Abstract
The average topological overlap of two graphs of two consecutive time steps measures the amount of changes in the edge configuration between the two snapshots. This value has to be zero if the edge configuration changes completely and one if the two consecutive graphs are identical. Current methods depend on the number of nodes in the network or on the maximal number of connected nodes in the consecutive time steps. In the first case, this methodology breaks down if there are nodes with no edges. In the second case, it fails if the maximal number of active nodes is larger than the maximal number of connected nodes. In the following, an adaption of the calculation of the temporal correlation coefficient and of the topological overlap of the graph between two consecutive time steps is presented, which shows the expected behaviour mentioned above. The newly proposed adaption uses the maximal number of active nodes, i.e. the number of nodes with at least one edge, for the calculation of the topological overlap. The three methods were compared with the help of vivid example networks to reveal the differences between the proposed notations. Furthermore, these three calculation methods were applied to a real-world network of animal movements in order to detect influences of the network structure on the outcome of the different methods.
Highlights
In contrast to the static situation, the time when edges are active and especially the chronological order of contacts play an important role in temporal networks
In previous studies dealing with network analysis, the temporal information has been partly neglected by an aggregation of contacts over specific observation windows, which have been analysed separately
The intention of this article was to eliminate uncertainties for the calculation of the topological overlap and the temporal correlation coefficient proposed by Nicosia et al (2013) and its extension proposed by Pigott and Herrera (2014) and to give clear definitions of the network parameters used for their calculations
Summary
In contrast to the static situation, the time when edges are active and especially the chronological order of contacts play an important role in temporal networks Both are essential elements for the representation of these dynamical systems (Holme and Saramäki 2012). The temporal correlation coefficient (hereinafter abbreviated C) is a measure of the overall average probability for an edge to persist across two consecutive time steps (Nicosia et al 2013; Tang et al 2010). For the calculation of the temporal correlation coefficient, the average topological overlaps of the graph which measures the amount of changes in the edge configuration between two consecutive time steps are determined. The newly proposed adaption, hereinafter referred to as Method 3, uses the maximal number of active nodes, i.e. the number of nodes with at least one edge, for the calculation of the topological overlap. Influences of the network structure on the differences between methods were statistically analysed
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