Memory and burstiness in dynamic networks

Abstract

A discrete-time random process is described which can generate bursty sequences of events. A Bernoulli process, where the probability of an event occurring at time t is given by a fixed probability x, is modified to include a memory effect where the event probability is increased proportionally to the number of events which occurred within a given amount of time preceding t. For small values of x the inter-event time distribution follows a power-law with exponent −2−x. We consider a dynamic network where each node forms, and breaks connections according to this process. The value of x for each node depends on the fitness distribution, ρ(x), from which it is drawn; we find exact solutions for the expectation of the degree distribution for a variety of possible fitness distributions, and for both cases where the memory effect either is, or is not present. This work can potentially lead to methods to uncover hidden fitness distributions from fast changing, temporal network data such as online social communications and fMRI scans. The mathematics of interactive complex systems has a vital role to play in the interpretation of large-scale social and biological data. Technology which facilitates the collection of vast amounts of information is increasingly becoming available for both academic and commercial purposes; however, in the absence of a detailed understanding of the underlying processes, there will always be a risk of deriving the wrong conclusion from the facts. Complexity science provides numerous models of social, biological, physical and economic systems which combine large numbers of individual components to reproduce the types of behaviour observed on the systemic level. The components in such systems are usually uninteresting in isolation, but when allowed to interact with each other they produce complex non-trivial patterns which in some cases agree very well with empirical results. This poses a challenge for data scientists: given information only about the system as a whole, with all its complex and interactive dynamics, how can one conclude anything about the individual components? To begin answering that question we need to understand, in mathematical terms, the form and extent of the biases that complexity creates.The purpose of the present work is to provide an understanding of how one very simple mechanism, a memory effect (brought about by interaction), will bias the statistical properties of a complex system such as the distribution of communication activity in a social network, or the distribution of brain activity of different cortical regions in a fMRI scan. We consider a hypothetical system of individual agents (nodes) and the instantaneous pairwise interactions which happen between them (edges). By aggregating all of the interactions that occur within some given time window, a network is formed whose structure can be analysed for a deeper understanding of the system. * E. Colman@Reading.ac.uk In general, the length of this time window determines the density of the network; as an incre...

Burstiness in activity-driven networks and the epidemic threshold

“…, N an activity a i drawn from a distribution ρ(a i ). The activation rate of node i is defined by its inter-event time distribution Ψ a i (τ ) [8][9][10][11][12]: this sets the statistic of time intervals between consecutive activations of node i, so that the average inter-event time equals the inverse of the activity of node i, i.e. τ i = a −1 i .…”

“…Besides, a large amount of work has been devoted to clarify the effects of intermittent patterns on the temporal structures of interactions, as described by temporal networks [7]. Bursty dynamics can indeed influence the structure of links on the local and on the global scale [8][9][10][11][12]. More importantly, heterogenous temporal patterns in the evolution of time-varying networks can affect in a non-trivial way dynamical processes.…”

Temporal pattern of online communication spike trains in spreading a scientific rumor: how often, who interacts with whom?

“…However, recent works have showed that further detail analysis is required to resolve temporal correlations [31,32], bursts [19][20][21][22], and cascading [53] driven by circadian rhythm [23,24], complex decision-making of individuals [3,27,54], and external factors [6] such as the announcement of discoveries, as considered in the current data [38].…”

“…The time series of user activities, e.g., posting a tweet and replying to a message, are quite distinct from uncorrelated (Poisson random) dynamics in the presence of burstiness [18][19][20], temporal correlations [6,21,22], and non-stationarity of human daily rhythm [23,24], which has significant implications. Diffusion on a temporal network cannot be accurately described by models on static networks and consequently the process presents non-Markovian features with strong influence on the time required to explore the system [25,26].…”

Aging and percolation dynamics in a Non-Poissonian temporal network model

“…Several models of temporal networks have been put forward in the literature [22][23][24][25][26], focused on different possible mechanisms to explain the empirically observed properties. Among those, it is noteworthy the recently proposed Non-Poissoinan activity driven (NoPAD) model [27].…”