Entropy of dialogues creates coherent structures in e-mail traffic

Abstract

We study the dynamic network of e-mail traffic and find that it develops self-organized coherent structures similar to those appearing in many nonlinear dynamic systems. Such structures are uncovered by a general information theoretic approach to dynamic networks based on the analysis of synchronization among trios of users. In the e-mail network, coherent structures arise from temporal correlations when users act in a synchronized manner. These temporally linked structures turn out to be functional, goal-oriented aggregates that must react in real time to changing objectives and challenges (e.g., committees at a university). In contrast, static structures turn out to be related to organizational units (e.g., departments).

Fig. 1.

Spike trains for the three communication channels determining the edges of a triangle formed by users A, B, and C. We have I_p(A,B) = 0.095, I_p(A,C) = 0.394, I_p(B,C) = 0.172, and I_T = 1.606. It is important to note that I_p and I_T capture the synchronization of the e-mail exchange at two different levels. I_T measures the coherence of the triangle as a whole and can take on high values, even though some of the I_p values are relatively small. The horizontal line at the bottom represents the whole period under analysis divided into days (and weekends) and was introduced to aid the visualization of the events determining the probabilities entering I_p and I_T (see Eqs. 1 and 2).

Fig. 2.

Probability distribution of the response time until a message is answered (as described in The Model). (Inset) The same probability distribution is measured in “ticks,” i.e., units of messages sent in the system. Binning is logarithmic. Solid lines approximately follow Δt^-1 and are meant as guides for the eye.

Fig. 3.

Graph of static and temporal statistical quantities. Probability distribution of the number of triangles T in which a user participates. Circles indicate static triangles, whereas crosses indicate temporally coherent triangles (i.e., mutual information I_T ≥ 0.1). Both lines are well fitby N ≈ T^-1.2, and the black line is provided as a guide for the eye with this slope.

Fig. 4.

The static structure of the graph of e-mail traffic obtained from our data, arranged according to curvature, based on triangles of mutual recognition. Time is thus not taken into account, and the graph of users arranges itself primarily according to departments, shown in various colors.

Fig. 5.

The conjugate graph, for a cutoff of 0.5. Each node is a triangle of three people conferring with temporal coherence I_T ≥ 0.5 (rather than a single user, as in Fig. 4), and each link connects two adjacent such triangles. The three colors of each node indicate the departments of the three people composing that node (we used the same color code as in Fig. 4). Note the strong clustering of the graph into very compact groups of people. The users cross departmental boundaries (their interests and connections are not shown, in consideration of privacy).

Fig. 6.

Cumulative probability graph for I_T. The solid lines are fits to two different exponentials, and they cross at the critical point of I_T ≈ 0.65. Axes are semilogarithmic. α is the threshold value.

Fig. 7.

Comparison of dynamic (red) and static (blue) groups. (Upper) Time sequence for the 83 days on which data were taken, showing two groups of comparable size (dynamic group, 6 participants and a total of 505 events over the whole period; static group, 10 participants and a total of 389 events), with the sum of the activity normalized to 1. (Lower Left) Graph of events for the two groups shown in Upper, after removing weekends. (Lower Right) Averaged graph for all dynamic and all static groups after rescaling (on the x axis) by the maximal number of events and normalizing the distribution (on the y axis) to unity. norm. units, Normalized units; arb. units, arbitrary units.