Continuous-time Networks Research Articles

Unsupervised Model Construction in Continuous-Time

Many of the advancements reconciling individual- and group-level results have occurred in the context of a discrete-time modeling framework. Discrete-time models are intuitive and offer relatively simple interpretations for the resulting dynamic structures; however, they do not possess the flexibility of models fitted in the continuous-time framework. We introduce ct-gimme, a continuous-time extension of the group iterative multiple model estimation (GIMME) procedure which enables researchers to fit complex, high-dimensional dynamic networks in continuous time. Our results indicate that ct-gimme outperforms N = 1 model fitting in continuous time by pooling information across multiple subjects. Likewise, ct-gimme outperforms group-level model fitting in the presence of within-sample heterogeneity. We conclude with an empirical illustration and highlight the limitations of the approach relating to the identification of meaningful starting values.

Bayesian inference and optimisation of stochastic dynamical networks

Network inference has been extensively studied in several fields, such as systems biology and social sciences. Learning network topology and internal dynamics is essential to understand mechanisms of complex systems. In particular, sparse topologies and stable dynamics are fundamental features of many real-world continuous-time (CT) networks. Given that usually only a partial set of nodes are able to observe, we consider linear CT systems to depict networks since they can model unmeasured nodes via transfer functions. Additionally, measurements tend to be noisy and with low and varying sampling frequencies. This paper applies dynamical structure functions (DSFs) derived from linear stochastic differential equations (SDEs) to describe networks of measured nodes. A numerical sampling method, preconditioned Crank–Nicolson (pCN), is used to refine coarse-grained trajectories to improve inference accuracy. The proposed method can handle sparsely sampled data and unmeasurable nodes. Monte Carlo simulations indicate that the proposed method outperforms state-of-the-art methods with various network topologies. The developed method can be applied under a wide range of contexts, such as gene regulatory networks, social networks and communication systems.

Continuous-Time Graph Representation with Sequential Survival Process

Over the past two decades, there has been a tremendous increase in the growth of representation learning methods for graphs, with numerous applications across various fields, including bioinformatics, chemistry, and the social sciences. However, current dynamic network approaches focus on discrete-time networks or treat links in continuous-time networks as instantaneous events. Therefore, these approaches have limitations in capturing the persistence or absence of links that continuously emerge and disappear over time for particular durations. To address this, we propose a novel stochastic process relying on survival functions to model the durations of links and their absences over time. This forms a generic new likelihood specification explicitly accounting for intermittent edge-persistent networks, namely GraSSP: Graph Representation with Sequential Survival Process. We apply the developed framework to a recent continuous time dynamic latent distance model characterizing network dynamics in terms of a sequence of piecewise linear movements of nodes in latent space. We quantitatively assess the developed framework in various downstream tasks, such as link prediction and network completion, demonstrating that the developed modeling framework accounting for link persistence and absence well tracks the intrinsic trajectories of nodes in a latent space and captures the underlying characteristics of evolving network structure.

Layer Compression of Deep Networks with Straight Flows

Very deep neural networks lead to significantly better performance on various real tasks. However, it usually causes slow inference and is hard to be deployed on real-world devices. How to reduce the number of layers to save memory and to accelerate the inference is an eye-catching topic. In this work, we introduce an intermediate objective, a continuous-time network, before distilling deep networks into shallow networks. First, we distill a given deep network into a continuous-time neural flow model, which can be discretized with an ODE solver and the inference requires passing through the network multiple times. By forcing the flow transport trajectory to be straight lines, we find that it is easier to compress the infinite step model into a one-step neural flow model, which only requires passing through the flow model once. Secondly, we refine the one-step flow model together with the final head layer with knowledge distillation and finally, we can replace the given deep network with this one-step flow network. Empirically, we demonstrate that our method outperforms direct distillation and other baselines on different model architectures (e.g. ResNet, ViT) on image classification and semantic segmentation tasks. We also manifest that our distilled model naturally serves as an early-exit dynamic inference model.

A continuous-time network evolution model describing $ N $-interactions

<p>We have introduced a new continuous-time network evolution model. We have described cooperation, so we have considered the cliques of nodes. The evolution of the network was based on cliques of nodes of the network and was governed by a branching process. The basic properties of the evolution process were described. Asymptotic theorems were proved for the number of cliques having a fixed size and the degree of a fixed node. The generating function was calculated, and then the probability of extinction was obtained. For the proof, advanced results of multi-type branching processes were used. Besides precise mathematical proofs, simulation examples also supported our results.</p>

Testing For Global Covariate Effects in Dynamic Interaction Event Networks

In statistical network analysis it is common to observe so called interaction data. Such data is characterized by actors forming the vertices and interacting along edges of the network, where edges are randomly formed and dissolved over the observation horizon. In addition, covariates are observed and the goal is to model the impact of the covariates on the interactions. We distinguish two types of covariates: global, system-wide covariates (i.e., covariates taking the same value for all individuals, such as seasonality) and local, dyadic covariates modeling interactions between two individuals in the network. Existing continuous time network models are extended to allow for comparing a completely parametric model and a model that is parametric only in the local covariates but has a global nonparametric time component. This allows, for instance, to test whether global time dynamics can be explained by simple global covariates like weather, seasonality etc. The procedure is applied to a bike-sharing network by using weather and weekdays as global covariates and distances between the bike stations as local covariates.

Temporal super-resolution traffic flow forecasting via continuous-time network dynamics

Temporal super-resolution traffic flow forecasting via continuous-time network dynamics

Synchronization of Delayed Complex Networks on Time Scales via Aperiodically Intermittent Control Using Matrix-Based Convex Combination Method.

This article reconsiders synchronization problem of linear complex networks with time-varying delay on time scales. For different types of time scales, aperiodically intermittent control scheme is established by using a matrix-based convex combination method, which has great potential in reducing control consumption and saving communication bandwidth. By employing a common Lyapunov function, aperiodically intermittent controllers are utilized successfully to achieve synchronization of linear delayed complex networks on special time scales onto an isolated node. Next, by constructing a special Lyapunov function with time-varying coefficients, sufficient criteria that consist of two linear matrix inequalities are demonstrated to make linear delayed complex networks on general time scales synchronized onto an isolated system with an exponential convergence rate given in advance. Due to delayed complex networks in this article defined on time scales, the proposed control schemes are applicable to continuous-time networks, their discrete-time forms, and any combination of them. Four numerical examples are offered to highlight the effectiveness and superiority of the proposed aperiodically intermittent control schemes at last.

Matrix-scaled resilient consensus of discrete-time and continuous-time networks

This paper studies the matrix-scaled resilient consensus problems over multi-agent networks as occurring in computer science and distributed control. Unlike existing works on consensus problems, where the states of agents converge to a common value or reach some prescribed proportions, we take a more general matrix-scaled approach to accommodate the interdependence of multi-dimensional states. We develop a unified analytical framework to deal with matrix-scaled resilient consensus of discrete-time and continuous-time dynamical agents, where the underlying communication network is modeled as a generic directed time-dependent random graph. We propose new distributed protocols to guarantee the matrix-scaled consensus of cooperative agents in the network in the presence of Byzantine agents, who have full knowledge of the system and pose a severe security threat to the collective consensus objective. The cooperative agents feature multiple input and multiple output, and the number and identities of Byzantine agents are not available to the cooperative ones. Our mathematical approach capitalizes on matrix analysis, control theory, graph theory, and martingale convergence. Some numerical examples are presented to demonstrate the effectiveness of our theoretical results.

Cluster consensus with first and higher-order antagonistic interaction dynamics

This paper studies the cluster consensus problem for networks with antagonistic interactions modelled by adjacency matrices with negative weights. By introducing an extended digraph representation consisting of purely cooperative interactions with lifting approach, we relate the trajectories of the system to those of an extended system with a positive digraph. The behaviours of agents in a signed network are extracted from its extended digraph. Consequently, the number of clusters and the cluster members are explicitly determined for any signed digraph by using primary and secondary layer subgraph concepts. The relation between the dynamics of a signed system and its extended representation is investigated for first and higher-order systems. The conditions that make both systems stable are stated. Additionally, the control parameters to achieve cluster consensus are derived explicitly for first, second and third order systems. The obtained results for continuous-time networks are subsequently extended to discrete-time networks. Finally, theoretical results are illustrated via numerical examples.

Continuous-Time Prediction of Industrial Paste Thickener System With Differential ODE-Net

It is crucial to predict the outputs of a thickening system, including the underflow concentration (UC) and mud pressure, for optimal control of the process. The proliferation of industrial sensors and the availability of thickening-system data make this possible. However, the unique properties of thickening systems, such as the non-linearities, long-time delays, partially observed data, and continuous time evolution pose challenges on building data-driven predictive models. To address the above challenges, we establish an integrated, deep-learning, continuous time network structure that consists of a sequential encoder, a state decoder, and a derivative module to learn the deterministic state space model from thickening systems. Using a case study, we examine our methods with a tailing thickener manufactured by the FLSmidth installed with massive sensors and obtain extensive experimental results. The results demonstrate that the proposed continuous-time model with the sequential encoder achieves better prediction performances than the existing discrete-time models and reduces the negative effects from long time delays by extracting features from historical system trajectories. The proposed method also demonstrates outstanding performances for both short and long term prediction tasks with the two proposed derivative types.

A Comparative Study of Some Point Process Models for Dynamic Networks

Modeling dynamic networks has attracted much interest in recent years, which helps understand networks’ behavior. Many works have been dedicated to modeling discrete‐time networks, but less work is done for continuous‐time networks. Point processes as powerful tools for modeling discrete events in continuous time have been widely used for modeling events over networks and their dynamics. These models have solid mathematical assumptions, making them interpretable but decreasing their generalizability for different datasets. Hence, neural point processes were introduced that don’t have strong assumptions on generative functions. However, these models can be impractical in the case of a large number of event types. This research presents a comparative study of different point process (Hawkes) models for continuous‐time networks. Furthermore, a previously introduced neural point process (neural Hawkes) model is applied for modeling network interactions. In this work, network clustering is used for specifying interaction types. These methods are compared using different synthetic and real‐world datasets, and their efficiency is evaluated on these datasets. The experiments represent that each model is appropriate for a group of datasets. In addition, the effect of clustering on results is discussed, and experiments for different clusters are presented.

A Continuous-Time Network Evolution Model Describing 2- and 3-Interactions

A continuous-time network evolution model is considered. The evolution of the network is based on 2- and 3-interactions. 2-interactions are described by edges, and 3-interactions are described by triangles. The evolution of the edges and triangles is governed by a multi-type continuous-time branching process. The limiting behaviour of the network is studied by mathematical methods. We prove that the number of triangles and edges have the same magnitude on the event of non-extinction, and it is eαt, where α is the Malthusian parameter. The probability of the extinction and the degree process of a fixed vertex are also studied. The results are illustrated by simulations.

A continuous-time network evolution model describing 3-interactions

A continuous-time network evolution model is studied. The basic units of the model are triangles describing 3-interactions. The evolution of the triangles is governed by a continuous-time branching process. The asymptotic behavior of the model is studied. It is proved that the number of triangles, edges and vertices have the same magnitude on the event of non-extinction, and it is where α is the Malthusian parameter.

Tie-Decay Networks in Continuous Time and Eigenvector-Based Centralities

Network theory is a useful framework for studying interconnected systems of interacting entities. Many networked systems evolve continuously in time, but most existing methods for the analysis of time-dependent networks rely on discrete or discretized time. In this paper, we propose an approach for studying networks that evolve in continuous time by distinguishing between interactions , which we model as discrete contacts, and ties , which encode the strengths of relationships over time. To illustrate our tie-decay network formalism, we adapt the well-known PageRank centrality score to our tie-decay framework in a mathematically tractable and computationally efficient way. We apply this framework to a synthetic example and then use it to study a network of retweets during the 2012 National Health Service controversy in the United Kingdom. Our work also provides guidance for similar generalizations of other tools from network theory to continuous-time networks with tie decay, including for applications to streaming data.

Hopf Bifurcation in a Delayed Single Species Network System

In this paper, we focus on a network system which describes spatiotemporal dynamics of single species population at different patches since species can have different features in various life stages and different behaviors in various spatial environments. With the effect of time delay and spatial dispersion, homogenous, periodic and spatiotemporally nonhomogeneous distributions are identified. The stability analysis is carried out for the discrete-space and continuous-time network on single species with time delay and the Hopf bifurcation of the single species population model in a network is explored. Formulas for determining the direction of Hopf bifurcation are derived by using the center manifold method and the normal form theorem. It is found that the network can generate spatial patterns only when time delay is present. Finally, numerical simulations are performed which agree well with our theoretical result, i.e. this discrete-space and continuous-time model admits regular temporal patterns since the delay induces Hopf bifurcations with network structure.

Models of continuous-time networks with tie decay, diffusion, and convection.

The study of temporal networks in discrete time has yielded numerous insights into time-dependent networked systems in a wide variety of applications. However, for many complex systems, it is useful to develop continuous-time models of networks and to compare them to associated discrete models. In this paper, we study several continuous-time network models and examine discrete approximations of them both numerically and analytically. To consider continuous-time networks, we associate each edge in a graph with a time-dependent tie strength that can take continuous non-negative values and decays in time after the most recent interaction. We investigate how the moments of the tie strength evolve with time in several models, and we explore-both numerically and analytically-criteria for the emergence of a giant connected component in some of these models. We also briefly examine the effects of the interaction patterns of continuous-time networks on the contagion dynamics of a susceptible-infected-recovered model of an infectious disease.

Inverse conic linear programs in Banach spaces

Given the costs and a feasible solution for a finite-dimensional linear program (LP), inverse optimization involves finding new costs that are close to the original and that also render the given solution optimal. Ahuja and Orlin employed the absolute sum norm and the maximum absolute norm to quantify distances between cost vectors, and applied duality to establish that the inverse LP problem can be formulated as another finite-dimensional LP. This was recently extended to semi-infinite LPs, countably infinite LPs, and finite-dimensional conic optimization problems. These works provide sufficient conditions so that the inverse problem also belongs to the same class as the forward problem. This paper extends this result to conic LPs in potentially infinite-dimensional Banach spaces. Moreover, the paper presents concrete derivations for continuous conic LPs, whose special cases include continuous linear programs and continuous conic programs; normed cone programs in Banach spaces, which include second-order cone programs as a special case; and semi-definite programs in Hilbert spaces. These derivations reveal the sharper result that, in each case, the inverse problem belongs to the same specific subclass as the forward problem. Instances where existing forward algorithms can then be adapted to solve the inverse problems are identified. Results in this paper may enable the application of inverse optimization to as yet unexplored areas such as continuous-time economic planning, continuous-time job-shop scheduling, continuous-time network flow, maximum flow with time-varying edge-capacities, and wireless optimization with time-varying coverage requirements.

A self-organizing recurrent fuzzy neural network based on multivariate time series analysis

Fuzzy neural networks (FNNs) have attracted considerable interest for modeling nonlinear dynamic systems in recent years. However, the recurrent design and the self-organizing design of FNNs generally lack adaptability, and their analyses on the change rule of networks in continuous time are insufficient. To solve these problems, a self-organizing recurrent fuzzy neural network based on multivariate time series analysis (SORFNN-MTSA) is proposed in this paper. First, a recurrent mechanism, based on wavelet transform fuzzy Markov chain algorithm, is introduced to obtain adaptive recurrent values and accelerate convergence speed of the network. Second, a self-organization mechanism, based on weighted dynamic time warping algorithm and sensitivity analysis algorithm, is presented to optimize the network structure. Third, the convergence of SORFNN-MTSA is theoretically analyzed to show the efficiency in both fixed structure and self-organizing structure cases. Finally, several benchmark nonlinear systems and a real application of wastewater treatment are used to verify the effectiveness of SORFNN-MTSA. Compared with other existing methods, the proposed SORFNN-MTSA performs better in terms of both high accuracy and compact structure.

Higher-order cluster consensus of a multi-agent network with continuous-time dynamics

This paper studies the higher-order cluster agreement problem for continuous-time networks evolving over any given directed graph. Necessary and sufficient conditions are derived to reach clusters that are not pre-determined as opposed to existing literature. The analysis shows that appropriate stabilizing choices of coupling strengths do not affect the number of clusters and the final values of the clusters. The results are verified by numerical examples.