Fig 1.
Three examples of networks where classical measures of degree and strength give same results.
Three different situations with the same DC and SC are illustrated: three focal nodes (X, Y, Z) are connected to the same number of nodes (DC = 5) with the same nodes’ strength (SC = 100). The strength of each focal node is distributed among linked nodes in different ways: uniformly (X node) or to a fewer number of nodes (case Y and Z nodes).
Fig 2.
Empirical cumulative weights’ distribution (Fc) for nodes X, Y, Z.
Graphical representation of the Area Under the Curve AUCFc of weights relative to nodes X, Y and Z of Fig 1. The AUCFc(3), the area related to the first 3 links of Y node is highlighted in grey.
Fig 3.
An example of weighted network and the related weighted adjacency matrix and transition matrix.
The transition matrix shows how the number of animals that come out from a node is distributed to the nodes with which it is connected.
Table 1.
Symbology adopted throughout the text to define a generic matrix A.
Fig 4.
DCIN and WDCIN values for IT2009 data.
The WDCIN (on the right) shows a more evident variability among regions than DCIN (on the left). The scale bars report the DCIN and WDCIN values for each region. In case of Friuli-Venezia-Giulia region, in the north-east part of Italy, the two measures are particularly different (DCIN = 17 and WDCIN = 1.73). Arrows representing in-going links are reported (17 regions), but only 5 regions already cover the 99% of the weights (fi values are reported on the corresponding region).
Fig 5.
Correlation between the two measures DC and WDC of the IT2009 network.
The correlation shows a moderate agreement between DC and WDC (0.55 ‘in’ and 0.31 ‘out’). For example, in 5 regions the same value of 19 for out-degree, is associated to a weighted out degree range of 2–7 and vice-versa 5 regions with values of weighted out-degree between 2 and 2.5, have values of out-degree ranging from 9 to 19.
Fig 6.
Correlation between simulation model results and degree centrality measures in the assumption that all nodes have the same population size.
In the upper side of the figure the correlation between the node’s risk (the mean number of infected nodes) and DCOUT (a) and WDCOUT (b) is showed; the last two graphs report the correlation between node’s vulnerability (the mean number of times a node gets infected) and the DCIN (c) and WDCIN (d).
Fig 7.
Correlation between model results and the strength centrality measures in the assumption that real values of population and number of moved animals are adopted.
In the upper side of the figure the correlation between the node’s risk (the mean number of infected nodes) and SCOUT (a) and WSCOUT (b) is showed; the last two graphs report the correlation between node’s vulnerability (the mean number of times a node gets infected) and the SCIN (c) and WSCIN(d).
Table 2.
Animals moved to Friuli-Venezia-Giulia region (percentages) in years 2008 to 2010.
Fig 8.
Correlation between model results and the strength centrality measures for the randomized network.
In the upper side of the figure the correlation between the node’s risk (the mean number of infected nodes) and SCOUT (a) and WSCOUT (b) is showed; the last two graphs report the correlation between node’s vulnerability (the mean number of times a node gets infected) and the SCIN (c) and WSCIN(d).
Table 3.
Limit values of R, varying SC and DC in their domain.