University of Chicago

Department of Ecology & Evolution

Matthew Michalska-Smith

Stefano Allesina
Trevor Price
Mercedes Pascual
Sarah Cobey
Amos Maritan

This talk was given on 2 May 2018 at the University of Chicago as the public defense of my Ph.D. Dissertation. Text here will supplement slide material.

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Ecological networks: a brief history
Are ecological networks structurally distinct?
Do trophic roles have structural signatures?
How ought competitive networks be structured?
Where do we go from here?

Ecological Networks: a brief history

Networks as a unifying methodology
Ecological networks
Network structure

Are ecological networks structurally distinct?
Do trophic roles have structural signatures?
How ought competitive networks be structured?
Where do we go from here?

Here we are reminded that in nature
nothing exists alone

—Rachel Carson, Silent Spring

Ings & Hawes 2018

Size (number of nodes)
Connectance (proportion of possible links realized)
Degree distribution (how many connections does each node have?)

The area of network science that most fascinates me is network structure. As networks became more commonplace, a natural question arose of how to compare networks to one another. Initially, scientists would just look at differences in the core components of networks, starting with size, that is, the number of things in the network. Represented here by these circles,

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The second core component of a network is connections between the nodes, usually termed "links" or "edges." Here, these are depicted with lines connecting the dots. notice that nodes can interact with themselves to produce what are called "self-loops" and that these edges (while undirected and all of the same strength here) can also convey this more sophisticated information in the general case. A summary statistic about edges that is often looked at in network science is the connectance which is just the proportion of all edges possible that are actually present in a given network. Here, that value is about $0.5$

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One step up in complexity of network structure is to look at properties of individual nodes in the network, for instance their degree which is just another way to say the number of links connected to that node.

Nestedness

Modularity

Bipartite

Weitz, et al. 2013
Wright & Reeves 1992
Ulrich, Almeida-Neto, & Gotelli 2008
Newman 2006

Especially in ecology, two "global" network structures have dominated investigations into network structure. These structures involve every node in the network, and are usually summarized with a single value.

The first of these popular ecological structures is "Nestedness" which is a attempt to quantify the degree to which the interactions of specialists are a subset of generalists' interactions. This results in a triangular network representation as seen in this modified figure of phages infecting bacteria from Joshua Weitz's group at Georgia Tech.

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The second structure is called "Modularity," also called compartmentalization. This structure asks to what degree can species be divided into subgroups which interact strongly within the group, but weakly (or not at all in this toy example) with members of other such groups.

For each of these structures, multiple methods of quantification have been proposed, but they are, in-general, computationally intensive to do well

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As a side note, these networks have an additional structure that comes up throughout this talk: they are bipartite. This means that, unlike the toy network in the previous slide, in which any node can be connected to any other node, in these networks, the nodes can be divided into two classes for which all interactions are between nodes of disparate class. In this example, all links connect a virus to a bacterium; there are no links between two viruses or between two bacteria.

Milo et al. 2002
Borrelli 2015

While the examination of the properties of small webs in isolation dates back decades, more recently these subgraphs or "motifs" have been interpreted as "building blocks" of larger networks.

For example, any network can be broken down into each possible combination of three connected nodes in the network, of which there are then only 13 possible configurations for the links to take. We can then tally up how many of each of these configurations there are in a given network or set of networks.

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For example, in this study by Jonathan Borrelli, now at SUNY Plattsburgh, we see that across 50 food-webs we find that the same subgraphs are more or less common in each of the analyzed webs.

Ecological Networks: a brief history
Are ecological networks structurally distinct?

Framing the question and the solution
Efforts thus far
A computational challenge

Do trophic roles have structural signatures?
How ought competitive networks be structured?
Where do we go from here?

In collaboration with:
Stefano Allesina

The Question:

Is it possible to distinguish networks of differing types by their structure?

Interaction Type:: the real-world-event being depicted by a network link
Network Structure:: the arrangement of connections within a network

The project arose from a fairly simple question, with some surprisingly complicated secondary considerations. First, we should probably clear up some of the terminology.

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By types, we are mainly thinking about the type of ecological interaction being represented, e.g. pollination, parasitism, seed-dispersal, etc..

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and by structure, we mean all of those arrangements of links mentioned in the introduction and any others that we can think of. Here we are only restricted by the computational effort required to measure something on many networks, some of which can be quite large.

Also, just to be clear, in this project we were only looking at bipartite networks, as this tends to be the structure of type-specific ecological interaction networks.

The Solution:

Seeking generality, specificity, and scalability.

Generality:: Will our model work on new data?
Specificity:: Will our model fail when it is supposed to?
Scalability:: What is the proper level of disambiguation?

In this and the following slides, I will be demonstrating a potential solution which has some success, but several shortcomings as well. For these figures, I am using a PCA that has been fit using a subset of non-ecological networks including webs of actors performing in the same movies, authors publishing in the same journals, crimes committed in different neighborhoods in different cities, legislature voting records, and microbiome OTU presence/absence data across sites on human hosts.

These non-ecological data have an advantage of being fairly large and consistent in their collection methods, as well as being very highly replicated compared to ecological data.

One of the best ways to demonstrate that you are not overfitting your model. That is, that you are not fitting your model to a particular set of data rather than to the general class of data you are interested in. Is to divide your data into fitting and testing sets. Here, we use $100$ networks from each type to fit the PCA and then ...

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we can add the leftover networks onto the same plot and see if they still fall within the same general areas.

The second concern is a tradeoff with generality. We want our model to be able to be broken when we know that the relevant structure is disrupted. Put another way, a model that is able to fit any set of networks will ...

To test this, we randomize the structure of each of our networks according to a randomization protocol and see whether or not this breaks the clustering. We do this randomization in two ways, first a randomization that preserved degree distribution: each node in the randomized web has the same number of connections as in the original network, but who those connections are with is different

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Second, we did a more conservative randomization in which only the total number of connections in the network is preserved.

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As you can see, in this latter case, there is a significant loss of separation, especially with regards to the authorship networks and (to a lesser degree) the legislature webs.

Data

Ecological webs (51, 126, 35)
Non-biological reference set
Erdős-Rényi and Configuration Model Randomizations

Code

Analysis pipeline
Easy-Add metrics

All hosted online at Github

Machine learning

Gradient Boosted Regression Trees

Continuing to add data

Ecological data as available
Enhancing/adding to non-ecological data

But why didn't our method work? Maybe we didn't measure the right metrics or maybe PCA is too simplistic of a clustering technique. Alternatively, maybe the ecological networks we have are of too-low quality to be able to get at the fine-scale structural differences between networks of mutualistic and antagonistic interactions.

An alternative interpretation is that it actually did work. Maybe it is just that the variation within ecological networks is greater than the variation between networks of differing interaction types.

To address this latter point, I will be continuing to add data to the repository, but I am also interested in trying out some of the machine learning techniques which are a bit more sophisticated than PCA and have been growing in reputation in recent years.

Ecological Networks: a brief history
Are ecological networks structurally distinct?
Do trophic roles have structural signatures?

Defining trophic roles
Introducing the methodology
Visualizing Imbalance

How ought competitive networks be structured?
Where do we go from here?

In collaboration with:
Liz Sander
Mercedes Pascual
Stefano Allesina

Michalska-Smith & Sander, et al. 2018

Camerano 1880

What are parasites?

Prey/predator body size ratio (?)
Intimate, durable interactions (?)
Number of hosts or prey attacked during a particular life-history stage

Lafferty 2008

Lafferty, in his contribution to the Encyclopedia of Ecology, explains some of the many complications that arise when trying to systematically differentiate parasites from other life-cycles we observe in nature He discounts the first option since many predators consume prey that are larger than themselves (e.g. wolves eating moose) and the second with a similar argument: lions spend weeks consuming a single antelope.

Though he settles on the number of hosts/prey attacked in a particular life-history stage as a good indicator, with parasites attacking fewer (usually only one), this is difficult to measure quickly and precisely using the sampling methods that are common when collecting food-web data.

This is a toy example of a food-web. Note that the links point from prey to predator. This allows us to distinguish between in-degree -- the number of prey a species has -- and out-degree -- the number of predators.

When viewed this way, we can use additional information to know that the owl is a predator and the tick a parasite, but theoretical ecologists tend to see this picture a little differently...

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When the additional information of ecological classifications is unavailable, e.g. with a previously unobserved species, parasites and consumers look much more similar.

This project started as a question of whether or not we can identify species roles based just on this information, that is, the network structure.

Put another way, can we do what an ecologist does when observing species extensively in the field by just looking at the binary network of interactions?

By approaching this problem in a more statistical way, we can more explicitly identify the important, distinguishing characteristics, and provide justification (or not) for the characteristics that are thought to be important.


1	1	1	0	0	1	0	0
0	1	1	0	0	1	0	0
0	0	0	1	1	1	1	1
0	0	0	1	1	1	0	1
0	0	0	0	0	1	0	1
0	0	0	0	1	1	1	1
0	0	0	0	0	0	0	0
0	0	0	0	0	0	1	1
0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0


1	1	1	0	0	1	0	0
0	1	1	0	0	1	0	0
0	0	0	1	1	1	1	1
0	0	0	1	1	1	0	1
0	0	0	0	0	1	0	1
0	0	0	0	1	1	1	1
0	0	0	0	0	0	0	0
0	0	0	0	0	0	1	1
0	0	0	1	1	1	0	0
0	0	0	1	1	1	0	0

Every network can also be viewed as a matrix. When we do this, the row and column labels are in the same order, and a 1 indicates that the column species is eating the row species.

One feature of food-webs viewed in this way is that they tend to have a "cascade structure" -- that is, the interactions tend to show a pattern in which the majority of the links fall above the diagonal (i.e. in the upper triangle).

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One unique feature of symbionts, such as parasites, is that they can be consumed by "accident" when their host is consumed. Note that these links tend to fall in the lower triangle. This difference is one of the structural cues we can use to distinguish species from one another.

Also known as the stochastic blockmodel
A generalized form of Modularity
Seeks to find groups of strongly connecting species

Allesina & Pascual 2009

Ecological classifications, such as herbivore or parasite, are assigned by ecologists following a detailed investigation of the species life history and, in particular, its interactions with other species. Another approach to group species is to look at the overall network structure and to group species that eat or are eaten by the same or similar prey or predators.

In 2009, Allesina & Pascual introduced the group model to ecology to make this concept statistically rigorous. The model had been used previously in the social sciences under a different name -- the stochastic blockmodel. This model works by re-arranging the rows/columns of the matrix in order to form blocks of species with similar interactions.


1	1	1	0	0	1	0	0
0	1	1	0	0	1	0	0
0	0	0	1	1	1	1	1
0	0	0	1	1	1	0	1
0	0	0	0	0	1	0	1
0	0	0	0	1	1	1	1
0	0	0	0	0	0	0	0
0	0	0	0	0	0	1	1
0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0


1	1	1	0	1	0	0	0
0	1	1	0	1	0	0	0
0	0	0	1	1	1	1	1
0	0	0	1	1	1	0	1
0	0	0	0	1	0	0	1
0	0	0	0	1	1	1	1
0	0	0	0	0	0	1	1
0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0


0	0	1	1	1	1	0	0
0	0	0	1	1	1	0	0
0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0
1	1	0	1	0	0	1	1
1	0	0	1	0	0	1	1
0	0	0	1	0	0	0	1
1	1	0	1	0	0	0	1
0	0	0	0	0	0	0	0
0	1	0	0	0	0	0	1

So, if we look back to our food-web from before, we can re-arrange the rows and columns and optimizing both the number and size of the groups involved ...

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... to find a grouping such as this, where there are many blocks with no links, and a few blocks with a high density of links.

One drawback of this model (in general, though not evident in this case) is that the degree -- i.e. the number of consumptive interactions a species has -- can influence the grouping in undesirable ways, such as artificially grouping specialists together just because they all have few prey.

To account for this, in this project, we also used a modified version of the group model that accounts for the differences in degree across species.

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Which, as you can see produces less intuitive and more evenly spaced groupings.

	ψ	p-value
Parasites	0.667	0.089
Consumers	0.4	0.533
Herbivores	0.6	0.158
Primary Producers	1.0	0.022
Overall	0.4	0.033

Parasites	0.5	0.8
Consumers	1	0.017
Herbivores	1	0.017
Primary Producers	0.5	0.8
Overall	0.5	0.004

Once we have these statistical groupings, we want to know how well these groupings correspond to ecological classifications.

Put another way, if we look at which species are parasites,

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we want to calculate how "imbalanced" the groupings are with respect to the provided groupings. Are all of the parasites in a group by themselves? or are they evenly spaced between the groups? Finally, using a permutation test, we can calculate explicitly

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how likely it is for such a level of imbalance to arise by chance.

We can apply this procedure for each ecological classification (as well as look at all of them at once).

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In this case, we see that the top grouping does well for producers and parasites, while the bottom grouping is more evenly spread (i.e. nonsignificant). Note that even though one of the degree-corrected groups is made up of entirely herbivores, the p-value is still quite high because it is not very unlikely for such a configuration to occur given this number of herbivores and groups.

7 Large, well-resolved, marine/estuarine food-webs
With/without Concomitant predation
Each species categorized:

Primary Producer
Herbivore
Parasite
Consumers

Dunne et al. 2013

Concomitant Links	Degree Corrected	g	Primary Producers	Herbivores	Consumers	Parasites	Overall
No	No	2	0.895***	0.737***	0.491***	0.421	0.246***
		3	0.826**	0.589	0.331***	0.260	0.120***
		5	0.757***	0.551***	0.628***	0.430***	0.307***
		10	0.585*	0.448***	0.439***	0.445***	0.214***
	Yes	2	0.874	0.692	0.660***	0.449***	0.397***
		3	0.815	0.565	0.381***	0.521***	0.250***
		5	0.689	0.350	0.289***	0.559***	0.109***
		10	0.487	0.145	0.244***	0.138***	0.027***
Yes	No	2	0.898**	0.745***	0.912***	0.555***	0.555***
		3	0.839**	0.598	0.835***	0.603***	0.370***
		5	0.731	0.440*	0.702***	0.710***	0.327***
		10	0.702***	0.284***	0.582***	0.343***	0.164***
	Yes	2	0.874	0.709***	0.812***	0.609***	0.537***
		3	0.833**	0.609***	0.564***	0.515***	0.282***
		5	0.716	0.360	0.365***	0.506***	0.173***
		10	0.451	0.198***	0.364***	0.445***	0.114***

Concomitant Links	Degree Corrected	g	Primary Producers	Herbivores	Consumers	Parasites	Overall
No	No	2	0.895***	0.737***	0.491***	0.421	0.246***
		3	0.826**	0.589	0.331***	0.260	0.120***
		5	0.757***	0.551***	0.628***	0.430***	0.307***
		10	0.585*	0.448***	0.439***	0.445***	0.214***
	Yes	2	0.874	0.692	0.660***	0.449***	0.397***
		3	0.815	0.565	0.381***	0.521***	0.250***
		5	0.689	0.350	0.289***	0.559***	0.109***
		10	0.487	0.145	0.244***	0.138***	0.027***
Yes	No	2	0.898**	0.745***	0.912***	0.555***	0.555***
		3	0.839**	0.598	0.835***	0.603***	0.370***
		5	0.731	0.440*	0.702***	0.710***	0.327***
		10	0.702***	0.284***	0.582***	0.343***	0.164***
	Yes	2	0.874	0.709***	0.812***	0.609***	0.537***
		3	0.833**	0.609***	0.564***	0.515***	0.282***
		5	0.716	0.360	0.365***	0.506***	0.173***
		10	0.451	0.198***	0.364***	0.445***	0.114***

This table summarizes how imbalanced each category was for a given model and number of groups (g), again looking at the Punta Banda network. The numbers are not as important as the p-values, so I will highlight here just according to significance, that is, imbalance values that are substantially lower than would be expected by chance.

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Red corresponds to an alpha value of one tenth of one percent.

Note that, despite the difference in degree we saw in the previous figure, the degree-corrected group model produced significantly more imbalanced groups, regardless of concomitant predation (paired t-test, estimated difference: .12, p < .001 and .067, p < .001, when including/excluding concomitant predation, respectively). This suggests that degree is not an essential distinguishing factor for parasites.

Including concomitant links also produced more imbalanced groups, under both the degree-corrected (paired t-test, estimated difference: .034, p = .0049) and uncorrected (estimated difference: 203 .085, p < .001) models.

For the following images, we will be looking at the case of including concomitant predation links, but not correcting for degree.

A more concise way to view these networks is to collapse the groups into a single node, colored according to the ecological classifications of the nodes within it.

In this case, for example, the species have been divided into two groups, with one of the groups consisting entirely of consumers. This suggests that consumers are behaving in a unique way, i.e. the network structure of a given consumer is more similar to other consumers than to non-consumers.

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Likewise, when we constrain the number of groups to 3, we still see consumers coming out, but now we see a group that is mostly parasites as well.

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When we have five groups, something interesting happens: the consumer block is split in two. This is suggestive of additional structure within these gross ecological categories. Put another way, there are sub-structures within the consumer class where the differences between the sub-categorizations are more explanatory than the difference between consumers at large and other ecological categories.

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Finally, the highest number of groups we looked at: 10 groups we get some uniform groups, and some groups that are still conglomerates. This shows that while our statistical approach does resemble the ecological categorizations, there are larger statistical differences between members of the same categorizations than between different ones. Some of the categories are better parsed than others, which suggests that these categories have a stronger foundation in network structure.

Ecological classifications have statistical meaning

Especially for Consumers and Parasites
(beyond simple degree distribution)

But this is not always the strongest source of variance

Further structure to be found

Concomitant links important
Next step: Identify the most explanatory properties of homogenous groups

So what can we take away from this?

Well, it is comforting to know that the classifications that ecologists use in the field have statistical meaning with regards to network structure, and that these differences extend beyond simple degree distribution (the degree-corrected model produced more imbalanced partitions, suggesting that discounting the effect of degree in fact improved the ability to distinguish ecological categorizations).

Interestingly, the intra-category differences were stronger than the inter-category differences in some cases, suggesting there might be more informative classifications than the one we used here.

Concomitant links are useful, but not the end-all in distinguishing parasites and consumers.

Finally, we can use these groupings (especially the homogenous ones) to identify the properties that differ between categorizations. These differences can then be used to produce more rigorous definitions and provide insight into the important distinctions between species roles in nature.

Ecological Networks: a brief history
Are ecological networks structurally distinct?
Do trophic roles have structural signatures?
How ought competitive networks be structured?

The traditional dogma
Generalizing for more species
Separating the effects of intra- vs. interspecific interactions

Where do we go from here?

In collaboration with:
György Barabás
Stefano Allesina

Barabás, Michalska-Smith, & Allesina 2017

Intraspecific competition must be greater than interspecific competition for stable coexistence

$\frac{\mathrm{d} N_1}{\mathrm{d} t}$$=$ $N_1$$($$r_1$$-$ $a_{12}$$N_2$$-$$a_{11}$ $N_1$$)$

$\frac{\mathrm{d} N_2}{\mathrm{d} t}$$=$ $N_2$$($$r_2$$-$ $a_{21}$$N_1$$-$$a_{22}$ $N_2$$)$

Condition for coexistence:

$a_{12}a_{21}$$<$$a_{11}a_{22}$

Chesson 2000

In almost every textbook on basic ecology, you will find a statement similar to this, stemming from the formulation of competition using the Lotka-Volterra competition equations.

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which can be written out in this way where the $r$'s are intrinsic growth rates, the $N$'s are population levels, and the $a$'s indicate the strength of interaction either with members of different species ($a_{ij}$, $i \neq j$) or the same species ($a_{ii}$)

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The stability conditions for two species interacting in this way can be depicted in phase diagrams such as these (in which I've fixed the intra-specific competition and growth rates to be $1$ for convenience). For this system, there are four possible cases. In the top right species $1$ outcompetes species $2$ and drives the former extinct. In the bottom left is the complement of that scenario

In th bottom right, the winner is determined by the starting conditions, and the only situation in which both species coexist is the top right where for both species, the strength of competition with the other species is less than one.

More formally,

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the geometric mean of the intraspecific competition must be less than the geometric mean of the interspecific competition

$\frac{\mathrm{d} \boldsymbol{N}}{\mathrm{d} t}$$=$ $\boldsymbol{N}$ $($$\boldsymbol{r}$ + $\boldsymbol{a}$$\boldsymbol{N}$$)$

$\boldsymbol{a}$=$- \begin{pmatrix} 9 & 1 \\ 13 & 2 \end{pmatrix}$

$a_{12}a_{21}$$<$$a_{11}a_{22}$

$\frac{\mathrm{d} \boldsymbol{N}}{\mathrm{d} t}$$=$ $\boldsymbol{N}$$($$\boldsymbol{r}$ +$\boldsymbol{a}$$\boldsymbol{N}$$)$

$\boldsymbol{a}$=$- \begin{pmatrix} 10 & 9 & 5 \\ 9 & 10 & 9 \\ 5 & 9 & 10 \end{pmatrix}$

Intra $>$ inter for all pairs ... yet no coexistence

$\frac{\mathrm{d} \boldsymbol{N}}{\mathrm{d} t}$$=$ $\boldsymbol{N}$$($$\boldsymbol{r}$ +$\boldsymbol{a}$$\boldsymbol{N}$$)$

$\boldsymbol{a}$=$- \begin{pmatrix} 10 & 7 & 12 \\ 15 & 10 & 8 \\ 7 & 11 & 10 \end{pmatrix}$

Intra$<$inter for species $1$ & $2$ ... yet coexistence

Are there generalizations for the relationship of intra- to interspecific competition for more than 2 species?

in particular, are their signatures for the most extreme cases?

We performed this experiment, taking a random interaction matrix and re-arranging the off-diagonal interaction coefficients in a matrix using a stochastic search algorithm to find the configuration that maximized stability

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we found that this nested configuration is most stabilizing.

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This could arise, for instance if species are able to coexist by either specializing more on a particular resource, or generalizing such that they can consume a wider diversity of resources.

In this figure, each curve indicates a diet of a different coexisting species.

In the least-stabilizing case, a different pattern emerges:

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The species interactions for a sort of "anti-modular" structure.

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You can imagine an ecological scenario that would lead to this situation, for instance in the case of allelopathy in plants, where species interact strongly with plants who produce very different toxins, but have resistance to toxins similar to their own.

In this figure the dots indicate which toxin is produced by each species and the colours indicate general clusters of toxin similarity.

InterspecificIntraspecific

StabilizingDestabilizing

We can repeat this procedure looking at the intra-specific interactions. In this case, we can take each of the inter-specific patterns as a background and modify the diagonal elements to see what patterns enhance or detract from the effect of the off-diagonal terms.

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Starting with the most-stabilizing case, we find that the diagonal elements enhance stability when they generally conform to the same pattern, with the same species that experience high inter-specific competition having the highest intra-specific competition.

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Unsurprisingly, the inverse is most destabilizing.

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For the least-stabilizing situation of interspecific competition, we see the same general pattern, with intraspecific competition mirroring the interspecific competition to enhance the effect and counter-pointing it to detract from the dominant effect of interspecific effects.

Two species coexistence: simple and intuitive
But this rule does not hold for more species
Patterns to exploit (for symmetric case):

Clustering most stable
Nested least stable
Intraspecific competition follows same patterns as inter-

Asymmetric competition much more difficult to study

Ecological Networks: a brief history
Are ecological networks structurally distinct?
Do trophic roles have structural signatures?
How ought competitive networks be structured?
Where do we go from here?

Synthesis
Extension

Ecological networks are diverse and complex
But there are detectable patterns with real, ecological implications

Ecological interaction networks have large within-type variation and are more structurally similar than previously appreciated
Trophic roles have structural signatures in food-webs
The extreme cases of competition network structure have clear ecological corollaries

While it is true that at first-glance these three projects appear to have little in common, each of them looks at the relationship between network structure and ecological processes underlying the construction and maintenance of those networks.

In particular, I want to highlight that there are patterns in ecological network structure that exist to be found and explored further. While the first project's main message was that such structures can be difficult to find, even in this "negative" result is information on the variance inherent to ecological systems. in the second project, we find that there are structural signatures for the commonly described trophic roles, even parasites which are notoriously difficult to define precisely. Finally, in this last section, I've shown the endpoints in a spectrum of stability for competitive networks, revealing that both ends have clear, interpretable structures.

Pilosof, et al. 2017

Acknowledgements

Adam	Brenon	Evan	Jody	Marcos	Sarah
Agata	Carlos	Finnigan	John	Mariah	Simon
Alex	Chad	Frank	Jose	Mark	Stefano
Allison	Cody	Grzesz	Katherine	Mary	Steve
Amos	Connie	Gyuri	Kimbra	Mercedes	Supriya
Angelo	Courtney	Iuri	Laura	Mike	Sylvia
Arvind	Dan	Jacopo	Lili	Monika	Trevor
Audrey	Dave	Jakez	Linnea	Phillip	Tyler
Ben	Elisa	Jared	Liz	Rahul	Yoseop
Bonnie	Emilio	Jeff	Madeline	Roberto	Zach	etc.

Questions?

Slides available on Michalska-Smith.com

Food-web animals ( 1, 2, 3, 4, 5 ) modified from images designed by Freepik and Snap2Objects.


1	1	1	0	0	1	0	0
0	1	1	0	0	1	0	0
0	0	0	1	1	1	1	1
0	0	0	1	1	1	0	1
0	0	0	0	0	1	0	1
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0	0	0	0	0	0	0	0
0	0	0	0	0	0	1	1
0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0


1	1	1	0	0	1	0	0
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0	0	0	1	1	1	0	1
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0	0	0	0	0	0	0	0
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0	0	0	1	1	1	0	0
0	0	0	1	1	1	0	0


1	1	1	0	0	1	0	0
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0	0	0	0	0	0	0	0
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1	1	1	0	1	0	0	0
0	1	1	0	1	0	0	0
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0	0	0	1	1	1	0	1
0	0	0	0	1	0	0	1
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0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0


0	0	1	1	1	1	0	0
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1	1	0	1	0	0	0	1
0	0	0	0	0	0	0	0
0	1	0	0	0	0	0	1


1	1	1	0	0	1	0	0
0	1	1	0	0	1	0	0
0	0	0	1	1	1	1	1
0	0	0	1	1	1	0	1
0	0	0	0	0	1	0	1
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0	0	0	0	0	0	1	1
0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0


1	1	1	0	0	1	0	0
0	1	1	0	0	1	0	0
0	0	0	1	1	1	1	1
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0	0	0	0	1	1	1	1
0	0	0	0	0	0	0	0
0	0	0	0	0	0	1	1
0	0	0	1	1	1	0	0
0	0	0	1	1	1	0	0


1	1	1	0	0	1	0	0
0	1	1	0	0	1	0	0
0	0	0	1	1	1	1	1
0	0	0	1	1	1	0	1
0	0	0	0	0	1	0	1
0	0	0	0	1	1	1	1
0	0	0	0	0	0	0	0
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0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0


1	1	1	0	1	0	0	0
0	1	1	0	1	0	0	0
0	0	0	1	1	1	1	1
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0	0	1	1	1	1	0	0
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1	1	0	1	0	0	1	1
1	0	0	1	0	0	1	1
0	0	0	1	0	0	0	1
1	1	0	1	0	0	0	1
0	0	0	0	0	0	0	0
0	1	0	0	0	0	0	1

University of Chicago

Department of Ecology & Evolution

Matthew Michalska-Smith

Outline

Outline

Networks as a unifying methodology

Networks as a unifying methodology

Ecological Networks

Network structure

Network structure: Global

Network structure: Local

Outline

The Question:

The Solution:

Generality

Specificity

Scalability

Efforts thus far

A computational challenge

All hosted online at Github

Future directions

Outline

Ecological Networks

introduction

What are parasites?

introduction: Food-Webs

methods: Food-Webs

methods: Group Model

methods: Group Model (Degree Corrected)

methods: Imbalance

methods: Data

results: Imbalance

results: Condensed Graphs

Summary

Outline

Dogma

Dogma

What about $3$ species?

What about $3$ species?

Generalization to large $n$

Generalization to large $n$

The additive effect of intra-specific competition

Summary

Outline

Synthesis

Extension

Acknowledgements

Questions?


1	1	1	0	0	1	0	0
0	1	1	0	0	1	0	0
0	0	0	1	1	1	1	1
0	0	0	1	1	1	0	1
0	0	0	0	0	1	0	1
0	0	0	0	1	1	1	1
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0	0	0	0	0	0	1	1
0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0


1	1	1	0	0	1	0	0
0	1	1	0	0	1	0	0
0	0	0	1	1	1	1	1
0	0	0	1	1	1	0	1
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0	0	0	0	0	0	1	1
0	0	0	1	1	1	0	0
0	0	0	1	1	1	0	0


1	1	1	0	0	1	0	0
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0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0


1	1	1	0	1	0	0	0
0	1	1	0	1	0	0	0
0	0	0	1	1	1	1	1
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0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0


0	0	1	1	1	1	0	0
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1	1	0	1	0	0	1	1
1	0	0	1	0	0	1	1
0	0	0	1	0	0	0	1
1	1	0	1	0	0	0	1
0	0	0	0	0	0	0	0
0	1	0	0	0	0	0	1