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Abstract To understand large, connected systems we cannot only zoom into the details. We also need to see the large-scale features from afar. One way to take a step back and get the whole picture is to model the systems as a network. However, many systems are not static, but consisting of contacts that are off and on as time progresses. This chapter is an introduction to the mathematical and computational modeling of such systems, and thus an introduction to the rest of the book. We will cover some of the earlier developments that form the foundation for the more specialized topics of the other chapters.

1 Introduction Life, at many levels, is about large connected systems. In the biological sense, life is a consequence of macromolecules building cells and carrying information. More mundanely, our everyday life happens in amid a network of friends, acquaintances and colleagues. To understand life, at every level, we need to zoom out from macromolecules or friendships and look at their global organization from a distance. Here, zooming out means discarding the less relevant information in a systematic way. One approach to this, successful the last decade, is network modeling. This means that one focusses on the units of the system, be it proteins or persons, and how they are connected, and nothing else. Of course, this is a very strong

P. Holme () IceLab, Department of Physics, Ume˚a University, 90187 Ume˚a, Sweden e-mail: [email protected] Department of Energy Science, Sungkyunkwan University, Suwon 440-746, Korea J. Saram¨aki Department of Biomedical Engineering and Computational Science, School of Science, Aalto University, FI-00076 AALTO, Espoo, Finland e-mail: [email protected] P. Holme and J. Saram¨aki (eds.), Temporal Networks, Understanding Complex Systems, DOI 10.1007/978-3-642-36461-7 1, © Springer-Verlag Berlin Heidelberg 2013

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simplification. One often has more information about a system that would enrich rather than obscure the picture. One such additional type of information regards when the interactions happen between the units. The essence of temporal-network modeling is to zoom out by excluding all information except which pairs of units that are in contact and when these contacts happen. There is a large number of systems that could, potentially, be modeled as temporal networks. In addition to the cellular processes and social communication mentioned above large technological infrastructures—technologies based on the Internet or mobile-phone networks for example—have both network and time aspects that make them interesting for temporal network modeling. Neural networks—perhaps primarily at a functional level of brain regions that are considered connected if there is a measurable information transfer between them—are another example. A third example is ecological networks and species and their interaction. Such networks—like food webs, depicting which species feed on which other species, or mutualistic networks of species providing mutual benefits,