Understanding the spatiotemporal patterns of human movement is important across diverse applications including urban design, disease control, social and cognitive science, and emergency response planning. Recently, multilayer mobility networks were used to study how movements between spatial units (e.g., census tracts) can significantly vary when they are stratified according to different industry sectors-e.g., movements to grocery stores, to schools, or to hospitals. Here, we study the predictability of movements across different industry sectors using statistical and machine learning models trained on demographic, socioeconomic, and infrastructure information. We compare ten predictive models and identify advantages for nonlinear models (with random forest regression being a consistent top performer). We identify the most important features enabling prediction (population size for outward movements from regions and industry-related infrastructure for movements into regions). Of the two, prediction for inward movements (i.e., in-degrees) is generally more difficult; however, the difference is small for movements associated with food services. We also compare the prediction of weekly and time-averaged movements, finding that with the addition of time-encoding input features, weekly movements are easier to predict than time-averaged values (at least for the nonlinear predictive models). These findings provide a practical step toward using machine learning for human movement modeling and the many downstream applications.

Abstract Understanding human mobility during disastrous events is crucial for emergency planning and disaster management. We develop a methodology to construct time-varying, multilayer networks where edges encode observed movements between spatial regions (census tracts) and network layers encode movement categories by industry sectors (e.g., schools, hospitals). Using the 2021 Texas winter storm as a case study, we find that people markedly reduced movements to ambulatory health care services, restaurants, and schools, but prioritized movements to grocery stores and gas stations. Additionally, we study the predictability of nodes’ in- and out-degrees in the multilayer networks, which encode movements into and out of census tracts. Inward movements prove harder to predict than outward movements, especially during the storm. This case study highlights our methodology’s effectiveness for detecting/characterizing mobility shifts, and our specific findings on the reduction, prioritization, and predictability of sector-specific movements aim to support mobility-related decisions during future extreme weather events.

Despite growing interest in synchronization dynamics over "higher-order" network models, optimization theory for such systems is limited. Here, we study a family of Kuramoto models inspired by algebraic topology in which oscillators are coupled over simplicial complexes (SCs) using their associated Hodge Laplacian matrices. We optimize such systems by extending the synchrony alignment function – an optimization framework for synchronizing graph-coupled heterogeneous oscillators. Computational experiments are given to illustrate how this approach can effectively solve a variety of combinatorial problems including the joint optimization of projected synchronization dynamics onto lower- and upper-dimensional simplices within SCs. We also investigate the role of SC homology and develop bifurcation theory to characterize the extent to which optimal solutions are contained within (or spread across) the three Hodge subspaces. Our work extends optimization theory to the setting of higher-order networks, provides practical algorithms for Hodge-Laplacian-related dynamics including (but not limited to) Kuramoto oscillators, and paves the way for an emerging field that interfaces algebraic topology, combinatorial optimization, and dynamical systems.

Network science has traditionally examined how structure determines dynamics. Here we invert this paradigm: we ask how functional dynamics and resource constraints shape network architecture. We introduce GradNet, an AI-enabled optimization framework that treats network topology as a continuously differentiable object. This allows designing networks that optimize arbitrary dynamical objectives, from synchronization to communication capacity, under realistic constraints. Applying this framework across diverse systems reveals that canonical network features emerge spontaneously from constrained optimization rather than requiring explicit imposition. Optimizing Kuramoto oscillator synchronization under fixed coupling budgets produces sparse, bipartite, frequency-disassortative architectures that eliminate classical synchronization thresholds. Minimizing social tension in opinion dynamics reproduces the empirically observed factional split in Zachary's karate club network. Maximizing entanglement distribution in spatial quantum networks under distance-dependent costs recovers minimum spanning tree architectures. These results demonstrate that optimization acts as both an engineering tool for network design, scalable to networks exceeding 10^5 nodes, and a scientific probe revealing fundamental structure-function relationships. By recasting network architecture as the solution to constrained optimization problems, this variational perspective offers a unified framework connecting network analysis, design, and inference across physical, biological, and technological systems.

Real-world networks, whether shaped by evolution or intelligent design, are typically optimized for functionality while adhering to physical, geometric, or budget constraints. Yet tools to identify such structures remain limited. We develop a gradient-based optimization framework to identify synchrony-optimal weighted networks under a constrained coupling budget. The resulting networks exhibit counterintuitive features: they are sparse, bipartite, elongated, and extremely monophilic (i.e., the neighbors of any node are similar to one another while differing from the node itself). These findings are matched by "constructive" theory: a nonlinear differential equation identifies which pairs of nodes are coupled, while a variational principle prescribes the budget allocated to each node. Dynamics unfolding over optimal networks provably lack a synchronization threshold; instead, as the budget exceeds a calculable critical value, the system globally phase-locks, exhibiting critical scaling at the transition. Our results have implications for power grids, neuromorphic computing, and other coupled oscillator technologies.

Many social and biological networks periodically change over time with daily, weekly, and other cycles. Thus motivated, we formulate and analyze susceptible-infectious-susceptible (SIS) epidemic models over temporal networks with periodic schedules. More specifically, we assume that the temporal network consists of a cycle of alternately used static networks, each with a given duration. We observe a phenomenon in which two static networks are individually above the epidemic threshold but the alternating network composed of them renders the dynamics below the epidemic threshold, which we refer to as a Parrondo paradox for epidemics. We find that network structure plays an important role in shaping this phenomenon, and we study its dependence on the connectivity between and number of subpopulations in the network. We associate such paradoxical behavior with anti-phase oscillatory dynamics of the number of infectious individuals in different subpopulations.

Graph-based representations of point-cloud data are widely used in data science and machine learning, including epsilon-graphs that contain edges between pairs of data points that are nearer than epsilon and kNN-graphs that connect each point to its k-nearest neighbors. Recently, topological data analysis has emerged as a family of mathematical and computational techniques to investigate topological features of data using simplicial complexes. These are a higher-order generalization of graphs and many techniques such as Vietoris-Rips (VR) filtrations are also parameterized by a distance epsilon. Here, we develop kNN complexes as a generalization of kNN graphs, leading to kNN-based persistent homology techniques for which we develop stability and convergence results. We apply this technique to characterize the convergence properties PageRank, highlighting how the perspective of discrete topology complements traditional geometrical-based analyses of convergence. Specifically, we show that convergence of relative positions (i.e., ranks) is captured by kNN persistent homology, whereas persistent homology with VR filtrations coincides with vector-norm convergence. Beyond PageRank, kNN-based persistent homology is expected to be useful to other data-science applications in which the relative positioning of data points is more important than their precise locations.

Networks are often interconnected, with one system wielding greater influence over another. However, the effects of such asymmetry on self-organized phenomena (e.g., consensus and synchronization) are not well understood. Here, we study collective dynamics using a generalized graph Laplacian for multiplex networks containing layers that are asymmetrically coupled. We explore the nonlinear effects of coupling asymmetry on the convergence rate toward a collective state, finding that asymmetry induces one or more optima that maximally accelerate convergence. When a faster and a slower system are coupled, depending on their relative timescales, their optimal coupling is either cooperative (network layers mutually depend on one another) or non-cooperative (one network directs another without a reciprocated influence). It is often optimal for the faster system to more-strongly influence the slower one, yet counter-intuitively, the opposite can also be true. As an application, we model collective decision-making for a human-AI system in which a social network is supported by an AI-agent network, finding that a cooperative optimum requires that these two networks operate on a sufficiently similar timescale. More broadly, our work highlights the optimization of coupling asymmetry and timescale balancing as fundamental concepts for the design of collective behavior over interconnected systems.

Multilayer networks describe the rich ways in which nodes are related by accounting for different relationships in separate layers. These multiple relationships are naturally represented by an adjacency tensor. In this work we study the use of the nonnegative Tucker decomposition (NNTuck) of such tensors under a KL loss as an expressive factor model that naturally generalizes existing stochastic block models of multilayer networks. Quantifying interdependencies between layers can identify redundancies in the structure of a network, indicate relationships between disparate layers, and potentially inform survey instruments for collecting social network data. We propose definitions of layer independence, dependence, and redundancy based on likelihood ratio tests between nested nonnegative Tucker decompositions. Using both synthetic and real-world data, we evaluate the use and interpretation of the NNTuck as a model of multilayer networks. Algorithmically, we show that using expectation maximization (EM) to maximize the log-likelihood under the NNTuck is step-by-step equivalent to tensorial multiplicative updates for the NNTuck under a KL loss, extending a previously known equivalence from nonnegative matrices to nonnegative tensors.

Despite the numerous ways now available to quantify which parts or subsystems of a network are most important, there remains a lack of centrality measures that are related to the complexity of information flows and are derived directly from entropy measures. Here, we introduce a ranking of edges based on how each edge's removal would change a system's von Neumann entropy (VNE), which is a spectral-entropy measure that has been adapted from quantum information theory to quantify the complexity of information dynamics over networks. We show that a direct calculation of such rankings is computationally inefficient (or unfeasible) for large networks: e.g.\ the scaling is $\mathcal{O}(N^3)$ per edge for networks with $N$ nodes. To overcome this limitation, we employ spectral perturbation theory to estimate VNE perturbations and derive an approximate edge-ranking algorithm that is accurate and fast to compute, scaling as $\mathcal{O}(N)$ per edge. Focusing on a form of VNE that is associated with a transport operator $e^{-\beta{ L}}$, where ${ L}$ is a graph Laplacian matrix and $\beta>0$ is a diffusion timescale parameter, we apply this approach to diverse applications including a network encoding polarized voting patterns of the 117th U.S. Senate, a multimodal transportation system including roads and metro lines in London, and a multiplex brain network encoding correlated human brain activity. Our experiments highlight situations where the edges that are considered to be most important for information diffusion complexity can dramatically change as one considers short, intermediate and long timescales $\beta$ for diffusion.

Views Icon Views Article contents Figures & tables Video Audio Supplementary Data Peer Review Share Icon Share Twitter Facebook Reddit LinkedIn Tools Icon Tools Reprints and Permissions Cite Icon Cite Search Site Citation Z. Gao, D. Ghosh, H. A. Harrington, J. G. Restrepo, D. Taylor; Dynamics on networks with higher-order interactions. Chaos 1 April 2023; 33 (4): 040401. https://doi.org/10.1063/5.0151265 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentAIP Publishing PortfolioChaos: An Interdisciplinary Journal of Nonlinear Science Search Advanced Search |Citation Search

Despite the vast literature on network dynamics, we still lack basic insights into dynamics on higher-order structures (e.g., edges, triangles, and more generally, k-dimensional "simplices") and how they are influenced through higher-order interactions. A prime example lies in neuroscience where groups of neurons (not individual ones) may provide building blocks for neurocomputation. Here, we study consensus dynamics on edges in simplicial complexes using a type of Laplacian matrix called a Hodge Laplacian, which we generalize to allow higher- and lower-order interactions to have different strengths. Using techniques from algebraic topology, we study how collective dynamics converge to a low-dimensional subspace that corresponds to the homology space of the simplicial complex. We use the Hodge decomposition to show that higher- and lower-order interactions can be optimally balanced to maximally accelerate convergence and that this optimum coincides with a balancing of dynamics on the curl and gradient subspaces. We additionally explore the effects of network topology, finding that consensus over edges is accelerated when two-simplices are well dispersed, as opposed to clustered together.

In composite materials composed of a soft polymer matrix and rigid, high aspect-ratio particles, the composite undergoes a transition in mechanical strength when the incorporated particle phase surpasses a critical density. This phenomenon, termed rheological percolation, is well known to occur in many polymer-rod and polymer-platelet composites at a critical density that exceeds the conductivity percolation threshold (which occurs when the conducting particles form a large connected component that spans the composite). Contact percolation in rod-like composites has been routinely exploited to engineer thermal or electrical conductivity in otherwise nonconducting polymers, and the characterization of contact percolation is well established. Mechanical or rheological percolation, however, has evaded a complete theoretical explanation and predictive description. A natural hypothesis is that rheological percolation is due to a rigidity phenomenon, whereby a large rigid component of inclusions spans the composite. Here we build an algorithm to detect the rigidity percolation threshold in rod-polymer composites. We model the composites as systems of randomly distributed, soft-core (intersecting at contact) rods and study the emergence of a giant (i.e., spanning) rigid component. Building on our previous results for two-dimensional composites, we develop an approximate algorithm that identifies spanning rigid components through hierarchically identifying and compressing provably rigid motifs???equivalently, decomposing a giant rigid component into rigid assemblies of a hierarchy of successively smaller rigid components. We apply this algorithm to random monodisperse systems that are generated in Monte Carlo simulations to estimate a rigidity percolation threshold (critical density) and explore its dependence on rod aspect ratio. We show that this transition point???like its contact percolation analogue???scales inversely with the excluded volume of a rod. Moreover, this implies that the critical contact number (i.e., the number of contacts per rod at the rigidity percolation threshold) is constant for aspect ratios above some relatively low value and is lower than the prediction from Maxwell???s isostatic condition.

Characterizing the importances of nodes in social, biological, information and technological networks is a core topic for the network-science and data-science communities. We present a linear-algebraic framework that generalizes eigenvector-based centralities---including PageRank---to provide a common framework for two popular classes of multilayer networks: multiplex networks with layers encoding different types of relationships and temporal networks in which the relationships change in time. Our approach involves the study of joint, marginal, and conditional supracentralities that can be obtained from the dominant eigenvector of a supracentrality matrix [Taylor et al., 2017], which couples centrality matrices that are associated with the individual network layers. We extend this prior work (which was restricted to temporal networks with layers that are coupled by undirected, adjacent-in-time coupling) by allowing the layers to be coupled through a (possibly asymmetric) interlayer adjacency matrix $\tilde{\bf A}$ in which each entry gives the coupling between layers $t$ and $t'$. Our framework provides a unifying foundation for centrality analysis of multiplex and/or temporal networks and reveals a complicated dependency of the supracentralities on the topology of interlayer coupling. We scale $\tilde{\bf A}$ by a coupling strength $\omega\ge0$ and develop singular perturbation theory for the limits of weak ($\omega\to0^+$) and strong coupling ($\omega\to\infty$), revealing an interesting dependence on the dominant eigenvectors of $\tilde{\bf A}$. We apply the framework to two empirical data sets: a multiplex network representation of airline transportation in Europe, and a temporal network encoding the graduation and hiring of mathematicians at U.S. colleges and universities.

Despite the prevalence of biological and physical systems for which synchronization is critical, existing theory for optimizing synchrony depends on global information and does not sufficiently explore local mechanisms that enhance synchronization. Thus, there is a lack of understanding for the self-organized, collective processes that aim to optimize or repair synchronous systems, e.g., the dynamics of paracrine signaling within cardiac cells. Here we present ``grass-roots'' optimization of synchronization, which is a multiscale mechanism in which local optimizations of smaller subsystems cooperate to collectively optimize an entire system. Considering models of cardiac tissue and a power grid, we show that grass-roots-optimized systems are comparable to globally optimized systems, but they also have the added benefit of being robust to targeted attacks or subsystem islanding. Our findings motivate and support further investigation into the physics of local mechanisms that can support self-optimization for complex systems.

Cascades over networks (e.g., neuronal avalanches, social contagions, and system failures) often involve higher-order dependencies, yet theory development has largely focused on pairwise-interaction models. Here, we develop a ‘simplicial threshold model’ (STM) for cascades over simplicial complexes that encode dyadic, triadic and higher-order interactions. Focusing on small-world models containing both short- and long-range k -simplices, we explore spatio-temporal patterns that manifest as a frustration between local and nonlocal propagations. We show that higher-order interactions and nonlinear thresholding coordinate to robustly guide cascades along a k -dimensional generalization of paths that we call ‘geometrical channels’. We also find this coordination to enhance the diversity and efficiency of cascades over a simplicial-complex model for a neuronal network, or ‘neuronal complex’. We support these findings with bifurcation theory and data-driven approaches based on latent geometry. Our findings provide fruitful directions for uncovering the multiscale, multidimensional mechanisms that orchestrate the spatio-temporal patterns of nonlinear cascades.

Convection is a well-studied topic in fluid dynamics, yet it is less understood in the context of network flows. Here, we incorporate techniques from topological data analysis (namely, persistent homology) to automate the detection and characterization of convective flows (also called cyclic or chiral flows) over networks, particularly those that arise for irreversible Markov chains. As two applications, we study convection cycles arising under the PageRank algorithm and we investigate chiral edge flows for a stochastic model of a bimonomer's configuration dynamics. Our experiments highlight how system parameters-e.g., the teleportation rate for PageRank and the transition rates of external and internal state changes for a monomer-can act as homology regularizers of convection, which we summarize with persistence barcodes and homological bifurcation diagrams. Our approach establishes a connection between the study of convection cycles and homology, the branch of mathematics that formally studies cycles, which has diverse potential applications throughout the sciences and engineering.

Collective behavior plays a key role in the function of a wide range of physical, biological, and neurological systems where empirical evidence has recently uncovered the prevalence of higher-order interactions, i.e., structures that represent interactions between more than just two individual units, in complex network structures. Here, we study the optimization of collective behavior in networks with higher-order interactions encoded in clique complexes. Our approach involves adapting the synchrony alignment function framework to a composite Laplacian matrix that encodes multiorder interactions including, e.g., both dyadic and triadic couplings. We show that as higher-order coupling interactions are equitably strengthened, so that overall coupling is conserved, the optimal collective behavior improves. We find that this phenomenon stems from the broadening of a composite Laplacian's eigenvalue spectrum, which improves the optimal collective behavior and widens the range of possible behaviors. Moreover, we find in constrained optimization scenarios that a nontrivial, ideal balance between the relative strengths of pairwise and higher-order interactions leads to the strongest collective behavior supported by a network. This work provides insight into how systems balance interactions of different types to optimize or broaden their dynamical range of behavior, especially for self-regulating systems like the brain.

Domestic violence (DV) is a serious public health issue, with 1 in 3 women and 1 in 4 men experiencing some form of partner-related violence every year. Existing research has shown a strong association between alcohol use and DV at the individual level. Accordingly, alcohol use could also be a predictor for DV at the neighborhood level, helping identify the neighborhoods where DV is more likely to happen. However, it is difficult and costly to collect data that can represent neighborhood-level alcohol use especially for a large geographic area. In this study, we propose to derive information about the alcohol outlet visits of the residents of different neighborhoods from anonymized mobile phone location data, and investigate whether the derived visits can help better predict DV at the neighborhood level. We use mobile phone data from the company SafeGraph, which is freely available to researchers and which contains information about how people visit various points-of-interest including alcohol outlets. In such data, a visit to an alcohol outlet is identified based on the GPS point location of the mobile phone and the building footprint (a polygon) of the alcohol outlet. We present our method for deriving neighborhood-level alcohol outlet visits, and experiment with four different statistical and machine learning models to investigate the role of the derived visits in enhancing DV prediction based on an empirical dataset about DV in Chicago. Our results reveal the effectiveness of the derived alcohol outlets visits in helping identify neighborhoods that are more likely to suffer from DV, and can inform policies related to DV intervention and alcohol outlet licensing.

Collective behavior plays a key role in the function of a wide range of physical, biological, and neurological systems where empirical evidence has recently uncovered the prevalence of higher-order interactions, i.e., structures that represent interactions between more than just two individual units, in complex network structures. Here, we study the optimization of collective behavior in networks with higher-order interactions encoded in clique complexes. Our approach involves adapting the Synchrony Alignment Function framework to a new composite Laplacian matrix that encodes multi-order interactions including, e.g., both dyadic and triadic couplings. We show that as higher-order coupling interactions are equitably strengthened, so that overall coupling is conserved, the optimal collective behavior improves. We find that this phenomenon stems from the broadening of a composite Laplacian's eigenvalue spectrum, which improves the optimal collective behavior and widens the range of possible behaviors. Moreover, we find in constrained optimization scenarios that a nontrivial, ideal balance between the relative strengths of pair-wise and higher-order interactions leads to the strongest collective behavior supported by a network. This work provides insight into how systems balance interactions of different types to optimize or broaden their dynamical range of behavior, especially for self-regulating systems like the brain.