Markov Random Field networks
The CRF models (with additional conditioning variables) clearly
outperformed the MRF (with only virus occurrences included): the AUC for
MRF was 0.69 while for the CRFs it varied between 0.87 and 0.89. Based
on cross-validation, there were no pronounced discrepancies between the
different CRFs, but the overall performance of the CRFs was better than
that of the MRF model: the 50% quantile for the mean for predicting
both true positives and negatives correctly for the MRF was 0.76,
whereas the corresponding value for the CRF variants was around 0.91.
The MRF predicted more false positives, whereas the CRFs predicted more
false negatives. The mean values for different performance measures are
reported in the Supplementary Results Table S1.
To understand the changes in the network resulting from the addition of
conditioning variables, we compared the virus-virus-associations between
viruses based on the MRF and the different CRF variants. The MRF
revealed mostly positive associations between the viruses (Figure 4A).
After including spatial, habitat and host-related variables (Table 1),
some of the associations between the viruses diminished or disappeared,
and all of the conditional associations were positive (Figure 4B). The
number of significant virus co-occurrences captured by the MRF model was
50 (Figure 4A). The corresponding number for the CRFfull model was 16
(Figure 4B). The CRFs incorporating subsets of conditioning variables
identified intermediate amounts of associations: 30 for CRFhost, 38 for
CRFspat, 28 for CRFhabitat, and 18 for CRFenv.
Although several associations could be explained exchangeably with
habitat- or host-related variables, many associations were also
explained solely by either habitat- or host-related variables (Figure
4C-D). For example, Bromoviridae showed a high number of associations
with other viruses (Figure 4A), but was not explained by host-related or
spatial variables (Figure 4C and E). However, several of its strong
associations with other viruses were explained by the habitat-related
effects (Figure 4D). The 11 association links captured by the CRFfull
model were captured by all the other conditional and unconditional model
variants as well (Figure 4B). We will refer to these associations as
‘permanent’. In this network, Bromoviridae and especially Secoviridae
appeared as hubs, with five association links to other viruses. These
permanent associations represented direct interactions between viruses
that could not be explained with indirect effects of the rest of the
virus community nor any combination of additional conditioning
variables.
Next, to understand how host- and habitat-related variables and spatial
configuration of the hosts influence virus community structure, we
investigated the direct effects of the additional conditioning
variables. All the significant direct effects of the environmental and
spatial variables were for either Caulimoviridae or Geminiviridae (Table
2): e.g., increasing agricultural land use in the surrounding landscape
increased the occurrence probability of Caulimoviridae, and host
population size predicted higher occurrence probability for
Geminiviridae.
None of the indirect effects of the additional conditioning variables
influenced the associations between viruses so that the direction of the
direct virus-virus association would change from positive to negative or
vice versa. However, all conditioning variables except host plant size
and agricultural land use had some effect(s) on some viral associations
(Table 2). In terms of the number of virus-virus-associations
influenced, the most influential indirect effects were the spatial
structure of the host populations (MEMs) and host population
connectivity. All the effects of the first, coarse scale spatial
variable (MEM1) were positive, whereas the effects of the spatial
variables at increasingly finer scales (MEM2-4) were all negative.
Increasing connectivity of the host population had both negative and
positive effects on the virus-virus associations: for example, higher
connectivity lowered the occurrence probability of Avsunoviridae in the
presence of Bromoviridae (and vice versa, symmetrically). There were
altogether eight significant herbivory-related effect, all of them
positive (Table 2).