Dynamic-mode decomposition (DMD) is a versatile framework for model-free analysis of time series that are generated by dynamical systems. We develop a DMD-based algorithm to investigate the formation of functional communities in networks of coupled,
heterogeneous Kuramoto oscillators. In these functional communities, the oscillators in the network have similar dynamics. We consider two common random-graph models (Watts--Strogatz networks and Barabasi--Albert networks) with different amounts of heterogeneities among the oscillators. In our computations, we find that membership in a community reflects the extent to which there is establishment and sustainment of locking between oscillators. We construct forest graphs that illustrate the complex ways in which the heterogeneous oscillators associate and disassociate with each other.
We study ``nanoptera, which are non-localized solitary waves with exponentially small but non-decaying oscillations, in two singularly-perturbed Hertzian chains with precompression. These two systems are woodpile chains (which we model as systems of
Hertzian particles and springs) and diatomic Hertzian chains with alternating masses. We demonstrate that nanoptera arise from Stokes phenomena and appear as special curves, called Stokes curves, are crossed in the complex plane. We use techniques from exponential asymptotics to obtain approximations of the oscillation amplitudes. Our analysis demonstrates that traveling waves in a singularly perturbed woodpile chain have a single Stokes curve, across which oscillations appear. Comparing these asymptotic predictions with numerical simulations reveals that this accurately describes the non-decaying oscillatory behavior in a woodpile chain. We perform a similar analysis of a diatomic Hertzian chain, that the nanpteron solution has two distinct exponentially small oscillatory contributions. We demonstrate that there exists a set of mass ratios for which these two contributions cancel to produce localized solitary waves. This result builds on prior experimental and numerical observations that there exist mass ratios that support localized solitary waves in diatomic Hertzian chains without precompression. Comparing asymptotic and numerical results in a diatomic Hertzian chain with precompression reveals that our exponential asymptotic approach accurately predicts the oscillation amplitude for a wide range of system parameters, but it fails to identify several values of the mass ratio that correspond to localized solitary-wave solutions.
Traveling to different destinations is a big part of our lives. We visit a variety of locations both during our daily lives and when were on vacation. How can we find the best way to navigate from one place to another? Perhaps we can test all of the
different ways of traveling between two places, but another method is to use mathematics and computation to find a shortest path. We discuss how to construct a shortest path and introduce Dijkstras algorithm to minimize the total cost of a path, where the cost may be the travel distance, travel time, or some other measurement. We also discuss how to use shortest paths in the real world to save time and increase traveling efficiency.
In social networks, interaction patterns typically change over time. We study opinion dynamics on tie-decay networks in which tie strength increases instantaneously when there is an interaction and decays exponentially between interactions. Specifica
lly, we formulate continuous-time Laplacian dynamics and a discrete-time DeGroot model of opinion dynamics on these tie-decay networks, and we carry out numerical computations for the continuous-time Laplacian dynamics. We examine the speed of convergence by studying the spectral gaps of combinatorial Laplacian matrices of tie-decay networks. First, we compare the spectral gaps of the Laplacian matrices of tie-decay networks that we construct from empirical data with the spectral gaps for corresponding randomized and aggregate networks. We find that the spectral gaps for the empirical networks tend to be smaller than those for the randomized and aggregate networks. Second, we study the spectral gap as a function of the tie-decay rate and time. Intuitively, we expect small tie-decay rates to lead to fast convergence because the influence of each interaction between two nodes lasts longer for smaller decay rates. Moreover, as time progresses and more interactions occur, we expect eventual convergence. However, we demonstrate that the spectral gap need not decrease monotonically with respect to the decay rate or increase monotonically with respect to time. Our results highlight the importance of the interplay between the times that edges strengthen and decay in temporal networks.
In the study of infectious diseases on networks, researchers calculate epidemic thresholds to help forecast whether a disease will eventually infect a large fraction of a population. Because network structure typically changes in time, which fundamen
tally influences the dynamics of spreading processes on them and in turn affects epidemic thresholds for disease propagation, it is important to examine epidemic thresholds in temporal networks. Most existing studies of epidemic thresholds in temporal networks have focused on models in discrete time, but most real-world networked systems evolve continuously in time. In our work, we encode the continuous time-dependence of networks into the evaluation of the epidemic threshold of a susceptible--infected--susceptible (SIS) process by studying an SIS model on tie-decay networks. We derive the epidemic-threshold condition of this model, and we perform numerical experiments to verify it. We also examine how different factors---the decay coefficients of the tie strengths in a network, the frequency of interactions between nodes, and the sparsity of the underlying social network in which interactions occur---lead to decreases or increases of the critical values of the threshold and hence contribute to facilitating or impeding the spread of a disease. We thereby demonstrate how the features of tie-decay networks alter the outcome of disease spread.
The study of motifs in networks can help researchers uncover links between the structure and function of networks in biology, sociology, economics, and many other areas. Empirical studies of networks have identified feedback loops, feedforward loops,
and several other small structures as motifs that occur frequently in real-world networks and may contribute by various mechanisms to important functions in these systems. However, these mechanisms are unknown for many of these motifs. We propose to distinguish between structure motifs (i.e., graphlets) in networks and process motifs (which we define as structured sets of walks) on networks and consider process motifs as building blocks of processes on networks. Using the steady-state covariances and steady-state correlations in a multivariate Ornstein--Uhlenbeck process on a network as examples, we demonstrate that the distinction between structure motifs and process motifs makes it possible to gain quantitative insights into mechanisms that contribute to important functions of dynamical systems on networks.
Suppose that youre going to school and arrive at a bus stop. How long do you have to wait before the next bus arrives? Surprisingly, it is longer - possibly much longer - than what the bus schedule suggests intuitively. This phenomenon, which is call
ed the waiting-time paradox, has a purely mathematical origin. Different buses arrive with different intervals, leading to this paradox. In this article, we explore the waiting-time paradox, explain why it happens, and discuss some of its implications (beyond the possibility of being late for school).
The study of network formation is pervasive in economics, sociology, and many other fields. In this paper, we model network formation as a choice that is made by nodes in a network to connect to other nodes. We study these choices using discrete-choi
ce models, in which an agent chooses between two or more discrete alternatives. One framework for studying network formation is the multinomial logit (MNL) model. We highlight limitations of the MNL model on networks that are constructed from empirical data. We employ the repeated choice (RC) model to study network formation cite{TrainRevelt97mixedlogit}. We argue that the RC model overcomes important limitations of the MNL model and is well-suited to study network formation. We also illustrate how to use the RC model to accurately study network formation using both synthetic and real-world networks. Using synthetic networks, we also compare the performance of the MNL model and the RC model; we find that the RC model estimates the data-generation process of our synthetic networks more accurately than the MNL model. We provide examples of qualitatively interesting questions -- the presence of homophily in a teen friendship network and the fact that new patents are more likely to cite older, more cited, and similar patents -- for which the RC model allows us to achieve insights.
Although social neuroscience is concerned with understanding how the brain interacts with its social environment, prevailing research in the field has primarily considered the human brain in isolation, deprived of its rich social context. Emerging wo
rk in social neuroscience that leverages tools from network analysis has begun to pursue this issue, advancing knowledge of how the human brain influences and is influenced by the structures of its social environment. In this paper, we provide an overview of key theory and methods in network analysis (especially for social systems) as an introduction for social neuroscientists who are interested in relating individual cognition to the structures of an individuals social environments. We also highlight some exciting new work as examples of how to productively use these tools to investigate questions of relevance to social neuroscientists. We include tutorials to help with practical implementation of the concepts that we discuss. We conclude by highlighting a broad range of exciting research opportunities for social neuroscientists who are interested in using network analysis to study social systems.
Animals use a wide variety of strategies to reduce or avoid aggression in conflicts over resources. These strategies range from sharing resources without outward signs of conflict to the development of dominance hierarchies, in which initial fighting
is followed by the submission of subordinates. Although models have been developed to analyze specific strategies for resolving conflicts over resources, little work has focused on trying to understand why particular strategies are more likely to arise in certain situations. In this paper, we use a model based on an iterated Hawk--Dove game to analyze how resource holding potentials (RHPs) and other factors affect whether sharing, dominance relationships, or other behaviours are evolutionarily stable. We find through extensive numerical simulations that sharing is stable only when the cost of fighting is low and the animals in a contest have similar RHPs, whereas dominance relationships are stable in most other situations. We also explore what happens when animals are unable to assess each others RHPs without fighting, and we compare a range of strategies for this problem using simulations. We find (1) that the most successful strategies involve a limited period of assessment followed by a stable relationship in which fights are avoided and (2) that the duration of assessment depends both on the costliness of fighting and on the difference between the animals RHPs. Along with our direct work on modeling and simulations, we develop extensive software to facilitate further testing; it is available at url{https://bitbucket.org/CameronLHall/dominancesharingassessmentmatlab/}.
