Temporal effects in the growth of networks

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 PA-based 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.

💡 Research Summary

The paper revisits the classic preferential attachment (PA) mechanism that underlies many models of growing complex networks and demonstrates that, on its own, PA cannot fully account for the empirical patterns observed in citation networks. The authors propose a parsimonious yet analytically tractable extension that incorporates two realistic ingredients: (i) heterogeneous node fitness (called “relevance”) and (ii) a systematic decay of this fitness over time.

Using the complete American Physical Society (APS) citation dataset (450,084 papers, 4.69 million citations spanning 1893–2009) and a focused subset of papers on network theory, the authors first define a node’s relevance X_i(t,Δt) as the ratio between the actual number of citations received in a short interval Δt and the number predicted by pure PA. Empirical analysis shows that relevance decays rapidly during the first few years after publication and then plateaus at a low, almost stationary value. Moreover, the total accumulated relevance X_T = Σ_t X_i(t) follows an exponential tail (α≈2.17×10⁻³).

The model builds a growing undirected network that starts with two linked nodes. At each time step a new node joins and attaches to an existing node i with probability

 P(i,t) = k_i(t) R_i(t) / Σ_j k_j(t) R_j(t) ,

where k_i(t) is the current degree and R_i(t) is the time‑dependent relevance of node i. Assuming that many nodes have non‑negligible k_i R_i, the denominator fluctuates little and can be approximated by a constant Ω*. Under this approximation the master equation yields a simple growth law

 d⟨k_i⟩/dt = k_i R_i / Ω* .

Integrating from the node’s birth time t_i gives the expected final degree

 ⟨k_i^F⟩ = exp