### Random Graph Embeddings with General Edge Potentials

Clayton Shonkwiler

http://shonkwiler.org

10.02.21

/css21

This talk!

### Collaborators

Jason Cantarella

U. of Georgia

Tetsuo Deguchi

Ochanomizu U.

Erica Uehara

Ochanomizu U.

Funding: Simons Foundation

### Linear polymers

A linear polymer is a chain of molecular units with free ends.

Polyethylene

Nicole Gordine [CC BY 3.0] from Wikimedia Commons

### Shape of linear polymers

In solution, linear polymers become crumpled:

Protonated P2VP

Roiter–Minko, J. Am. Chem. Soc. 127 (2005), 15688-15689

[CC BY-SA 3.0], from Wikimedia Commons

### Topological polymers

A topological polymer joins monomers in some graph type:

Petersen graph

### Synthetic topological polymers

The Tezuka lab in Tokyo can now synthesize many topological polymers in usable quantities

Y. Tezuka, Acc. Chem. Res. 50 (2017), 2661–2672

### Main Question

What is the probability distribution on the shapes of topological polymers in solution?

### Ansatz

Linear polymers

Topological polymers

Random walks with independent steps

Random walks with steps conditioned on ???

### Graphs and the Graph Laplacian

Suppose $$\mathcal{G}$$ is a directed graph with $$\mathbf{v}$$ vertices and $$\mathbf{e}$$ edges.

$$L:\mathbb{R}^{\mathbf{v}}\to \mathbb{R}^{\mathbf{v}}$$ given by $$L = \partial \partial^T$$ is the graph Laplacian.

\mathbb{R}^{\mathbf{e}} \stackrel{\partial}{\longrightarrow} \mathbb{R}^{\mathbf{v}}
\partial(e_4)=v_4-v_2
L(v_4)=3v_4-v_1-v_2-v_5

Think of $$x=(x_1,\dots,x_{\mathbf{v}})\in \mathbb{R}^{\mathbf{v}}$$ as a realization of $$\mathcal{G}$$ in $$\mathbb{R}$$.

E=\frac{1}{2}\sum_{j=1}^{\mathbf{e}}\|x(\operatorname{head}(e_j))-x(\operatorname{tail}(e_j))\|^2 = \frac{1}{2}\langle x, L x\rangle
p(x) \propto e^{-\frac{1}{2} \langle x, L x\rangle}

Harmonic bonds $$\Rightarrow$$ Gaussian vertex distribution:

### Realizations

This is James–Guth–Flory phantom network theory!

### Some More Careful Linear Algebra

Define $$\widetilde{L}:=L+\frac{1}{\mathbf{v}} \mathbf{1}\mathbf{1}^T$$.

Proposition.

1. $$\partial:(\operatorname{EC},\langle \cdot,\cdot\rangle) \to (\operatorname{VC},\langle \cdot,\widetilde{L}^{-1}\cdot\rangle)$$ is a partial isometry.
2. $$\partial:(\operatorname{VP},\langle \cdot,\widetilde{L}^*\cdot\rangle)\to(\operatorname{ED},\langle\cdot,\cdot\rangle)$$ is a partial isometry.

$$\mu_{\mathcal{G}}$$: $$O(d)$$-invariant measure concentrated on $$\operatorname{im}\partial^* \subset \operatorname{ED}$$

$$\nu_{\mathcal{G}}=(\partial^{*+})_\sharp \mu_{\mathcal{G}}$$, its pushforward to $$\operatorname{VP}$$.

\operatorname{VP} \stackrel{\partial^*}{\longrightarrow} \operatorname{ED}

$$\Sigma_{\mathbf{v}}:=$$ matrix of expectations of products of a fixed coordinate of the vertices of $$\mathcal{G}$$

$$\Sigma_{\mathbf{e}}$$ likewise for edges.

Theorem. The expected radius of gyration is

$$\langle \operatorname{R}_g^2\rangle_{\nu_{\mathcal{G}}} = \frac{d}{\mathbf{v}}\operatorname{tr}(\Sigma_{\mathbf{v}}) = \frac{d}{\mathbf{v}} \operatorname{tr}(\partial^{T+}\Sigma_{\mathbf{e}}\partial^+)$$

When $$\mu_{\mathcal{G}}$$ is a standard Gaussian on $$\operatorname{im}\partial^*$$,

$$\langle\operatorname{R}_g^2\rangle_{\nu_{\mathcal{G}}}= \frac{d}{\mathbf{v}}\operatorname{tr}\partial^{T+}(\partial^T\partial^{T+})\partial^T = \frac{d}{\mathbf{v}} \operatorname{tr} \partial^{T+}\partial^T = \frac{d}{\mathbf{v}} \operatorname{tr}L^+$$.

### Main Theorem

Theorem. Suppose $$f_0$$ and $$f_1$$ are injective chain maps between graphs $$\mathcal{G}'$$ and $$\mathcal{G}$$ of the same cycle rank. Given $$O(d)$$- and translation-invariant $$g:\operatorname{VP} \to \mathbb{R}$$ which is expressible in terms of $$\mathcal{G}'$$, then

$$\mathcal{E}_{\nu_{\mathcal{G}}}(g) = \mathcal{E}_{\nu'_{\mathcal{G}'}}(g')$$.

### Consequences

$$\mathcal{G}$$

$$\mathcal{G}_n$$

Proposition. In phantom network theory, the joint distribution of squared distances between junction vertices in $$\mathcal{G}_n$$ is the joint distribution of $$n$$ times the squared distances between vertices in $$\mathcal{G}$$.

### Consequences

In the freely-jointed chain, each edge is a unit vector.

$$\langle \operatorname{squared\, junction-junction\, distance}\rangle_{\nu_{\mathcal{G}_n}}$$ is in principle an $$(18n-12)$$-dimensional integral

Using the theorem, can reduce to a $$6$$-dimensional integral.

### Questions

What about including distributions on bond angles in addition to bond lengths?

Sampling algorithms?

Steric constraints and the complete graph?

# Thank you!

### References

J. Cantarella, T. Deguchi, C. Shonkwiler, E. Uehara, Gaussian random embeddings of multigraphs, preprint, 2020, arXiv:2001.11709​

J. Cantarella, T. Deguchi, C. Shonkwiler, E. Uehara, Radius of gyration, contraction factors, and subdivisions of topological polymers, preprint, 2020, arXiv:2004.06199

J. Cantarella, T. Deguchi, C. Shonkwiler, E. Uehara, Random graph embeddings with general edge potentials, preprint, 2021, on arXiv soon!