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Register a new userModels of Continuous-Time Networks with Tie Decay, Diffusion, and Convection
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The study of temporal networks in discrete time has yielded numerous insights into time-dependent networked systems in a wide variety of applications. For many complex systems, however, 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 mean tie strength evolves 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 interaction patterns of our continuous-time networks on contagion dynamics in a susceptible-infected-recovered model of an infectious disease.
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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. Specifically, 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 fundamentally 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.
We consider similarity solutions of the generalized convection-diffusion-reaction equation with both space- and time-dependent convection, diffusion and reaction terms. By introducing the similarity variable, the reaction-diffusion equation is reduced to an ordinary differential equation. Matching the resulting ordinary differential equation with known exactly solvable equations, one can obtain corresponding exactly solvable convection-diffusion-reaction systems. Some representative examples of exactly solvable systems are presented. We also describe how an equivalent convection-diffusion-reaction system can be constructed which admits the same similarity solution of another convection-diffusion-reaction system.
The diffusion of an artificial active particle in a two-dimensional periodic pattern of stationary convection cells is investigated by means of extensive numerical simulations. In the limit of large Peclet numbers, i.e., for self-propulsion speeds below a certain depinning threshold and weak roto-translational fluctuations, the particle undergoes asymptotic normal diffusion with diffusion constant proportional to the square root of its diffusion constant at zero flow. Chirality effects in the propulsion mechanism, modeled here by a tunable applied torque, favors particles jumping between adjacent convection rolls. Roll jumping is signaled by an excess diffusion peak, which appears to separate two distinct active diffusion regimes for low and high chirality. A qualitative interpretation of our simulation results is proposed as a first step toward a fully analytical study of this phenomenon.
Networks are used as highly expressive tools in different disciplines. In recent years, the analysis and mining of temporal networks have attracted substantial attention. Frequent pattern mining is considered an essential task in the network science literature. In addition to the numerous applications, the investigation of frequent pattern mining in networks directly impacts other analytical approaches, such as clustering, quasi-clique and clique mining, and link prediction. In nearly all the algorithms proposed for frequent pattern mining in temporal networks, the networks are represented as sequences of static networks. Then, the inter- or intra-network patterns are mined. This type of representation imposes a computation-expressiveness trade-off to the mining problem. In this paper, we propose a novel representation that can preserve the temporal aspects of the network losslessly. Then, we introduce the concept of constrained interval graphs (CIGs). Next, we develop a series of algorithms for mining the complete set of frequent temporal patterns in a temporal network data set. We also consider four different definitions of isomorphism to allow noise tolerance in temporal data collection. Implementing the algorithm for three real-world data sets proves the practicality of the proposed algorithm and its capability to discover unknown patterns in various settings.
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