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Reallocating connections between individuals.: (a) The manner in which the (undirected) friendships between 34 people in “Zachary’s Karate Club” are reconnected by increasing or decreasing C, respectively for inciting or impeding cascades of communications. (b) The change in the potential for the cascades C with the steepest ascent (rightward) or descent (leftward) based on equation (7), (c) Average clustering coefficient.
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There is a commonality among contagious diseases, tweets, urban crimes, nuclear reactions, and neuronal firings that past events facilitate the future occurrence of events. The spread of events has been extensively studied such that the systems exhibit catastrophic chain reactions if the interaction represented by the ratio of reproduction exceeds...
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... While these models are able to reproduce macro statistical laws at the population level, they have been shown to be insufficient at reproducing the local microstructural interactions of human activity [12]. The incorporation of random novelties and preferential attachment alone is not sufficient to capture the open-ended nature [13][14][15] of human activity such as waves of novelty, the emergence of new ideas, technologies, or trends that spread rapidly through a population or system, often replacing or supplanting existing ones [16][17][18][19]. ...
Understanding the growth and evolution of social networks is an important area of study, as these networks form the foundation for many popular online services such as social networking sites (SNS) and online games. However, previous models developed to explain the growth mechanisms of these networks have struggled to accurately reproduce certain behaviors that are frequently observed in real data, such as waves of novelty, in which new individuals or topics receive more attention than existing ones for a short period of time. In this study, we introduce a new model that incorporates context information into existing agent-based models in order to more accurately capture the structure and growth dynamics of these networks. Context information is introduced through labels based on the timing of appearance and relationships with antecedent agents. New agents are first added to the network when they are called by existing agents, and at this time they are also given a label. Agents added to the network at the same time by the same agent will have the same label. These labels are used to classify agents and give them different selection probabilities. This newly introduced selection probability creates a mechanism in which new agents receive attention beyond preferential attachment. By comparing the results of our model with real data on ten metrics, we demonstrate that it is able to produce behavior that more closely resembles real data. This improved understanding of the dynamics of social networks has important implications for designing effective interventions, including strategies for user acquisition and retention.
... To answer these questions, a model of neuronal interactions needs to be settle properly. Hawkes processes and variants have been used in many articles as a model for interacting neurons [23,25]. Indeed, the shape of their intensity mimics approximately the synaptic integration. ...
Functional connectivity is a neurobiological notion, informally stating that there would be a strong dependence between neurons and that this dependence might be useful in understanding the way the brain encodes stimuli, programs actions, etc. However, in practice such strong dependencies are often reconstructed via Hawkes processes based on an amazingly small number of neurons, because of the very scarce observation of this very complex and huge network. We derive new simple equations, which explain how the ideal Hawkes reconstruction is linked to the covariance between the observed neurons. These equations help us in particular to understand what the Hawkes reconstruction does in two settings, synchronization and classical point process asymptotics. Moreover they might help us to also understand what is qualitatively happening at the scale of the huge unobserved network, paving the path for a possible mathematical definition of functional connectivity.
... However, the Hawkes process does not account for the finite size effect, in which infected and recovered people represent a finite fraction of the entire population. There have been some models that incorporate the finite population size effect into the Hawkes process, as has been done with the SIR or SEIR models [8,29]. To analyze the current status of COVID-19, however, we do not take the finite size effect into account, as the fraction of the recovered or removed people is still less than a few % of the entire population. ...
After slowing down the spread of the novel coronavirus COVID-19, many countries have started to relax their confinement measures in the face of critical damage to socioeconomic structures. At this stage, it is desirable to monitor the degree to which political measures or social affairs have exerted influence on the spread of disease. Though it is difficult to trace back individual transmission of infections whose incubation periods are long and highly variable, estimating the average spreading rate is possible if a proper mathematical model can be devised to analyze daily event-occurrences. To render an accurate assessment, we have devised a state-space method for fitting a discrete-time variant of the Hawkes process to a given dataset of daily confirmed cases. The proposed method detects changes occurring in each country and assesses the impact of social events in terms of the temporally varying reproduction number, which corresponds to the average number of cases directly caused by a single infected case. Moreover, the proposed method can be used to predict the possible consequences of alternative political measures. This information can serve as a reference for behavioral guidelines that should be adopted according to the varying risk of infection.
... The critical reproduction number R c is identical to that obtained for the principled histogram method, such that the selected bin size diverges above this reproduction number R c [44,45]. As shown in Fig. 3(c), the Bayesian rate estimator cannot capture the rate fluctuation if the endogenous self-excitation R * or exogenous fluctuation σ * 2 τ * e /μ * is small. ...
Event occurrence is not only subject to the environmental changes, but is also facilitated by the events that have occurred in a system. Here, we develop a method for estimating such extrinsic and intrinsic factors from a single series of event-occurrence times. The analysis is performed using a model that combines the inhomogeneous Poisson process and the Hawkes process, which represent exogenous fluctuations and endogenous chain-reaction mechanisms, respectively. The model is fit to a given dataset by minimizing the free energy, for which statistical physics and a path-integral method are utilized. Because the process of event occurrence is stochastic, parameter estimation is inevitably accompanied by errors, and it can ultimately occur that exogenous and endogenous factors cannot be captured even with the best estimator. We obtained four regimes categorized according to whether respective factors are detected. By applying the analytical method to real time series of debate in a social-networking service, we have observed that the estimated exogenous and endogenous factors are close to the first comments and the follow-up comments, respectively. This method is general and applicable to a variety of data, and we have provided an application program, by which anyone can analyze any series of event times.
... The critical reproduction ratio α c is identical to that obtained for the principled histogram method, such that the selected bin size diverges above this reproduction ratio α c [41,42]. As shown in Fig. 3 (c), the Bayesian rate estimator cannot capture the rate fluctuation if the endogenous selfexcitation α * or exogenous fluctuation σ * 2 τ * e /µ * is small. ...
Event occurrence is not only subject to the environmental changes, but is also facilitated by the events that have occurred in a system. Here, we develop a method for estimating such extrinsic and intrinsic factors from a single series of event-occurrence times. The analysis is performed using a model that combines the inhomogeneous Poisson process and the Hawkes process, which represent exogenous fluctuations and endogenous chain-reaction mechanisms, respectively. The model is fit to a given dataset by minimizing the free energy, for which statistical physics and a path-integral method are utilized. Because the process of event occurrence is stochastic, parameter estimation is inevitably accompanied by errors, and it can ultimately occur that exogenous and endogenous factors cannot be captured even with the best estimator. We obtained four regimes categorized according to whether respective factors are detected. By applying the analytical method to real time series of debate in a social-networking service, we have observed that the estimated exogenous and endogenous factors are close to the first comments and the follow-up comments, respectively. This method is general and applicable to a variety of data, and we have provided an application program, by which anyone can analyze any series of event times.
... Specifically, we use the fitted parameters of the multivariate Hawkes process and simulate the model to compute the degree of impact caused by deleting each thread from the system. This is done by computing the activity level of the entire bulletin board (Onaga and Shinomoto, 2016), comparing the activity levels from before and after the removal, and regarding the difference as the impact of the removal of each thread. If a thread has a high degree of impact despite having a small density (i.e., the number of users participating in the thread), then we identify the thread as a keystone thread. ...
... We define this burstiness to represent the activity level of the network. Onaga and Shinomoto (2016) also theoretically showed that when the value of the cascading condition C is greater than two, the event occurrence of the entire network undergoes a phase transition to a bursty, non-stationary state. Conversely, if the value of C is less than two, the event occurrence state of the network is a stationary state. ...
... Specifically, we use the fitted parameters of the multivariate Hawkes process and simulate the model to compute the degree of impact caused by deleting each thread from the system. This is done by computing the activity level of the entire bulletin board [16] and comparing the activity levels from before and after the removal and regard that as an impact of the removal of each thread. If a thread has a high degree of impact despite a small density (i.e., the number of users participating in the thread), then we identify the thread as a keystone thread. ...
... To calculate keystone threads, we need to calculate the impact I i on the ecosystem, viz., the thread network, when thread i is removed. Here, we use cascading condition C, which represents the activity level of network [16] ...
Research is being conducted to understand social and innovative behavior in human interactions on the Web as a biological ecosystem. Keystone species in a biological ecosystem are defined as a set of species that significantly impacts the ecosystem if they are removed from the system, irrespective of its small biomass. Identifying keystone species is an important problem as they play an important role in maintaining diversity and stability in ecosystems. A human community in the web system also possesses keystone species. They can be influential users or contents on the web systems, even though their commitments to the web are relatively small. We use data from an online bulletin board, and identify keystone threads (= "species") that have a large impact if they are removed or become unpopular, despite their small population size. Here, the removal of threads can be regarded as a state in which there is no attention or actions by users on the thread. The multivariate Hawkes process is used to measure the degree of influence among all threads and calculate the overall activity level on the online bulletin board. Our analysis confirms that keystone threads do exist in the system. Apparently, the number of keystone species increases along with the service maturation. The keystone concept in online services proposed in this study gives a new viewpoint for their stable operation.
... Several works have focused on the structure and dynamics of the resulting information cascades, from their characterization in empirical data to the design of machine learning algorithms and mathematical models to predict their behavior [9][10][11][12][13][14][15][16][17]. Mathematically, information cascades are often modeled by self-exciting point processes [18][19][20][21], as previous events may trigger new events, in a way that generalizes the standard Hawkes process [22]. In their simplest instance, Hawkes processes are linear self-reinforced processes, where the occurrence of an event increases the * fujita.kazuki.37n@st.kyoto-u.ac.jp † alexey.medvedev@unamur.be ...
... We generated events with the nonlinear Hawkes process given by Eq. (11) with the self-excitation term α 0 greater than in the case (b), so that event occurrence exhibits large fluctuations. By applying the optimal histogram method, we obtained a fluctuating rate (i.e., the optimal bin size was finite), implying that the nonlinear Hawkes process may also exhibit the stationary-nonstationary (SN) transition that was found in the linear Hawkes process [20,21]: significant fluctuations appear even in the absence of external modulation. Although the rate estimation method suggested that the rate is fluctuating, our GLM was able to reveal that exogenous forcing was absent, and we conclude that the fluctuations appeared solely due to the self-excitation [ Fig. 2(c), right panel]. ...
The occurrence of new events in a system is typically driven by external causes and by previous events taking place inside the system. This is a general statement, applying to a range of situations including, more recently, to the activity of users in online social networks (OSNs). Here we develop a method for extracting from a series of posting times the relative contributions that are exogenous, e.g., news media, and endogenous, e.g., information cascade. The method is based on the fitting of a generalized linear model (GLM) equipped with a self-excitation mechanism. We test the method with synthetic data generated by a nonlinear Hawkes process, and apply it to a real time series of tweets with a given hashtag. In the empirical dataset, the estimated contributions of exogenous and endogenous volumes are close to the amounts of original tweets and retweets respectively. We conclude by discussing the possible applications of the method, for instance in online marketing.
... The rate of event occurrences in these models is partitioned into two components: a background rate, which describes exogenous effect (e.g., trends); and a mutually exciting component, in which events trigger an increase in the process rate. Owing to the latter component, Hawkes processes exhibit rich dynamical behavior, in terms of event cascades [18,19]. Much attention has recently been paid to the field of social data mining [10,25]. ...
... (16) and (17), we see that the variance of the DM distribution is greater than that of the multinomial distribution. The DM distribution (19) converges to the multinomial distribution (15) as φ → 0. The properties of the DM distribution are summarized in Appendix C. ...
... Note that our model with the Poisson distributions becomes the Hawkes process in the continuous time limit. The Hawkes process is known to exhibit "bursting" when the self-exciting effect is strong [18,19], resulting that the variance of the event count in the finite interval becomes greater than the mean. Thus, our model using the NB distributions would be more adequate for a discrete time counterpart than that using the Poisson distributions, when the underlying process has strong self-excitations. ...
We propose a statistical model for networks of event count sequences built on a cascade structure. We assume that each event triggers successor events, whose counts follow additive probability distributions; the ensemble of counts is given by their superposition. These assumptions allow the marginal distribution of the count sequences and the conditional distribution of the event cascades to have analytic forms. We present our model framework using Poisson and negative binomial distributions as the building blocks. Based on this, we describe a statistical method for estimating the model parameters and the event cascades from the observed count sequences.
... The majority of work use network models (for example see (Huo et al., 2016, Chierichetti et al., 2011, Onaga et al., 2016). This is an application of graph theory in mathematics. ...
Studies on how rumor spreads have attracted increasing attention since rumor can shape public opinion
and affects the beliefs of individuals, which can change individual’s attitude towards social, economic and political
aspects. Rumor has negative impact on society and hence its spread needs to be clearly understood. However,
understanding the spread of rumors in the society is challenging as many factors are involved, and their effects have to
be determined. The use of mathematical models for analysing the dynamics of the rumor spread is common, and a
number of mathematical models have been developed to examine its spreading dynamics. Therefore, a systematic
review about the available models is required. This paper reviews mathematical models for rumor spread in order to
summarise several significant results, and to identify important aspects that require further investigation to advance our
understanding of the dynamics of rumor spread.
