Tore Opsahl

Longitudinal Networks

Networks evolve over time by the addition and removal of nodes, and the forming, strengthening, weakening, and ultimately, the severing of ties. For example, one network that is currently receiving a great deal of attention in the literature is the network of commercial airports (the nodes) that are tied together by scheduled flights (Barrat et al., 2004). This network grows when new airports open and shrinks when old ones close down. Ties are created when new routes are started and reinforced if the capacity of an existing route is increased. Weakening of a tie occurs when airlines cut capacity, which ultimately results in the severing of the tie if all airlines terminate flights on the route.

Datasets are often collected through surveys or interviews at a single point in time. Consequently, the evolution of networks is typically not recorded (i.e., static). As such, most network measures have been developed for the purpose of analysing cross-sectional networks. However, for certain types of analyses, knowing the evolution is key. This is particularly the case when studying tie formation in networks. In a static network, it is not appropriate to model the existence of ties based the neighbouring ties as ties are not independent of each other. In fact, the network is a reflection of the dependency structure. One way to overcome this obstacle is to employ highly complex simulations to infer the likelihood of tie formation based on structural configurations (Robins and Morris, 2007). Another approach is to observe the network at multiple times and simulating the changes between panels using stochastic actor-oriented models (SAOMs; Snijders, 1996, 2001). This latter approach has allowed for studies of both selection and influence in social networks, and thereby, giving insights into whether behavioral process emerge from or contribute to network formation. One known limitation of SAOMs is that they infer continuous time process as it only observes discrete snapshots of the network.

In recent years, there has been a growing use of networks where the time of tie creation is recorded (Brandes et al., 2009; Butts, 2008; de Nooy, 2011; Kossinets and Watts, 2006; Opsahl and Hogan, 2011; Stadtfeld, 2010). These networks are often referred to as relational event, timestamped, continuously-observed, or longitudinal networks. There are multiple types of longitudinal networks. They might be one-mode or two-mode, binary or weighted, have time information for ties only or also for nodes, and information of tie creation/reinforcement but not on weakening/severing. Read more…

Create Snapshots
A common way of analysing richer types of networks is to create static snapshots at a fixed interval (e.g., daily, weekly, or yearly; Kossinets and Watts, 2006; Panzarasa et al., 2009). This process generates insights into how network properties change over time, such as the average degree or clustering coefficient. A key benefit of this approach is that any static network measure can be applied to the network. This page shows how to apply some common network measures. Read more…

Sliding Window
Many longitudinal networks are collected using archival data. These datasets often do not contain information on the weakening and severing of ties. If this is the case and the network is analysed directly, there is an assumption that relationships, once established, never decay. This can be overcome by introducing a sliding window that removes ties after a set amount of time (Kossinets and Watts, 2006; Panzarasa et al., 2009). The length of the window is crucial in determining which past events are taken into account to generate the network structure at a given point in time. By analysing which past events are relevant to the current state of the network, the length of the window can be defined. An ill-defined sliding window will have the effect of, either breaking continuous social interactions into independent sets of interactions, or combining two separate interactions into a single one. Read more…

Network Evolution
Understanding how networks evolve has been a quest in network analysis. However, static network data is far from ideal when trying to achieve this goal. It is first with longitudinal data that we are able to detect the underpinning principles of tie formation in large-scale networks. In recent years, a number of frameworks has been developed for various types of longitudinal data (Brandes et al., 2009; Butts, 2008; de Nooy, 2011; Kossinets and Watts, 2006; Opsahl and Hogan, 2011; Stadtfeld, 2010). Although this page will focus on my work in this field (Opsahl and Hogan, 2011), it will try to include the other frameworks were applicable. Read more…

Random Longitudinal Networks
Similarly to weighted networks and two-mode networks, random networks play an important role when analysing longitudinal networks. After estimating growth parameters on an observed networks, it is possible to create random networks based on these parameters and then test whether the random networks resemble the observed networks. This would be similar to goodness-of-fit for SIENA models. Read more…

References

Barrat, A., Barthelemy, M., Pastor-Satorras, R., Vespignani, A., 2004. The architecture of complex weighted networks. Proceedings of the National Academy of Sciences 101 (11), 3747-3752. arXiv:cond-mat/0311416

Brandes, U., Lerner, J., Snijders, T.A.B., 2009. Networks evolving step by step: Statistical analysis of dyadic event data. In: 2009 International Conference Advances in Social Network Analysis and Mining (ASONAM 2009), IEEE Computer Society, pp. 200-205.

Butts, C.T., 2008. A relational event framework for social action. Sociological Methodology 38, 155-200.

de Nooy, W., 2011. Networks of action and events over time: A multilevel discrete-time event history model for longitudinal network data. Social Networks 33, 31-40.

Frank, O., Strauss, D., 1986. Markov graphs. Journal of the American Statistical Association 81, 832-842.

Kossinets, G., Watts, D. J., 2006. Empirical analysis of an evolving social network. Science 311, 88-90.

Opsahl, T., Hogan, B., 2011. Growth mechanisms in continuously-observed networks: Communication in a Facebook-like community. arXiv:1010.2141.

Panzarasa, P., Opsahl, T., Carley, K.M., 2009. Patterns and dynamics of users’ behavior and interaction: Network analysis of an online community. Journal of the American Society for Information Science and Technology 60 (5), 911-932.

Robins, G.L., Morris, M., 2007. Advances in exponential random graph (p*) models. Social Networks 29 (2), 169-172.

Snijders, T.A.B., 1996. Stochastic actor-oriented dynamic network analysis. Journal of Mathematical Sociology 21, 149-172.

Snijders, T.A.B., 2001. The statistical evaluation of social network dynamics. Sociological Methodology 31, 361-395.

Stadtfeld, C., 2010. Who communicates with whom? measuring communication choices on social media sites. Proceedings of the 2010 IEEE Second International Conference on Social Computing (socialcom), Minneapolis, MN, 564 – 569.

Wasserman, S., Pattison, P.E., 1996. Logit models and logistic regression for social networks: I. An introduction to Markov graphs and p*. Psychometrika 61, 401-425.