Scientific Method / Science & Exploration

Mathematics of text messaging could help cellcos manage networks

Half of the cellphone users out there display a pattern of communications that …

Human communication patterns, particularly in the form of text messages, appear to follow a pattern that's a fusion of two mathematical models. In a paper published in PNAS on Monday, researchers analyzed a large set of text messages from service providers and found most users trade over 90 percent of their messages in bursts with only one other person, followed by an exponential drop into silence. The pattern may be due to interplay between the number and priority of tasks the two parties are trying to accomplish together, and the math behind them may help service providers manage their resources.

Scientists have lately been swimming in human communication data, from emails to texts and phone calls. As more and more people get cell phones and computers, all this information is begging to be analyzed for the benefit of mobile and Internet service providers, not to mention for its value as a wider indicator of human behavior.

To see if they could identify a communication pattern, a group of researchers culled data on text messages from three different mobile service companies over the course of a month. Specifically, they were interested in the length of time between messages, and wanted to know if there is a typical pace.

When they looked at the data, they noticed that 50 percent of users send 90 percent of their text messages to the same person. Since this covered a large chunk of all the text message business, they used this subset to see how time between messages was distributed.

They found that text-messaging patterns of individual users are bimodal, or are described by two different mathematical models. Text message sets often start off with a burst: the times between messages are short and follow a power-law distribution (that is, there are a lot of text messages with short intervals between them).


This may sound like a familiar pattern to anyone who carries on conversations: you come up with a topic, talk about it until you both run out of things to say, and taper off into silence. But the authors went a step further, and tried to reverse engineer the power law/exponential distributions of these text message conversations to better grasp why conversations happen this way.

They learned that the graphs could be reproduced if they assumed that the two correspondents were talking about some kind of task that they could both contribute to. A conversation would start when there was a "task," or topic of discussion, that required the action of both members, and one conversant brought it up with the other (for instance, two people in a relationship deciding what to eat for dinner).

The initial text message puts the task on both conversants' to-do lists. The authors next assigned a probability that the recipient of the first message would care about the task and respond. A 100 percent chance of caring would result in endless text messages back and forth; likewise, a zero percent probability never produces a response.

In the model, the size of the text message bursts were tunable by this response probability. The larger the burst, the higher priority the task for both parties, with all exchanges of a priority between 100 and zero eventually tapering off.

This also may explain why most texters overwhelmingly favor one texting partner: the two may happen to share a large number of high-priority tasks. Even the best of friends run out of chatting steam regularly (a few times a day, according to the records) but, because they have so many conversations, the texting still manages to occupy a lot of each others' time.

The data and analysis probably doesn't cover the majority of the text messages traded over a network. But it does describe a sizeable chunk of them. The authors speculate that this burst-and-taper pattern of text messages also applies to other information ecosystems like e-mail and trading, and that service providers could potentially allocate their bandwidth management or phone line availability based on this pattern.

PNAS, 2010. DOI: 10.1073/pnas.1013140107  (About DOIs).

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