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<!DOCTYPE html>
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<head>
<title>Connectal Coding</title>
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name:opening
**An Introduction to Graph Statistics**<br>
Joshua Vogelstein |
{[BME](https://www.bme.jhu.edu/),[CIS](http://cis.jhu.edu/), [KNDI](http://kavlijhu.org/)}@[JHU](https://www.jhu.edu/)
<a href="https://neurodata.io"><img src="images/neurodata_purple.png" style="height:430px;"/></a>
<!-- <img src="images/funding/jhu_bme_blue.png" STYLE="HEIGHT:95px;"/> -->
<!-- <img src="images/funding/KNDI.png" STYLE="HEIGHT:95px;"/> -->
<!-- <font color="grey"></font> -->
.foot[[jovo@jhu.edu](mailto:[email protected]) | <http://neurodata.io/talks> | [@neuro_data](https://twitter.com/neuro_data)]
---
class: center, middle
## .center[https://neurodata.io/graspy/]
---
## Outline
- Background
- Statistical Models of Connectomes
- Statistical Models of Populations of Connectomes
- Applications
- Discussion
---
class: middle
## .center[.k[Background]]
---
### What is a Connectome?
- .r[Network] of a brain, at a spatiotemporal precision & extent
- .r[Nodes] are distinct biophysical entities
- .r[Edges] indicate the presence of a connection/communication between nodes
- .r[Attributes] of the network, nodes, or edges are possible
--
<br>
- Example nodes: cells, cellular compartment, cellular ensembles
- Example edges: synapses, gap junction, fiber bundles
---
## Many adjacency matrices, one graph
Drosophila larva connectome, Eichler et al. 2017
<img src="images/multi-adj-dros.png" style="width: 700px;"/>
---
### Definitions
.r[Neural (activity) coding]: inferring the relationships between neural *activity* and past, present, or future events, states, or traits
--
.r[Connectal coding]: inferring the relationships between neural *connectivity* and past, present, or future events, states, or traits
--
.s[events]: genetic, developmental, experiential (stimuli/behavior)
.s[traits]: IQ, sex, personality, learning disabled
.s[states]: happy, resting, manic
<!-- ---
### Implications
- A brain could have many different connectomes, at different times and/or resolutions, and measured in different ways (e.g. structural or functional)
- The definition of node can mean different things at different scales
- We measure properties of the brain to .r[estimate] connectomes
- Estimates are always .r[noisy] -->
---
### Why Use Statistical Models?
- Connectome estimates are noisy
- Connectomes, and their relationships to events, states, or traits, can be complicated
- We wish to
- quantify uncertainty
- incorporate domain knowledge to the extent possible
- summarize natural phenomena in a "simple" way
- understand limitations of analyses
### Implications
- All inferences about population .r[depend on model]
---
### Connectome Analysis Styles
- Bag of edges
- Bag of features
- Bag of parameters
---
### Bag of Edges
- Treat each edge as independent
- Implicit model: independent edge model
---
### Bag of Features
- Choose $m$ features and compute them per graph
- Characterize connectome with this set of "parameters"
- Implicit model: exponential random graph model
---
### Bag of Parameters
- Build a **statistical parametric model** of brain network
- Can encode some domain knowledge explicitly
- Can model edges, nodes, communities
- Implicit model: latent structure model
---
### Limitations of approaches
---
### Limitations of Bag of Edges
- Completely ignores graph structure of data
- Too simple for many questions
---
#### Sometimes the signal is in the node
<img src="images/pop1_p_mat.png" style="width: 350px;"/>
<img src="images/pop2_p_mat.png" style="width: 350px;"/>
---
#### Edge-wise stats show no significance
<img src="images/edgewise_p_vals.png" style="width: 600px;"/>
---
#### Node-wise test finds the signal
<img src="images/nodewise_p_vals.png" style="width: 750px;"/>
---
### Limitations of Bag of Features
- how do I choose which features (hint: arbitrary)?
- how many features are possible given a graph with $n$ nodes (hint: many)?
- do these features characterize the brain (hint: no)?
- can we make causal claims using these features (hint: no)?
- are these features independent (hint: no)?
- least well understood of the approaches, but very common
---
### Same Stats, Different Graphs
<div>
<img src="images/same_stats_diff_graphs.png" style="height: 400px;" align="right"/>
</div>
- num vertices = 12
- num edges = 21
- number of triangles = 10
- global clustering coefficient = 0.5
<br><br><br><br><br><br><br><br><br>
.foot[[Chen et al.](https://link.springer.com/chapter/10.1007/978-3-030-04414-5_33)]
---
### Distribution of Features, n=10
<img src="images/j1c-all-graphs-hexbin.png" style="height: 500px;"/>
---
### Condition on "close" to base graph
<img src="images/j1c_hexbin_31_base.png" style="height: 500px;" />
.footnote[(edges=31, threshold=3, n=200k)]
<!-- ER image & IER image (random p_ij's) -->
<!-- Model of 1 graph
- for each of the 3 approaches (IE, ERGM, LSM)
- define different approaches
- give examples
- demonstrate pro's and con's of different approaches
- prob with ER: too simple
- prob with IE: sometimes signal is in the nodes
- prob with ERGM: j1c's thingy
- prob with latent positions: less intuitive
- value of latent positions:
- characterize the data
- model of \geq 2 graphs
- correlated ER
- COSIE
- MRDPG
- applications of multigraph models
- test for equality, mouse example
- test for indepedence, bear example
- test for heritability
- graph matching
- mic drop
-->
<!--
- requires multiple hypothesis correction for valid tests
- does anybody know a good way to correct (hint: no)?
- BH: way under-conservative (false positives)
- Bonferroni: way over-conservative (false negatives) -->
---
### Limitations of Bag of Parameters
- Conceptually less intuitive
---
class: middle
## .center[.k[Statistical Models of Connectomes]]
---
### Erdos-Renyi (ER)
- akin to assuming a neuron's spike rate is Poisson with a fixed rate.
- all edges independent
- all edges sampled from identical distribution
- only 1 parameter: prob of an edge
- $\mathbb{P}[A_{i,j}] = p$
Notes
- <b> Simplest random graph model; lacks descriptive power </b>
---
### Drosophila Connectome ER
<br>
<img src="images/dros_er_model.png" style="width: 800px;" />
- p = 0.166
---
### Degree Corrected Erdos-Renyi (DCER)
- edges are independent
- edges are sampled from .r[different] distributions
- .r[n+1 parameters]: degree correction for each node
- $\mathbb{P}[A_{i,j}] = \theta_i\theta_jp$
Notes
- n+1 paramers is much larger than 1
- still ignores structure
---
### Drosophila Connectome DCER
<br>
<img src="images/dros_dcer_model.png" style="width: 800px;" />
---
### Stochastic Block Model (SBM)
- akin to assuming a neuron's are in different states, which determine Poisson rate.
- edges are .r[conditionally] independent
- each node has a class assignment
- $\mathbb{P}[A_{i,j}]$ = $B$(class i, class j)
Notes
- simplest >2 parameter model
---
### Drosophila Connectome SBM
<img src="images/dros_sbm_model.png" style="width: 800px;"/>
---
### Degree-corrected Stochastic Block Model (DCSBM)
- edges are .r[conditionally] independent
- each node has a class assignment
- $\mathbb{P}[A_{i,j}]$ = $\theta_i\theta_jB$(class i, class j)
Notes
- simplest >2 parameter model
---
## Drosophila Connectome DCSBM
<img src="images/dros_dcsbm_model.png" style="width: 800px;" />
---
### Random Dot Product Graphs (RDPG)
- akin to latent state models in population coding
- edges are conditionally independent
- each node has a .r[latent position in d-dimensions]
- $\mathbb{P}[A_{i,j}]$ = f(latent position i, latent position j)
- for example, $\mathbb{P}[A_{i,j}]$ is the dot product of latent positions
Notes
- generalizes previous models
---
### Latent positions allow for more general relationships
<img class="left" src="images/RDPGrank6latent.png" style="width: 400px;" />
<img class="right" src="images/legendn copy.png" style="width: 150px;"/>
---
### Drosophila Connectome RDPGs
<img src="images/dros_rdpg_model.png" style="width: 750px;"/>
---
### Model considerations/extensions
- Directed extensions exist
- Loopy extensions exist
- Weighted extensions exist (mostly)
<!-- ---
### What can we do with parameters?
- Node clustering or classification
- Bootstrap? -->
---
class: middle
## .center[.k[Statistical Models of] .r[Populations of] .k[ Connectomes]]
<!--
## Population Graph Models
- Joint RDPG
- Common Subspace Independent Edge Graph (COSIE) -->
---
### Joint Random Dot Product Graphs
- All nodes have a latent position in d-dimensional space
- Each graph has a latent position matrix
<img src="images/omni_method.png" style="width: 650px;" />
---
### Common Subspace Independent Edge Graph (COSIE)
- Common latent position matrix shared across all graphs
- Individual graphs are transformation of the common matrix
<img src="images/mase_method.png" style="width: 650px;" />
---
class: middle
## .center[.k[Application of Population Models]]
---
### Mouse Connectomes From Same Genotype are Similar
<img src="images/mouse_connectomes.png" style="width: 750px;" />
---
### Structural Connectomes are Heritable
<img src="images/heritability.png" style="width: 750px;" />
- MRDPG model
---
### COSIE Model Can Recover Bilateral Separation
<img src="images/HNU1-latentpositions-plot.png" style="width: 700px;" />
- HNU1 Dataset
---
### COSIE Model Can Recover Clusters of Same Test-Retest Scans
<img src="images/HNU1-mds123-cbpalette.png" style="width: 750px;" />
- HNU1 Dataset
---
### COSIE Model Can Perfectly Classify of Subjects
<img src="images/HNU1-classerror.png" style="width: 750px;" />
---
class: middle
## .center[.k[Discussion]]
---
## Summary and Next Steps
- Connectomes are the mechanistic link:
.center[.r[genotype --> phenotype]]
- Extend ideas from coding theory to support these analyses
- Connectomes, genetic and phenotypic data are available
---
### References
- Connectal Coding [[1]](https://doi.org/10.1016/j.conb.2019.04.005)
- Description of GraSPy [[2]](https://arxiv.org/abs/1904.05329)
- Statistics on RDPG [[3]](https://dl.acm.org/citation.cfm?id=3242083)
- Two-sample hypothesis testing for RDPG [[4]](https://arxiv.org/abs/1403.7249)
- Two-sample hypothesis testing for two random graphs [[5]](https://projecteuclid.org/euclid.bj/1489737619)
- COSIE model and estimation [[6]](https://arxiv.org/abs/1906.10026)
- Omnibus Embedding for JRDPG estimation [[7]](https://ieeexplore.ieee.org/document/8215766)
- Mouse Connectome Heritability [[8]](https://www.biorxiv.org/content/10.1101/701755v1)
- Connectome smoothing [[9]](https://ieeexplore.ieee.org/document/8570772)
---
### Acknowledgements
<div class="small-container">
<img src="faces/jovo.png" />
<div class="centered">Joshua Vogelstein</div>
</div>
<div class="small-container">
<img src="faces/jaewon.jpg" />
<div class="centered">Jaewon Chung</div>
</div>
<div class="small-container">
<img src="faces/pedigo.jpg"/>
<div class="centered">Ben Pedigo</div>
</div>
<div class="small-container">
<img src="faces/ebridge.jpg"/>
<div class="centered">Eric Bridgeford</div>
</div>
<div class="small-container">
<img src="faces/hayden.png"/>
<div class="centered">Hayden Helm</div>
</div>
<div class="small-container">
<img src="faces/jesus.jpg"/>
<div class="centered">Jesus Arroyo</div>
</div>
<div class="small-container">
<img src="faces/ronak.jpg"/>
<div class="centered">Ronak Mehta</div>
</div>
<div class="small-container">
<img src="faces/cep.png"/>
<div class="centered">Carey Priebe</div>
</div>
<div class="small-container">
<img src="faces/randal.jpg"/>
<div class="centered">Randal Burns</div>
</div>
<div class="small-container">
<img src="faces/mim.jpg"/>
<div class="centered">Michael Miller</div>
</div>
<div class="small-container">
<img src="faces/dtward.jpg"/>
<div class="centered">Daniel Tward</div>
</div>
<div class="small-container">
<img src="faces/vikram.jpg"/>
<div class="centered">Vikram Chandrashekhar</div>
</div>
<div class="small-container">
<img src="faces/drishti.jpg"/>
<div class="centered">Drishti Mannan</div>
</div>
<div class="small-container">
<img src="faces/jesse.jpg"/>
<div class="centered">Jesse Patsolic</div>
</div>
<div class="small-container">
<img src="faces/falk_ben.jpg"/>
<div class="centered">Benjamin Falk</div>
</div>
<div class="small-container">
<img src="faces/loftus.jpg"/>
<div class="centered">Alex Loftus</div>
</div>
<div class="small-container">
<img src="faces/bcaffo.jpg"/>
<div class="centered">Brian Caffo</div>
</div>
<div class="small-container">
<img src="faces/minh.jpg"/>
<div class="centered">Minh Tang</div>
</div>
<div class="small-container">
<img src="faces/avanti.jpg"/>
<div class="centered">Avanti Athreya</div>
</div>
<div class="small-container">
<img src="faces/vince.jpg"/>
<div class="centered">Vince Lyzinski</div>
</div>
<div class="small-container">
<img src="faces/dpmcsuss.jpg"/>
<div class="centered">Daniel Sussman</div>
</div>
<div class="small-container">
<img src="faces/youngser.jpg"/>
<div class="centered">Youngser Park</div>
</div>
<div class="small-container">
<img src="faces/cshen.jpg"/>
<div class="centered">Cencheng Shen</div>
</div>
<div class="small-container">
<img src="faces/shangsi.jpg"/>
<div class="centered">Shangsi Wang</div>
</div>
<div class="small-container">
<img src="faces/ronan.jpg"/>
<div class="centered">Ronan Perry</div>
</div>
<div class="small-container">
<img src="faces/vivek.jpg"/>
<div class="centered">Vivek Gopalakrishnan</div>
</div>
<div class="small-container">
<img src="faces/tommy_athey.jpg"/>
<div class="centered">Tommy Athey</div>
</div>
<div class="small-container">
<img src="faces/patsolic_heather.jpg"/>
<div class="centered">Heather Patsolic</div>
</div>
<div class="small-container">
<img src="faces/bijan.jpg"/>
<div class="centered">Bijan Varjavand</div>
</div>
<span style="font-size:200%; color:red;">♥, 🦁, 👪, 🌎, 🌌</span>
<img src="images/funding/nsf_fpo.png" STYLE="HEIGHT:95px;"/>
<img src="images/funding/nih_fpo.png" STYLE="HEIGHT:95px;"/>
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<img src="images/funding/schmidt.jpg" STYLE="HEIGHT:95px;"/>
---
class: middle
## .center[.k[Additional Information]]
---
## Adjacency Spectral Embedding
- Method for estimating parameter for RDPG model (for single graph)
- $\hat{X} = UD^{1/2}$ where $U, D, V = SVD(A)$.
---
## Omnibus Embedding
- Method for estimating parameters for Joint RDPG model
<img src="images/omni_method.png" style="width: 750px;" />
---
## Multiple Adjacency Spectral Embedding (MASE)
- Method for estimating parameters for COSIE model
<img src="images/mase_method.png" style="width: 750px;" />
---
### Genotype, Phenotype, Connectotype
- .r[Phenotype]: a description of an individual's properties with regard to a phenomenon of interest
- .r[Genotype]: a set of genes and associated variants associated with that phenotype
- .r[Connectotype]: a set of nodes, edges, and their properties that are associated with that phenotype
--
<br><br>
.center[Genotype --> Connectotype --> Phenotype]
**Connectotypes are the implementation-level mechanisms linking genotypes to phenotypes**
---
</textarea>
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