Temporal Effects in the Growth of Networks

Abstract

We show that to explain the growth of the citation network by preferential attachment (PA), one has to accept that individual nodes exhibit heterogeneous fitness values that decay with time. While previous PAbased models assumed either heterogeneity or decay in isolation, we propose a simple analytically treatable model that combines these two factors. Depending on the input assumptions, the resulting degree distribution shows an exponential, log-normal or power-law decay, which makes the model an apt candidate for modeling a wide range of real systems.Over the years, models with preferential attachment (PA) were independently proposed to explain the distribution of the number of species in a genus [1], the power-law distribution of the number of citations received by scientific papers [2], and the number of links pointing to World Wide Web (WWW) pages [3]. A theoretical description of this class of processes and the observation that they generally lead to power-law distributions are due to Simon [4]. Notably, the application of PA to WWW data by Barabási and Albert helped to initiate the lively field of complex networks [5]. Their network model, which stands at the center of attention of this work, was much studied and generalized to include effects such as presence of purely random connections [6], nonlinear dependence on the degree [7], node fitness [8], and others ([9], Chap. 8).Despite its success in providing a common roof for many theoretical models and empirical data sets, preferential attachment is still little developed to take into account the temporal effects of network growth. For example, it predicts a strong relation between a node's age and its degree. While such first-mover advantage [10] plays a fundamental role for the emergence of scale-free topologies in the model, it is a rather unrealistic feature for several real systems (e.g., it is entirely absent in the WWW [11] and significant deviations are found in citation data [10,12]). This motivates us to study a model of a growing network where a broad degree distribution does not result from strong time bias in the system. To this end we assign fitness to each node and assume that this fitness decays with time-we refer it as relevance henceforth. Instead of simply classifying the vertices as active or inactive, as done in [13,14], we use real data to investigate the relevance distribution and decay therein and build a model where decaying and heterogeneous relevance are combined.Models with decaying fitness values (''aging'') were shown to produce narrow degree distributions (except for very slow decay) [15] and widely distributed fitness values were shown to produce extremely broad distributions or even a condensation phenomenon where a single node attracts a macroscopic fraction of all links [16]. We show that when these two effects act together, they produce various classes of behavior, many of which are compatible with structures observed in real data sets.Before specifying a model and attempting to solve it, we turn to data to provide sup...

Attention decay in science

“…The typical approach is to study the evolution of the number of citations received by a paper in a given time frame (usually one year), since its publication. The nature of the decay has been controversial, between claims of an exponential trend (Avramescu, 1979;Nakamoto, 1988;Medo et al, 2011) and analyses supporting a slower power law curve (Pollman, 2000;Redner, 2005;Bouabid, 2011;Bouabid and Larivière, 2013). This is partly due to the different types of analysis and the use of distinct data sources.…”

Temporal networks

“…There are likely other economic systems that would benefit from temporal network modeling. Networks that, like citation networks [105] are normally thought of as strictly growing, could show temporal effects in the growth that could benefit from being studied in a temporal-network framework.…”

Model-based evaluation of scientific impact indicators

“…In this work, we use a model that extends the one suggested in [20]. The system is composed of researchers (or authors) and papers, indicated by Latin and Greek letters, respectively.…”

“…In particular, we extend a previous model constrained on the citation dynamics of scientific papers [20] by assuming that researchers are endowed with intrinsic productivity and ability levels-the latter determining the fitness of their authored papers-that in turn make connections to the existing body of literature according to the modified preferential attachment mechanism described above. The artificial bibliometric data generated by the model then allow us to compute a variety of impact indicators, which can be compared with the individual traits of researchers in order to determine what these indicators actually capture.…”

“…Importantly, the original version of the model predicts a strong relation between a paper's age and its citation count, but significant deviations from this behavior are found in bibliometric data [19]. It has been recently pointed out that to model citation dynamics well, preferential attachment has to be combined with intrinsic paper relevance: a heterogeneously distributed "quality" (fitness) that decays with time [20][21][22]. These models are then capable of generating artificial data that closely resemble real citation networks [23].…”