Fig 1.
Time evolution of connections’ weights.
The weight ωij of a connection between users (i, j) corresponds to the number of emails exchanged by i and j during a whole year. We only consider connections with ω ≥ 12 (see text) (A) Distributions of weights for each one of the years in our dataset (2007-2010). Note that the distribution is stable in time. (B) Distribution of the centered weight logarithmic growth rates for Δt = 1, 2, 3 (dots, squares and diamonds, respectively). Lines show fits to the convolution of a Laplace distribution and a Gaussian distributed noise (see Eq (5)) (parameters Δt = 1: σexp = 0.43, and σG = 0.35, Δt = 2: σexp = 0.50, and σG = 0.47 and Δt = 3: σexp = 0.50, and σG = 0.60). Note that as Δt increases the peaks are rounder and the distributions are slightly wider (see Fig D in S2 File). See Fig B in S2 File for values of the distribution modes μ(t, Δt).
Fig 2.
Time evolution of nodes’ strengths.
The strength si of node i is the number of emails that user i exchanged with other users during one year. (A) Distributions of strengths for each one of the years in our dataset (2007-2010). Note that the distribution is stable in time. (B) Distribution of centered strength logarithmic growth rates for Δt = 1, 2, 3 years (dots, squares and diamonds, respectively). Lines show fits to a Laplace distribution (parameters Δt = 1: σexp = 0.57, Δt = 2: σexp = 0.74 and Δt = 3: σexp = 0.83). Note that as Δt increases the distributions are wider (see Fig D in S2 File). For the specific values of the distribution modes μ(t, Δt) see Fig B in S2 File.
Fig 3.
Predictability of logarithmic growth rates for connection weight rω(t + 1) (A, C, E) and user strength rs(t + 1) (B, D, F).
(A) Joint probability density of rω(t + 1), the logarithmic growth rate of weights at time t + 1, and rω(t), the logarithmic growth rate of weights at time t. (B) Joint probability density of rs(t + 1), the logarithmic growth rate of strengths at time t + 1, and rs(t), the logarithmic growth rate of strengths at time t. (C) Joint probability density of rω(t + 1), the logarithmic growth rate of weights at time t + 1, and ω(t), the weight at time t. The area shaded in grey area is no allowed since rω(t + 1)≥ − log ω(t). (D) Joint probability density of rs(t + 1), the logarithmic growth rate of strengths at time t + 1, and s(t), the strength at time t. The area shaded in grey is forbidden since rs(t + 1)≥ − log s(t). In plots (A-D), circles and error bars show the mean and one standard error of the mean for values binned along the X axis. It is visually apparent that ω(t) and s(t) are more informative about rω(t + 1) and rs(t + 1), respectively, than rω(t) and rω(t) (as confirmed by Spearman’s ρ and p-values, displayed inside each graph). (E, F) Root mean squared error (MSE) of the predictions of the logarithmic growth rates at time t + 1 obtained from leave-one-out experiments. As predictors, we use: (E) ω(t), rω(t), and μω(t) (see Eq (5)); (F) s(t), rs(t), and μs(t) (see Eq (3)). Additionally, in both cases we try to predict the logarithmic growth rate using a Random Forest regressor [29]. Note that a simple approach (i.e. considering the weight/strength at time t) performs significantly better than a well-performing machine learning algorithm such as the Random Forest. In any case, and despite being the most predictive, weight/strength at time t only provide moderate improvements over predictions made using the mean value μω for all connections and μs for all users.
Fig 4.
Stability of social signatures.
(A) Distribution of the standardized Shannon entropy Si (see text) for users in the period 2007–2010. Entropy quantifies the extent to which and individual’s communication efforts are distributed among her contacts, so that Si = 1 when user i exchanges the same number of emails with all her contacts and Si ≈ 0 when she exchanges almost all of her emails with a single contact. Distributions for all years collapse onto a single curve. The line shows a kernel density estimation of the four yearly datasets pooled together. (B) Distributions of the change of individual standardized Shannon entropy ΔSi(Δt) = Si(t + Δt) − Si(t), ∀i for Δt = 1, 2, 3 years (dots, squares and diamonds, respectively). The lines show the Laplace best fits based on BIC for the three distributions (Δt = 1 σ = 0.065; Δt = 2 σ = 0.075; and Δt = 3σ = 0.085). (C) Comparison between the absolute difference in individual social signatures |ΔSi(Δt)|self = |Si(t + Δt) − Si(t)| and the typical absolute difference of entropies between individuals |ΔSij|ref = |Si(t) − Sj(t)|. The boxplot shows unambiguously that users have stable social signatures.
Fig 5.
Stability of individual communication strategies.
(A) Distribution of the fraction of emails sent by users to pre-existing contacts fi (see text). The line shows the kernel density estimation of the three yearly datasets pooled together. Most users exchange most of their emails with preexisting contacts. with the maximum at . (B) Distribution of the change of fi, Δfi(Δt) = fi(t + Δt) − fi(t) for Δt = 1, 2 years (dots and squares, respectively). The lines show the Laplace best fits based on BIC for the two distributions (P(Δfi)∼exp(−|Δfi − μ|/σ); Δt = 1 σ = 0.18 μ = 0.046; and Δt = 2 σ = 0.19 μ = 0.062). Most of the users keep the number of emails sent to preexisting contacts constant in time, and the distributions are quite stable in time despite a slight shift towards larger changes for larger Δt. (C) Comparison between yearly absolute individual change in the fraction of emails sent to preexisting contacts |Δfi(Δt)|self and the typical differences between users |Δfij|ref = |fi(t) − fj(t)|, ∀j ≠ i. The boxplot shows unambiguously that individual users have a stable communication strategy over time.