynamics with the tools of perturbation theory. Specifically, we exploit the separation of timescales between the fast oscillations of the individual pendulums and the much slower adjustments of their amplitudes and phases. By scaling the equations appropriately and applying the method of multiple timescales, we derive explicit formulas for the regimes in parameter space where either antiphase or in-phase synchronization are stable, or where both are stable. Although this sort of perturbative analysis is standard in other parts of nonlinear science, it has been applied surprisingly rarely in the context of Huygens's clocks. Unusual features of our approach include its treatment of the escapement mechanism, a small-angle approximation up to cubic order, and both a two- and three-timescale asymptotic analysis., Comment: 15 pages, 8 figures

For any network of identical Kuramoto oscillators with identical positive coupling, there is a critical connectivity above which the system is guaranteed to converge to the in-phase synchronous state, for almost all initial conditions. But the precise value of this critical connectivity remains unknown. In 2018, Ling, Xu, and Bandeira proved that if each oscillator is coupled to at least 79.29 percent of all the others, global synchrony is ensured. In 2019, Lu and Steinerberger improved this bound to 78.89 percent. Here, by focusing on circulant networks, we find clues that the critical connectivity may be exactly 75 percent. Our methods yield a slight improvement on the best known lower bound on the critical connectivity, from $68.18\%$ to $68.28\%$. We also consider the opposite end of the connectivity spectrum, where the networks are sparse rather than dense. In this regime, we ask how few edges one needs to add to a ring of $n$ oscillators to turn it into a globally synchronizing network. We prove a partial result: all the twisted states in a ring of size $n=2^m$ can be destabilized by adding just $\mathcal{O}(n \log_2 n)$ edges. To finish the proof, one also needs to rule out all other candidate attractors. We have done this for $n=8$ with computational algebraic geometry, but the problem remains open for larger $n$. Thus, even for systems as simple as Kuramoto oscillators, much remains to be learned about dense networks that do not globally synchronize and sparse ones that do., Comment: 6 pages, 4 figures

In 1992, a puzzling transition was discovered in simulations of randomly coupled limit-cycle oscillators. This so-called volcano transition has resisted analysis ever since. It was originally conjectured to mark the emergence of an oscillator glass, but here we show it need not. We introduce and solve a simpler model with a qualitatively identical volcano transition and find that its supercritical state is not glassy. We discuss the implications for the original model and suggest experimental systems in which a volcano transition and oscillator glass may appear.

Understanding conformational change is crucial for programming and controlling the function of many mechanical systems such as allosteric enzymes and tunable metamaterials. Of particular interest is the relationship between the network topology or geometry and the specific motions observed under controlling perturbations. We study this relationship in mechanical networks of 2-D and 3-D Maxwell frames composed of point masses connected by rigid rods rotating freely about the masses. We first develop simple principles that yield all bipartite network topologies and geometries that give rise to an arbitrarily specified instantaneous and finitely deformable motion in the masses as the sole non-rigid body zero mode. We then extend these principles to characterize networks that simultaneously yield multiple specified zero modes, and create large networks by coupling individual modules. These principles are then used to characterize and design networks with useful material (negative Poisson ratio) and mechanical (targeted allosteric response) functions., Comment: Main text: 14 pages, 5 figures, article, with supplement

Synchronization occurs in many natural and technological systems, from cardiac pacemaker cells to coupled lasers. In the synchronized state, the individual cells or lasers coordinate the timing of their oscillations, but they do not move through space. A complementary form of self-organization occurs among swarming insects, flocking birds, or schooling fish; now the individuals move through space, but without conspicuously altering their internal states. Here we explore systems in which both synchronization and swarming occur together. Specifically, we consider oscillators whose phase dynamics and spatial dynamics are coupled. We call them swarmalators, to highlight their dual character. A case study of a generalized Kuramoto model predicts five collective states as possible long-term modes of organization. These states may be observable in groups of sperm, Japanese tree frogs, colloidal suspensions of magnetic particles, and other biological and physical systems in which self-assembly and synchronization interact., Collective self-organized behavior can be observed in a variety of systems such as colloids and microswimmers. Here O’Keeffe et al. propose a model of oscillators which move in space and tend to synchronize with neighboring oscillators and outline five types of collective self-organized states.

The incubation period for typhoid, polio, measles, leukemia and many other diseases follows a right-skewed, approximately lognormal distribution. Although this pattern was discovered more than sixty years ago, it remains an open question to explain its ubiquity. Here, we propose an explanation based on evolutionary dynamics on graphs. For simple models of a mutant or pathogen invading a network-structured population of healthy cells, we show that skewed distributions of incubation periods emerge for a wide range of assumptions about invader fitness, competition dynamics, and network structure. The skewness stems from stochastic mechanisms associated with two classic problems in probability theory: the coupon collector and the random walk. Unlike previous explanations that rely crucially on heterogeneity, our results hold even for homogeneous populations. Thus, we predict that two equally healthy individuals subjected to equal doses of equally pathogenic agents may, by chance alone, show remarkably different time courses of disease., eLife digest When one child goes to school with a throat infection, many of his or her classmates will often start to come down with a sore throat after two or three days. A few of the children will get sick sooner, the very next day, while others may take about a week. As such, there is a distribution of incubation periods – the time from exposure to illness – across the children in the class. When plotted on a graph, the distribution of incubation periods is not the normal bell curve. Rather the curve looks lopsided, with a long tail on the right. Plotting the logarithms of the incubation periods, however, rather than the incubation periods themselves, does give a normal distribution. As such, statisticians refer to this kind of curve as a “lognormal distribution". Remarkably, many other, completely unrelated, diseases – like typhoid fever or bladder cancer – also have approximately lognormal distributions of incubation periods. This raised the question: why do such different diseases show such a similar curve? Working with a simple mathematical model in which chance plays a key role, Ottino-Löffler et al. calculate how long it takes for a bacterial infection or cancer cell to take over a network of healthy cells. The model explains why a lognormal-like distribution of incubation periods, modeled as takeover times, is so ubiquitous. It emerges from the random dynamics of the incubation process itself, as the disease-causing microbe or mutant cancer cell competes with the cells of the host. Intuitively, this new analysis builds on insights from the “coupon collector’s problem”: a classical problem in mathematics that describes the situation where a person collects items like baseball cards, stamps, or cartoon monsters in a videogame. If a random item arrives every day, and the collector’s luck is bad, they may have to wait a long time to collect those last few items. Similarly, in the model of Ottino-Löffler et al., the takeover time is dominated by dramatic slowdowns near the start or end of the infection process. These effects lead to an approximately lognormal distribution, with long waits, as seen in so many diseases. Ottino-Löffler et al. do not anticipate that their findings will have direct benefits for medicine or public health. Instead, they believe their results could help to advance basic research in the fields of epidemiology, evolutionary biology and cancer research. The findings might also make an impact outside biology. The term “contagion” has now become a familiar metaphor for the spread of everything from computer viruses to bank failures. This model sheds light on how long it takes for a contagion to take over a network, for a variety of idealized networks and spreading processes.

Taxi services are a vital part of urban transportation, and a considerable contributor to traffic congestion and air pollution causing substantial adverse effects on human health. Sharing taxi trips is a possible way of reducing the negative impact of taxi services on cities, but this comes at the expense of passenger discomfort quantifiable in terms of a longer travel time. Due to computational challenges, taxi sharing has traditionally been approached on small scales, such as within airport perimeters, or with dynamical ad-hoc heuristics. However, a mathematical framework for the systematic understanding of the tradeoff between collective benefits of sharing and individual passenger discomfort is lacking. Here we introduce the notion of shareability network which allows us to model the collective benefits of sharing as a function of passenger inconvenience, and to efficiently compute optimal sharing strategies on massive datasets. We apply this framework to a dataset of millions of taxi trips taken in New York City, showing that with increasing but still relatively low passenger discomfort, cumulative trip length can be cut by 40% or more. This benefit comes with reductions in service cost, emissions, and with split fares, hinting towards a wide passenger acceptance of such a shared service. Simulation of a realistic online system demonstrates the feasibility of a shareable taxi service in New York City. Shareability as a function of trip density saturates fast, suggesting effectiveness of the taxi sharing system also in cities with much sparser taxi fleets or when willingness to share is low., Main text: 6 pages, 3 figures, SI: 24 pages

We study a stochastic model of infection spreading on a network. At each time step a node is chosen at random, along with one of its neighbors. If the node is infected and the neighbor is susceptible, the neighbor becomes infected. How many time steps $T$ does it take to completely infect a network of $N$ nodes, starting from a single infected node? An analogy to the classic "coupon collector" problem of probability theory reveals that the takeover time $T$ is dominated by extremal behavior, either when there are only a few infected nodes near the start of the process or a few susceptible nodes near the end. We show that for $N \gg 1$, the takeover time $T$ is distributed as a Gumbel for the star graph; as the sum of two Gumbels for a complete graph and an Erd\H{o}s-R\'{e}nyi random graph; as a normal for a one-dimensional ring and a two-dimensional lattice; and as a family of intermediate skewed distributions for $d$-dimensional lattices with $d \ge 3$ (these distributions approach the sum of two Gumbels as $d$ approaches infinity). Connections to evolutionary dynamics, cancer, incubation periods of infectious diseases, first-passage percolation, and other spreading phenomena in biology and physics are discussed., Comment: 19 pages, 10 figures

We show that self-organization occurs in the phase dynamics of soliton modelocking in paramet- ric frequency combs. Reduction of the Lugiato-Lefever equation (LLE) to a simpler set of phase equations reveals that this self-organization arises via mechanisms akin to those in the Kuramoto model for synchronization of coupled oscillators. In addition, our simulations show that the phase equations evolve to a broadband phase-locked state, analogous to the soliton formation process in the LLE. Our simplified equations intuitively explain the origin of the pump phase offset in soliton- modelocked parametric frequency combs. They also predict that the phase of the intracavity field undergoes an anti-symmetrization that precedes phase synchronization, and they clarify the role of chaotic states in soliton formation in parametric combs.

We consider a mean-field model of coupled phase oscillators with quenched disorder in the natural frequencies and coupling strengths. A fraction $p$ of oscillators are positively coupled, attracting all others, while the remaining fraction $1-p$ are negatively coupled, repelling all others. The frequencies and couplings are deterministically chosen in a manner which correlates them, thereby correlating the two types of disorder in the model. We first explore the effect of this correlation on the system's phase coherence. We find that there is a a critical width $\gamma_c$ in the frequency distribution below which the system spontaneously synchronizes. Moreover, this $\gamma_c$ is independent of $p$. Hence, our model and the traditional Kuramoto model (recovered when $p=1$) have the same critical width $\gamma_c$. We next explore the critical behavior of the system by examining the finite-size scaling and the dynamic fluctuation of the traditional order parameter. We find that the model belongs to the same universality class as the Kuramoto model with deterministically (not randomly) chosen natural frequencies for the case of $p, Comment: 7 pages, 6 figures

Sharing rides could drastically improve the efficiency of car and taxi transportation. Unleashing such potential, however, requires understanding how urban parameters affect the fraction of individual trips that can be shared, a quantity that we call shareability. Using data on millions of taxi trips in New York City, San Francisco, Singapore, and Vienna, we compute the shareability curves for each city, and find that a natural rescaling collapses them onto a single, universal curve. We explain this scaling law theoretically with a simple model that predicts the potential for ride sharing in any city, using a few basic urban quantities and no adjustable parameters. Accurate extrapolations of this type will help planners, transportation companies, and society at large to shape a sustainable path for urban growth., National Science Foundation (U.S.) (DMS-1513179), National Science Foundation (U.S.) (CCF-1522054)