Matt Michalska-Smith
Lindsey Otto-Hanson, Georgiana May
Elizabeth Borer, Eric Seabloom
Linda Kinkel

This talk was presented at the annual meeting of the Ecological Society of America in Louisville on 14 August, 2019. Text here will supplement slide material.

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Experiment overview
Network analysis
Summary / Conclusions

Control

NPK+ Supplemented

Control

NPK Supplemented

Experiment overview
Network analysis

Degree distributions
Network structures
Triads

Summary / Conclusions

Perhaps the simplest measure of network structure is the distribution of interaction strengths, termed the degree distribution. These histograms include all of the pairwise interaction strengths, that is, the degree of niche overlap between each pair of isolates. We subdivide them according to node type (columns) and host plant treatment (rows).

Of course, we can subdivide this a little more, if we consider the kingdoms of both the source and the target isolate:

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Note that, except in the case of the Bacterial effect on Fungal isolates, each of these distributions shows a strong right skew — there are more high values than low values. Also note that, with the exception of the effect of bacterial isolates on one another, these distributions all differ significantly between the two treatments (You can see the results of the relevant statistical tests in a supplemental slide here, by pressing Spc, PgDn, or n)

Interaction Type	NPK Supplemented	Control	T-test p-value	KS-test p-value
Bacteria → Bacteria	0.661	0.659	0.896	0.116
Bacteria → Fungi	0.494	0.408	< 0.001	< 0.001
Fungi → Bacteria	0.731	0.831	< 0.001	< 0.001
Fungi → Fungi	0.656	0.724	< 0.001	< 0.001

One step up in complexity of network structure is to consider the relationship between incoming and outgoing links in the same node. Here we plot the in-degree against the outdegree for each node in our netwokrks, dividing nodes according to the node's kingdom and the treatment of the host plant.

Note that all of these lines have negative slopes — nodes with high indegrees tend to have low outdegree and vice versa. Put another way, there are some isolates which we think of as strong competitors: they overlap many other isolates' niches, but do not often have their own niche significantly overlapped; and there are some isolates we could think of as weak competitors: overlapping few others, yet having many overlap their own niche.

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Similarly to the histograms before, we can break these trends down by the type of interaction as well. Here, we color the points as either between two bacterial isolates (blue), between two fungal isolates (red), or an inter-kingdom link (purple). All slopes are still significantly less than 0. Moreover, while the slopes differ significantly between interaction types, they do not differ between treatment.

Treatment	Interaction Type	Formula	p-value	Adjusted $R^2$
Control	Bacteria → Any	indegree ~ 19.252 · outdegree - 0.508	< 0.001	0.767
	Bacteria → Bacteria	indegree ~ 9.467 · outdegree - 0.597	< 0.001	0.743
	Bacteria → Fungi	indegree ~ 10.072 · outdegree - 0.449	< 0.001	0.733
	Fungi → Any	indegree ~ 19.996 · outdegree - 0.649	< 0.001	0.623
	Fungi → Bacteria	indegree ~ 10.582 · outdegree - 0.79	< 0.001	0.511
	Fungi → Fungi	indegree ~ 9.991 · outdegree - 0.558	< 0.001	0.643
NPK Supplemented	Bacteria → Any	indegree ~ 17.442 · outdegree - 0.385	< 0.001	0.546
	Bacteria → Bacteria	indegree ~ 8.765 · outdegree - 0.474	< 0.001	0.589
	Bacteria → Fungi	indegree ~ 8.839 · outdegree - 0.31	< 0.001	0.369
	Fungi → Any	indegree ~ 18.355 · outdegree - 0.569	< 0.001	0.673
	Fungi → Bacteria	indegree ~ 8.676 · outdegree - 0.512	< 0.001	0.471
	Fungi → Fungi	indegree ~ 9.546 · outdegree - 0.617	< 0.001	0.772

Term	df	sum squared error	mean squared error	p-value
treatment	1	3.989	3.989	0.419
interaction_type	3	150.053	50.018	< 0.001
treatment:interaction_type	3	6.4	2.133	0.789
Residuals	432	2636.528	6.103
Adjusted $R^2$	0.04209

While this framework of measuring the degree of niche overlap between each pair of isolates provides a high level of detail about the structure of iteractions across the community, many of the most popular metrics of network structure are defined only in the case of binary networks, that is networks where the links only indicate presence or absence of an interaction, and do not provide information on the strength of any given interaction.

To convert our weighted interactions into binary ones, we need only define a threshold

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Above which, we say the degree of niche overlap is meaningful, drawing a link between the two isolates, and below which we ignore the interaction as being too weak to be biologically relevant. For the following analyses, we use a, somewhat arbitrary, cutoff of 0.75.

Control

NPK Supplemented

If we look back at our networks from before, applying this threshold severely reduces the number of links in each network:

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and allows us to better visualize the structures that are present. (though this is, in general a rather unreliable way to evaluate the presence or absence of complex network structures). For instance, it appears that nodes of the same kingdom are clustering in these layouts. We can test for the significance of such arrangements by first clustering the nodes of the network using a method agnostic to node kingdom, and then checking whether or not the kingdoms are disproportionally represented in these clusters.

We find that for control plants, nodes cluster according to kingdom in the weighted networks, but not in the binary networks (despite the appearance of the binary networks inspiring this analysis!). For the NPK plants, we see signficant clustering in both the binary and weighted networks but again, more consistently in the weighted case.

Metrics of local network structure are calcuated for each node of the network, so we can use then to compare bacterial and fungal isolates within each network.

One of the most commonly utilized network structures is that of centrality. There are many different formulations for centrality, but they all boil down to assigning importance to each node in the network. Where they differ is in how they define "importance". Here, we report two measures of centrality that have some biological relevance to the system we are working with: degree and closeness.

An isolate with high degree centrality has the potential to directly influence many of its neighbors, but an isolate with high closeness centrality has greater potential to spread its influence throughout the network as a whole.

We also report differences in the clustering coefficient of each node. This value is calculated by counting how many of an isolate's partners interact with each other, forming triangular patterns in the interaction network.

Each box in this table is a network, colored according to the sign and significance of differences between isolates of each of the two kingdoms: blue indicates that bacteria have larger values for this metric, red indicates fungal isolates do. Increasing darkness signifies smaller p-values: < 0.05 (light), < 0.01 (medium), and < 0.001 (dark). As can be seen, bacteria and fungi differ often in the control plants, but very rarely in the nutrient supplemented ones. This suggests some sort of equalizing force being applied in the NPK treatment.

For network-wide structures, we calculate a single value for each network, precluding inter-kingdom comparisons, so here we are instead comparing metrics across treatments. Because of this, we only have two comparisons for each metric: the binary and the weighted case. As before, we color each box in favor of the larger value, with increasing significance being depicted by increasing color intensity.

When we compare the 6 values for each of the control networks to the 6 values for NPK in each of these metrics (using t-tests), we see very few differences in structure between treatments. That is, control and NPK plants tend to have similarly structured networks, despite the observed differences in degree distribution.

(Note that these results are qualitatively independent from the inclusion/exclusion of the outlying control network)

One of the issues with comparing networks directly is that there can be confounding differences for instance in size and connectance that are not as biologically relevant. to get around, do randomizations
explain table
many "significant" differences, but they are quite small

Whle comparing two sets of networks directly has the potential to provide insights about differences between treatments, there is also a risk of detecting differences that are less biologically relevant due simply to differences in size or connectance between the groups.

This can be overcome by comparing networks against randomizations of their own structure, where the randomization protocol perserves some aspects of the structure, but not others. Here, we use a configuration model randomization, which preserves the degree distribution, but allows larger scale structures to vary more freely.

Again, we indicate each network by one icon in each row. There are twenty-four icons because we count each network twice: once to depict its p-value (that is, what proportion of random networks are less than or greater than the empirically observed value for that web), and again to depict the z-score (that is, how much larger or smaller than the mean of the randomizations is the empirically observed value). Empirical values larger than the expectation are depicted with a or , for p-values > 0.95 or > 0.99, respectively and z-scores > 2 or > 4, respectively. Analogously, empirical values smaller than the expectation are depicted using or . Hyphens indicate nonsignificantly different empirical values and x's indicate cases where there were insufficient unique randomizations to calculate p-values or z-scores (here we use a cutoff of 20 unique randomizations).

Note that, while there are many significant differences in terms of p-values, they are almost all very small in terms of z-score.

Experiment overview
Network analysis
Summary / Conclusions

Kingdom comparisons
Treatment comparisons
Take-home

Degree distributions differ significantly between interaction types

in particular, fungi tend to have a larger impact on bacteria than vice versa (especially for control plants)

Isolates cluster by kingdom

especially in weighted networks (though more noticeable in binary!)

Bacteria and fungi differ in their node-specific structure only in control plants

Degree distributions also differ significantly beween treatments

But...

There are few / weak differences in network-wide structure between treatments
(Binary) networks show many significant deviations from their random expectations

but these differences tended to be small

Nutrient ammendments level the playing field

Fungi tend to have wider niches and greater overlap on bacterial isolates
But these differences are attenuated in NPK plants

Despite pairwise differences, differences not visible in larger network structure
These small resource competition networks seem to be dominated by pairwise effects

No evidence of higher-order interactions

Control

NPK+ Supplemented

Control

NPK Supplemented

Control

NPK Supplemented

Comparing isolate kingdoms

Comparing network treatments

Comparing network treatments — Triad counts

Comparing network treatments — against random

Triad counts — against random

Comparing Kingdom

Comparing Treatment

But...

Conclusions

Thank you!