Figure 1

Illustration of the various sets defined to determine flow persistence and the usefulness of the threshold \(\mathcal{W}\). From the data for flows in the time period \(\mathcal{T}_{<}=[t-\Delta t+1,t]\), we identify those node pairs with a flow equal or exceeding \(\mathcal{W}\) (width of lines represents flow amounts) and construct a set \(\mathcal{E}_{\mathcal{W}}\) whose elements are those pairs. Similarly, in the time period \(\mathcal{T}_{>}=[t+1,t+\Delta t]\) we identify node pairs with flow and build the set of pairs \(\bar{\mathcal{E}}_{1}\). The set of nodes present in the pairs \(\mathcal{E}_{\mathcal{W}}\) is \(\mathcal{N}_{\mathcal{W}}\), and the set of those present in \(\bar{\mathcal{E}}_{1}\) is \(\bar{\mathcal{N}}_{1}\). The common nodes (in red) are \(\mathcal{N}_{\mathcal{W}}^{*}=\mathcal{N}_{\mathcal{W}}\cap \bar{\mathcal{N}}_{1}\). The node pairs of \(\mathcal{E}_{\mathcal{W}}\) and \(\bar{\mathcal{E}}_{1}\) that involve exclusively the nodes \(\mathcal{N}_{\mathcal{W}}^{*}\) are, respectively, \(\mathcal{E}^{*}_{\mathcal{W}}\) and \(\bar{\mathcal{E}}^{*}_{1}\). The intersection (in blue) \(\mathcal{E}^{*}_{\mathcal{W}}\cap \bar{\mathcal{E}}^{*}_{1}\) is the set of node pairs both in \(\mathcal{E}^{*}_{\mathcal{W}}\) and \(\bar{\mathcal{E}}^{*}_{1}\). Our statistical model tests whether the observed \(|\mathcal{E}^{*}_{\mathcal{W}}\cap \bar{\mathcal{E}}^{*}_{1}|\) is larger than expected from chance