The complex networks perspective has become an important methodological framework for both the modeling and analysis of complex systems. However, the vast majority of studies on complex networked systems is based on a static network perspective, in which links and nodes are ssumed to exist at any point in time. In contrast to this view, the availability of high-resolution time-stamped data on networked systems has recently validated the intuition that real complex systems are often highly dynamic, i.e. links are not active continuously but rather occur in specific temporal patterns. The following video shows two examples for temporal networks, which have the same time-aggregated topology, but different temporal patterns. The example further shows that these different temporal patterns can have dramatic effects on dynamical processes like, e.g., diffusion.
The fact that we currently lack a suitable methodological framework which integrates both the temporal and the topological dimension of complex systems severely limits the understanding of dynamic complex systems. Motivated by this limitation, empirical studies of temporal networks with time-varying topologies enter the focus of a growing research community. The few existing studies on temporal networks occurring in social, economic and technological systems have generally highlighted the presence of statistical inhomogeneities in terms of broad distributions of the waiting times between consecutive node or link activities or the duration of individual contacts. Through our own research, we recently added an important additional perspective, namely that interactions in many complex systems occur in specific temporal orders and that this ordering is essential to understand dynamical processes on temporal networks. For example in communication processes, in order for an information to be transferred from node A via node B to node C, it is crucial that the interaction between A and B happens before the interaction between B and C.
To study effects that are due to the order in which interactions occur, in our most recent research we have investigated models that preserve non-Markovian properties of contact sequences, i.e. the fact that in many real-world systems the next interaction of a node is not independent of with whom the node has interacted shortly before. We have shown that such non-Markovian properties give rise to effective interaction topologies that significantly differ from what would be expected from a static network representation. We have further demonstrated that, due to these order correlations, extrapolating findings from time-aggregated networks to time-varying networks can lead to significantly wrong statements about dynamical processes like diffusion or information propagation, as well as about the importance of individual nodes. To quantify this discrepancy, we developed an information-theoretic measure (betweenness preference) which allows to quantify the presence and strength of "preferred contact sequences" in longitudinal network data and how this influences dynamical processes. We further demonstrated the usability of this novel measure in a number of real-world data sets from social, technical and biological contexts.
Most recently, we have developed a powerful framework for the modeling and analysis of temporal networks based on so-called "higher-order aggregate networks", which can be seen as a generalisation of the static abstraction commonly used in network analysis. This approach allows to study dynamical processes in temporal networks using standard techniques like, for example, a dynamic analysis in terms of the eigenvalues of a modified Laplacian matrix, without losing information on the ordering of interactions which are hidden in the time dimension.
Our research on temporal networks has been featured in top-tier scientific journals like Physical Review Letters and Nature Communications. Our unique approach not only allows us to quantitatively study a previously unexplored temporal-topological dimension of complexity in time-stamped data. It also provides a methodological framework that offers broad perspectives for the development of new data mining and visualisation techniques which improve our ability to extract knowledge from dynamic networked systems.
Apart from this fundamental research, we are also actively involved in the actual (technical) implementation of our methods in terms of software packages. We have recently released the open source data mining package pathpy, which implements our methods. We further have a dedicated third-party funded project targeting at making our research usable in industrial applications.