Funded Grants

The nature of time in complex systems

In 1917 the Danish mathematician and engineer Agner Erlang published a landmark manuscript, which had a fundamental impact on today's telecommunication industry. The paper addressed an apparently simple question: how many phone lines a company needs to the outside world? Consider an institution with 2000 employees, each with a phone on its desk. If all employees were to call outside at the same time, the institution would need 2000 lines to serve them all. In reality, however, only a small fraction of the employees talk at any given moment. Erlang made the assumption that the chance that we are on the phone in a given moment has a fixed probability, concluding that if everybody talks on average for two minutes with probability 1/30, only 87 lines are sufficient. Today engineers use Erlang's formula each time they install a new phone system or design a wireless antenna for mobile phones in a neighborhood.

Erlang's formula is based on the simple but widely held assumption that the events taking place in society and nature are uniformly distributed in time, well described by Poisson processes. This means that my chance of making a call in the next five minutes is the same as in a five minute interval two hours from now. Rate equation models of reactions assume that molecules in a cell bump into each other at a uniform rate. The spread of viruses is modeled under the assumption that an infected person's probability to pass on the virus to a healthy individual follows a Poisson process. The bandwidth of Internet routers is determined using the assumption that our emails are sent at a uniform rate. Yet, recently there is increasing evidence that the timing of events in complex systems is far from being uniform. For example, we send several emails in close succession, followed by long periods of no email activity. Seabirds fly long stretches between rapid burst of fishing patterns. Intensive burts of earthquakes are followed by long quiet periods. Thus for all practical purposes time does not flow uniformly in complex systems. It matters when things happen.

The observed inhomogeneous time distributions indicate a fundamental lack of understanding of what determines the timing of events in a complex system. We tend to believe that molecules in the cell react in an unpredictable fashion, that people spontaneously engage in conversations or that a computer virus spontaneously spreads to other users. But spontaneity isn't random. Bacteria in a glucose rich environment cannot create a fructose-6-phosphate molecule until glucose-6-phosphate has been has been synthesized. You will not make that phone call or send that email until you have all the information for the communication. In most complex systems events are highly interdependent, and this interdependency drives their timing as well.

The questions we plan to ask during this research are apparently simple: Are there laws that govern the timing of events taking place in complex systems? What are the fundamental limits of event predictability? To answer them we plan to study three classes of systems, offering an increasing range of challenges and research opportunities. We first plan to focus on designed systems, like a computer chip or a software package, whose internal dynamics is fully deterministic, but whose activity patterns are driven by external inputs with unknown nature. Extensive datasets capturing the time dependent activity of millions of gates and the activity patterns of several large software packages will allow us to study the nature of time and predictability in designed systems using a combination of empirical and modeling tools. Second, we will focus on human activity patterns, as recorded by anonymized telephone, email and web browsing records, allowing us to explore the coupled activity of millions of individuals over considerable time frames. The wealth of the available empirical data will allow us to uncover the universal statistical signatures of event timing. The experience we gain from exploring these well mapped designed and social systems will be used to study the timing of the molecular events within the cell, focusing on selected regulatory pathways whose reaction parameters and mechanisms are known. Current advances in fluorescent microscopy that can capture reactions at the single molecule level offer hope that the predictions we develop can be experimentally tested.

While predicting the timing of events may sound like science fiction, in reality our goal is somewhat more limited: we are not planning to predict events of unknown nature, but focus only on events that are known to take place with some regularity in a given system. In this respect the real challenge is to develop the tools to approach the timing of events in a systematic fashion. A successful research program in this direction could take complexity research into a new direction, moving from the static network based approaches that have dominated our thinking about complex systems in the past five years to uncovering what is going on in real systems, and understanding how the components of the system are utilized and activated in time to generate the observed complex behavior.

We often think of complex systems as networks whose links support material or information flow. Yet, given the small copy number of transcription factors in a cell, this flow picture is highly unrealistic and should be replaced by rarely occurring molecular interactions. Similarly, the strong ties characterizing social or professional networks in reality represent the sum of brief communications. How do we rethink complexity to take into account the discreteness of the individual events? How do we approach the nature of time in a complex system? While today these questions are rarely asked, I believe that they will dominate our thinking about complexity in the coming decade. It is the goal of the research proposed here to lay the scientific groundwork for such thinking, potentially impacting a number of fields that regularly deal with complexity, from human dynamics to cell biology.