Input-output relationship in social communications characterized by spike train analysis

We study the dynamical properties of human communication through different channels, i.e., short messages, phone calls, and emails, adopting techniques from neuronal spike train analysis in order to characterize the temporal fluctuations of successive interevent times. We first measure the so-called local variation (LV) of incoming and outgoing event sequences of users and find that these in- and out-LV values are positively correlated for short messages and uncorrelated for phone calls and emails. Second, we analyze the response-time distribution after receiving a message to focus on the input-output relationship in each of these channels. We find that the time scales and amplitudes of response differ between the three channels. To understand the effects of the response-time distribution on the correlations between the LV values, we develop a point process model whose activity rate is modulated by incoming and outgoing events. Numerical simulations of the model indicate that a quick response to incoming events and a refractory effect after outgoing events are key factors to reproduce the positive LV correlations.

Figure 1

Three sequences of events with different temporal fluctuations (right) that are generated from the same set of the IETs (left).

Figure 2

(a) Distribution of the IETs of outgoing events of the SMS data set. Scatter plots of the LV and CV values for (b) outgoing events and (c) incoming events. The histograms of the measures are also shown. The same set of the plots for (d)–(f) the Phone data set and for (g)–(i) the Email data set.

Figure 3

(a) Histogram of LVs, and the correlations between (b) ${\mathrm{LV}}^{\mathrm{in}}$ and ${\mathrm{LV}}^{\mathrm{out}}$ and (c) ${\mathrm{CV}}^{\mathrm{in}}$ and ${\mathrm{CV}}^{\mathrm{out}}$ for the SMS data set. The same set of the plots for [(d), (e), and (f)] the Phone data set and [(g), (h), and (i)] the Email data set.

Figure 4

Schematic of computation of the response-time distribution.

Figure 5

Response-time distributions for the SMS, the Phone, and the Email data sets. (a) Normal plots. (b) Comparisons with the response-time distribution obtained from the randomized data sets. (c) The net response functions to an incoming event after subtracting the null-model baseline. (d) Semilog plots fitted by an exponential function.

Figure 6

[(a) and (b)] Phase diagrams of the correlation coefficient $R$ between the ${\mathrm{LV}}^{\mathrm{in}}$ and ${\mathrm{LV}}^{\mathrm{out}}$ values obtained by the numerical simulations of the model [Eq. (4)]. The parameters that are not indicate by the axis are set to $(u_0,a_{\mathrm{self}},{\tau }_{\mathrm{ex}},{\tau }_{\mathrm{self}})=(0.2,-2,0.2,0.5)$. Scatter plots of the ${\mathrm{LV}}^{\mathrm{in}}$ and ${\mathrm{LV}}^{\mathrm{out}}$ values obtained with (c) $(u_0,a_{\mathrm{ex}},a_{\mathrm{self}},{\tau }_{\mathrm{ex}},{\tau }_{\mathrm{self}})=(0.2,1.2,-2,0.2,0.5)$, (d) $(0.2,0.1,1,0.2,0.5)$, and (e) $(0.2,2,0,0.2,0.5)$. These parameter settings are indicated by cross points in the diagrams (a) and (b).

Figure 7

Effect of the number of IETs on the standard deviation of the LV values. For each point, the standard deviation was calculated over 100 instances of event sequences generated from the gamma process with a given number of IETs. The results were well fitted with a power-law function $1.2n^{-0.5}$.

Figure 8

Correlation coefficient $R$ between ${\mathrm{LV}}^{\mathrm{in}}$ and ${\mathrm{LV}}^{\mathrm{out}}$ of the users who have both incoming and outgoing events no less than the threshold value of the number of IETs. The dotted lines show the $95\%$ confidence intervals of the $R$ values.