Copula-based algorithm for generating bursty time series

Temporal scaling theory for bursty time series with clusters of arbitrarily many events

Long-term temporal correlations in time series in a form of an event sequence have been characterized using an autocorrelation function that often shows a power-law decaying behavior. Such scaling behavior has been mainly accounted for by the heavy-tailed distribution of interevent times, i.e., the time interval between two consecutive events. Yet, little is known about how correlations between consecutive interevent times systematically affect the decaying behavior of the autocorrelation function. Empirical distributions of the burst size, which is the number of events in a cluster of events occurring in a short time window, often show heavy tails, implying that arbitrarily many consecutive interevent times may be correlated with each other. In the present study, we propose a model for generating a time series with arbitrary functional forms of interevent time and burst size distributions. Then, we analytically derive the autocorrelation function for the model time series. In particular, by assuming that the interevent time and burst size are power-law distributed, we derive scaling relations between power-law exponents of the autocorrelation function decay, interevent time distribution, and burst size distribution. These analytical results are confirmed by numerical simulations. Our approach helps to rigorously and analytically understand the effects of correlations between arbitrarily many consecutive interevent times on the decaying behavior of the autocorrelation function.

Burst and Memory-aware Transformer: capturing temporal heterogeneity

Burst patterns, characterized by their temporal heterogeneity, have been observed across a wide range of domains, encompassing event sequences from neuronal firing to various facets of human activities. Recent research on predicting event sequences leveraged a Transformer based on the Hawkes process, incorporating a self-attention mechanism to capture long-term temporal dependencies. To effectively handle bursty temporal patterns, we propose a Burst and Memory-aware Transformer (BMT) model, designed to explicitly address temporal heterogeneity. The BMT model embeds the burstiness and memory coefficient into the self-attention module, enhancing the learning process with insights derived from the bursty patterns. Furthermore, we employed a novel loss function designed to optimize the burstiness and memory coefficient values, as well as their corresponding discretized one-hot vectors, both individually and jointly. Numerical experiments conducted on diverse synthetic and real-world datasets demonstrated the outstanding performance of the BMT model in terms of accurately predicting event times and intensity functions compared to existing models and control groups. In particular, the BMT model exhibits remarkable performance for temporally heterogeneous data, such as those with power-law inter-event time distributions. Our findings suggest that the incorporation of burst-related parameters assists the Transformer in comprehending heterogeneous event sequences, leading to an enhanced predictive performance.

Copula-based analysis of the generalized friendship paradox in clustered networks

A heterogeneous structure of social networks induces various intriguing phenomena. One of them is the friendship paradox, which states that on average, your friends have more friends than you do. Its generalization, called the generalized friendship paradox (GFP), states that on average, your friends have higher attributes than yours. Despite successful demonstrations of the GFP by empirical analyses and numerical simulations, analytical, rigorous understanding of the GFP has been largely unexplored. Recently, an analytical solution for the probability that the GFP holds for an individual in a network with correlated attributes was obtained using the copula method but by assuming a locally tree structure of the underlying network [Jo et al., Phys. Rev. E 104, 054301 (2021)]. Considering the abundant triangles in most social networks, we employ a vine copula method to incorporate the attribute correlation structure between neighbors of a focal individual in addition to the correlation between the focal individual and its neighbors. Our analytical approach helps us rigorously understand the GFP in more general networks, such as clustered networks and other related interesting phenomena in social networks.

Analytical approach to the generalized friendship paradox in networks with correlated attributes

One of the interesting phenomena due to the topological heterogeneities in complex networks is the friendship paradox, stating that your friends have on average more friends than you do. Recently, this paradox has been generalized for arbitrary nodal attributes, called a generalized friendship paradox (GFP). In this paper, we analyze the GFP for the networks in which the attributes of neighboring nodes are correlated with each other. The correlation structure between attributes of neighboring nodes is modeled by the Farlie-Gumbel-Morgenstern copula, enabling us to derive approximate analytical solutions of the GFP for three kinds of methods summarizing the neighborhood of the focal node, i.e., mean-based, median-based, and fraction-based methods. The analytical solutions are comparable to simulation results, while some systematic deviations between them might be attributed to the higher-order correlations between nodal attributes. These results help us get deeper insight into how various summarization methods as well as the correlation structure of nodal attributes affect the GFP behavior, hence better understand various related phenomena in complex networks.

Efficient simulation of non-Markovian dynamics on complex networks

We study continuous-time multi-agent models, where agents interact according to a network topology. At any point in time, each agent occupies a specific local node state. Agents change their state at random through interactions with neighboring agents. The time until a transition happens can follow an arbitrary probability density. Stochastic (Monte-Carlo) simulations are often the preferred-sometimes the only feasible-approach to study the complex emerging dynamical patterns of such systems. However, each simulation run comes with high computational costs mostly due to updating the instantaneous rates of interconnected agents after each transition. This work proposes a stochastic rejection-based, event-driven simulation algorithm that scales extremely well with the size and connectivity of the underlying contact network and produces statistically correct samples. We demonstrate the effectiveness of our method on different information spreading models.