Fig 1.
Schematic of pork production chains (solid arrows) in the pig-trade network.
Different production chains can be connected by additional cross links (dashed arrows).
Table 1.
Standard network properties of the static German pig trade network.
Diameter and shortest path length are computed for the GSCC.
Fig 2.
Component structure of directed networks.
The giant strongly connected component (GSCC) forms the center of the network (red box). Nodes of the GSCC and the giant in-component (GIC, dashed blue box) have a high spreading potential, whereas all other nodes (GOC—giant out component, dashed yellow box; TEN—tendril; EXT—external nodes, grey dashed box) cannot reach a significant fraction of the network. Box sizes do not reflect the actual sizes of the components. The giant weekly connected component (GWCC) is given by the grey dotted box.
Fig 3.
Relative sizes of the large scale components of the pig trade network.
Sizes are normalized to the total number of nodes in the network.
Fig 4.
Range for every node in the pig trade network.
About 50% of the nodes have a long range of approximately 40,000 nodes, i.e. an infection starting here could reach almost half the network (under the assumption of time-independent links). The other 50% of the network nodes have a short range and cannot cause large outbreaks (maximum short range: 70). The probability distribution of the ranges is shown in the right panel.
Table 2.
Maximum ranges for the large scale structures of the network.
Table 3.
Assortativity coefficients φ of the static German pig trade network.
The network is assortative with respect to federal state, district and municipality and disassortative with respect to node degree. σ is a statistical error estimate. Assortativity coefficients for node categories are computed using Eq (2), for the degree Eq (3) has been used.
Fig 5.
Trade between districts in Germany.
Node sizes correspond to the degree. Edges are bundled with respect to the federal states. Trade is dominated by links between districts in NW and NI and BY and BW, respectively. Self-loops (intra-state trade) are not shown.
Fig 6.
Degree distribution of the livestock trade network.
The out-degree distribution can be approximated by a power law of the form pk ∼ k−μ with μ ≈ 2.7 (estimated using a maximum likelihood approach [49]). The figure shows the cumulative distribution to minimize fluctuations.
Fig 7.
Impact of centrality based node removal, when up to 1% of the nodes are removed.
CD-Degree centrality, CB-Betweenness, CC-Closeness. Size of giant strongly connected component is normalized to unity.
Fig 8.
A temporal network (left) and its static counterpart (right).
Fig 9.
Development of the edge density.
There is a clear trend to edge reduction over time. The edge density of the static network is 3 × 10−5.
Fig 10.
Relative size of the GSCC for different partial aggregation windows.
The sizes show stronger seasonal fluctuations on small time scales (red), but remain rather constant over large time scales (grey). Size is normalized by the number of nodes in G.
Fig 11.
Distribution of node and edge waiting times.
The empirical waiting times cover values over three orders of magnitude. Dashed lines show mean and standard deviation of the node waiting times, respectively. Solid lines show median and 95% quantile of the node waiting time distribution.
Fig 12.
Causal path between nodes i and j in a temporal network.
Although the path Pi⇝k does not exist in the temporal network, this path exists in the static case.
Fig 13.
Range for every node in the temporal network.
The right panel shows the histogram of the y-axis values on a log scale. In contrast to the static case (see Fig 4) the values cover the whole spectrum from minimum to maximum range.
Fig 14.
Accessibility graphs of the network of Fig 8.
All nodes in the static accessibility graph (left) have a path back to themselves (i.e. self-loops, not shown). Note that although the underlying temporal network is undirected, the temporal accessibility graph is directed.
Fig 15.
Unfolding accessibility for node 1 in the network shown in Fig 8.
a Accessibility of node 1 including time stamps when nodes are accessed. b Number of infected nodes over time. c Infection curve (i.e. range) for source node 1.
Fig 16.
Unfolding accessibility of the temporal pig trade network.
The mean herd prevalence (grey solid line) is given by the path density. The probability density function (PDF) of shortest path durations (blue dotted line) shows a global maximum at about 120 days.
Fig 17.
Causal fidelity for different aggregation windows.
For aggregation windows < 365 days, the static network representation should be used with caution. A static network view is also adequate for very short aggregation windows (inset). The static network considered in this work corresponds to the rightmost datapoint, i.e. the maximum possible aggregation window.