Evidence for a bimodal distribution in human communication

Abstract

Interacting human activities underlie the patterns of many social, technological, and economic phenomena. Here we present clear empirical evidence from Short Message correspondence that observed human actions are the result of the interplay of three basic ingredients: Poisson initiation of tasks and decision making for task execution in individual humans as well as interaction among individuals. This interplay leads to new types of interevent time distribution, neither completely Poisson nor power-law, but a bimodal combination of them. We show that the events can be separated into independent bursts which are generated by frequent mutual interactions in short times following random initiations of communications in longer times by the individuals. We introduce a minimal model of two interacting priority queues incorporating the three basic ingredients which fits well the distributions using the parameters extracted from the empirical data. The model can also embrace a range of realistic social interacting systems such as e-mail and letter communications when taking the time scale of processing into account. Our findings provide insight into various human activities both at the individual and network level. Our analysis and modeling of bimodal activity in human communication from the viewpoint of the interplay between processes of different time scales is likely to shed light on bimodal phenomena in other complex systems, such as interevent times in earthquakes, rainfall, forest fire, and economic systems, etc.

Fig. 1.

Typical patterns of SMs activity of a pair of users. The users send more than 95% of the messages to each other. (A) Succession of events by user A (blue) and B (red). The horizontal axis denotes time (in 1 s) and each vertical line corresponds to an event of sending an SM. (B) An enlargement of a short period where the events of A (blue) and B (red) are put together, showing clearly a sending-response pattern by the alternating blue and red colors. The interval between two consecutive lines with the same color is the interevent time τ and that between two consecutive lines with different colors is the waiting time τ_w. (C) and (D) are the distributions P(τ) of the interevent times for the users A and B, respectively. P(τ) is binned in the log-log scale. The upper inset displays the corresponding accumulative distribution F(τ). The vertical dotted line indicates τ₀ = 780, which is used to separate the event sequence into independent bursts (see Materials and Methods and SI Text). The lower inset shows the exponential tails of P(τ) in the linear-log plot. The straight lines are the power-law and exponential fitting functions, which are correspondingly shown by the red line and red curve in the upper inset. The exponents are: γ_A = 1.79 ± 0.01, β_A = (3.78 ± 0.02) × 10^-4 and γ_B = 1.93 ± 0.05, β_A = (3.90 ± 0.03) × 10^-4. (E) and (F) as (C) and (D), but for the distributions P(τ_w) of the waiting times τ_w. The exponents are γ_wA = 2.12 ± 0.01, β_wA = (4.34 ± 0.04) × 10^-4 and γ_wB = 1.90 ± 0.02 ,β_wB = (3.63 ± 0.03) × 10^-4. All the exponents in this work are obtained by the least square method.

Fig. 2.

Separation of bursts and estimation of parameters from data. (A) Communication patterns within the separated bursts (obtained at certain τ₀) that are initiated by user A. The index i denotes the position of the message in a burst, and the height of the bar (n_i) is the number of bursts having a message at position i by user A (blue) or user B (red). (B) Relative error E(τ₀) (see Eq. S2 for definition) displays a minimum where the initiations of bursts in the two users are best approximated by independent Poisson random processes.

Fig. 3.

Separation of the initiative and passivity messages with the most suitable τ₀. (A) Accumulative distributions F(τ) for the interval between two consecutive bursts that are initiated by the same user A (with rate λ_A) and the interval between two consecutive bursts that can be either initiated by A or the response to an initiative message of B (with rate δ_A). These rates of the two users satisfy the relationships δ_A = λ_A + P_Aλ_B and δ_B = λ_B + P_Bλ_A, implying that the initiation of communications in the two users are independent Poisson processes. (B) The distribution of the size n_b of the separated bursts, i.e., the number of messages sent by a user within a burst. The solid line is the exponential fitting . (C, D) Distributions P(τ_w) of the waiting time τ_w obtained only from the messages within the separated bursts, for the user A and B, respectively. The solid lines are the power-law fitting with γ_wA = 2.05 ± 0.01, γ_wB = 1.89 ± 0.01.

Fig. 4.

Fitting of the model (red) to empirical data (blue). The model is simulated for various processing time t_p, using all the other parameters obtained from the data, to generate the same number of events as in the data. The relative difference between the accumulative distributions F(τ) of the interevent times τ in the model and data, is obtained as a function of t_p (averaging over 10 realizations of independent model simulations, insets of (A, B)). E is minimal at t_p = 10 for both users, yielding very accurate fitting of F(τ) (A, B), P(τ) (C, D) and P(τ_w) (insets of (C, D)), except for a few points with the minimal and maximal intervals mainly due to finite size fluctuations.

Fig. 5.

Effect of interaction on human activity patterns in the model. The cumulative distribution F(τ) of the interevent times obtained at various response rates P₁. The other parameters are fixed as λ = 1.5 × 10^-4, α = 1.0, and t_p = 1. The inset shows the exponential tails in the linear-log plot.