Graph Analysis via
\(p\)-Modulus
Nathan Albin
K-State Mathematics
Theory and Applications
Shameless Plug
Modulus in the Continuum
\begin{equation*}
\begin{cases}
\Delta u = 0\quad\text{in }\Omega,\\
u\big|_A = 0,\\
u\big|_B = 1,\\
\frac{\partial u}{\partial n}\big|_{C\cup D} = 0
\end{cases}
\end{equation*}
\begin{equation*}
\begin{split}
\underset{u}{\text{minimize}}\quad&\int_\Omega|\nabla u|^2\;dx\\
\text{subject to}\quad& u\big|_A=0\\
&u\big|_B=1
\end{split}
\end{equation*}
\int_\gamma |\nabla u|\;ds \ge 1
\begin{equation*}
\begin{split}
\underset{\rho}{\text{minimize}}\quad&\int_\Omega\rho^2\;dx\\
\text{subject to}\quad&\rho\ge 0,\\
& \int_\gamma\rho\;ds\ge 1\quad\forall\;\gamma\in\Gamma(A,B)
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
\underset{u}{\text{minimize}}\quad&\int_\Omega|\nabla u|^2\;dx\\
\text{subject to}\quad& u\big|_A=0\\
&u\big|_B=1
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
\underset{\rho}{\text{minimize}}\quad&\int_\Omega\rho^2\;dx\\
\text{subject to}\quad&\rho\ge 0,\\
& \int_\gamma\rho\;ds\ge 1\quad\forall\;\gamma\in\Gamma(A,B)
\end{split}
\end{equation*}
All paths connecting A to B
\begin{equation*}
\begin{split}
\underset{\rho}{\text{minimize}}\quad&\int_\Omega\rho^2\;dx\\
\text{subject to}\quad&\rho\ge 0,\\
& \int_\gamma\rho\;ds\ge 1\quad\forall\;\gamma\in\Gamma(A,B)
\end{split}
\end{equation*}
All paths connecting A to B
\text{Mod}(\Gamma(A,B))
Modulus on a Graph
\begin{equation*}
\begin{split}
\underset{\rho\in\mathbb{R}^E_{\ge 0}}{\text{minimize}}\quad&\sum_{e\in E}\rho(e)^2\\
\text{subject to}\quad&\rho\ge 0,\\
& \sum_{e\in\gamma}\rho(e)\ge 1\quad\forall\;\gamma\in\Gamma(a,b)
\end{split}
\end{equation*}
\text{Mod}(\Gamma(a,b))
\frac{1}{2}
\frac{1}{2}
\frac{1}{4}
\frac{1}{4}
\mathcal{E}(\rho) := \sum_{e\in E}\rho(e)^2 = \frac{1}{4} + \frac{1}{4} + \frac{1}{16} + \frac{1}{16} = \frac{5}{8}
\frac{1}{2}
\frac{1}{2}
\frac{1}{4}
\frac{1}{4}
\mathcal{E}(\rho) := \sum_{e\in E}\rho(e)^2 = \frac{1}{4} + \frac{1}{4} + \frac{1}{16} + \frac{1}{16} = \frac{5}{8}
admissible but not optimal
\alpha
1-\alpha
\frac{1-\alpha}{2}
\frac{1-\alpha}{2}
\mathcal{E}(\rho) = \alpha^2 + (1-\alpha)^2 + 2\left(\frac{1-\alpha}{2}\right)^2
\frac{3}{5}
\frac{2}{5}
\frac{1}{5}
\frac{1}{5}
\mathcal{E}(\rho) = \frac{9}{25} + \frac{4}{25} + 2\frac{1}{25} = \frac{3}{5} = \text{Mod}(\Gamma(a,b))
Effective Resistance
\(1\)
\(1\)
\(1\)
\(1\)
\(1\)
\(1\)
\(2\)
\(1\)
\(\frac{2}{3}\)
\(\frac{5}{3}\)
Effective Resistance
\text{Mod}(\Gamma(a,b)) = R_{\text{eff}}(a,b)^{-1}
\(1\)
\(1\)
\(1\)
\(1\)
\(1\)
\(1\)
\(2\)
\(1\)
\(\frac{2}{3}\)
\(\frac{5}{3}\)
\begin{equation*}
\begin{split}
\underset{\phi\in\mathbb{R}^V}{\text{minimize}}\quad&\sum_{\{x,y\}\in E}(\phi(x)-\phi(y))^2\\
\text{subject to}\quad&\phi(a)=0,\\
&\phi(b)=1
\end{split}
\end{equation*}
\text{Mod}(\Gamma(a,b))
The Probabilistic Interpretation
\(\underline{\gamma},\underline{\gamma'}\) are iid random paths
\(\mathbb{P}\left(\underline{\gamma}=\gamma_i\right) = \mu(\gamma_i)\)
Minimum Expected Overlap
\(\underline{\gamma},\underline{\gamma'}\) are iid random paths
\(\mathbb{P}\left(\underline{\gamma}=\gamma_i\right) = \mu(\gamma_i)\)
Minimum Expected Overlap
\mathbb{E}_\mu|\underline{\gamma}\cap\underline{\gamma'}|
= \alpha^2(2) + 2\alpha(1-\alpha)(1) + (1-\alpha)^2(3)
\(\underline{\gamma},\underline{\gamma'}\) are iid random paths
\(\mathbb{P}\left(\underline{\gamma}=\gamma_i\right) = \mu(\gamma_i)\)
Minimum Expected Overlap
\mathbb{E}_\mu|\underline{\gamma}\cap\underline{\gamma'}|
= \alpha^2(2) + 2\alpha(1-\alpha)(1) + (1-\alpha)^2(3)
\alpha=\frac{2}{3}\quad\implies\quad\mathbb{E}_\mu|\underline{\gamma}\cap\underline{\gamma'}| = \frac{5}{3} = \text{Mod}(\Gamma)^{-1}
Minimum Expected Overlap
\text{Mod}(\Gamma)^{-1} = \min_{\mu\in\mathcal{P}(\Gamma)}\mathbb{E}_\mu|\underline{\gamma}\cap\underline{\gamma'}|
pmfs on \(\Gamma\)
modulus prefers short paths + variety
Canonical Example
\(k\) parallel paths of length \(\ell\)
\(\mu(\gamma_i)=\frac{1}{k}\) for \(i=1,2,\ldots,k\)
\text{Mod}(\Gamma)= \left(\min_{\mu\in\mathcal{P}(\Gamma)}\mathbb{E}_\mu|\underline{\gamma}\cap\underline{\gamma'}|\right)^{-1}
= \left(\frac{1}{k}(\ell) + \frac{k-1}{k}(0)\right)^{-1} = \frac{k}{\ell}
Generalizing the Energy
\begin{equation*}
\begin{split}
\underset{\rho\in\mathbb{R}^E_{\ge 0}}{\text{minimize}}\quad&\mathcal{E}_{p,\sigma}(\rho)\\
\text{subject to}\quad&\rho\ge 0,\\
& \sum_{e\in\gamma}\rho(e)\ge 1\quad\forall\;\gamma\in\Gamma(a,b)
\end{split}
\end{equation*}
\mathcal{E}_{p,\sigma}(\rho) = \sum_{e\in E}\sigma(e)\rho(e)^p,\quad 1\le p<\infty
\mathcal{E}_{\infty,\sigma}(\rho) = \max_{e\in E}\sigma(e)\rho(e)
\text{Mod}_{p,\sigma}(\Gamma)
Shortest Path
\text{Mod}_{\infty,\sigma}(\Gamma(a,b)) = \text{shortest}(a,b;\sigma^{-1})^{-1}
Max-Flow Min-Cut
\text{Mod}_{1,\sigma}(\Gamma(a,b)) = \text{min-cut}(a,b;\sigma)
Effective Resistance
\text{Mod}_{2,\sigma}(\Gamma(a,b)) = R_\text{eff}(a,b;\sigma)^{-1}
Modulus Metrics
(a,b)\mapsto \text{Mod}_{p,\sigma}(\Gamma(a,b))^{-\frac{q}{p}}
is a metric for \(1<p<\infty\)
Generalizing the Family of Objects
Objects are Vectors
e_1
e_2
e_3
e_4
\gamma_1 =
\begin{bmatrix}
1 \\ 0 \\ 1 \\ 1
\end{bmatrix}
\gamma_2 =
\begin{bmatrix}
1 \\ 1 \\ 0 \\ 0
\end{bmatrix}
Objects are Vectors
e_1
e_2
e_3
e_4
\gamma_1 =
\begin{bmatrix}
1 \\ 0 \\ 1 \\ 1
\end{bmatrix}
\gamma_2 =
\begin{bmatrix}
1 \\ 1 \\ 0 \\ 0
\end{bmatrix}
In general, \(\Gamma\subset\mathbb{R}^E_{\ge 0}\)
\(\text{Mod}(\Gamma)\)
\begin{equation*}
\begin{split}
\underset{\rho\in\mathbb{R}^E_{\ge 0}}{\text{minimize}}\quad&\sum_{e\in E}\rho(e)^2\\
\text{subject to}\quad&\rho\ge 0,\\
& \gamma^T\rho\ge 1\quad\forall\;\gamma\in\Gamma
\end{split}
\end{equation*}
\(\text{Mod}(\Gamma)\)
\begin{equation*}
\begin{split}
\underset{\rho\in\mathbb{R}^E_{\ge 0}}{\text{minimize}}\quad&\sum_{e\in E}\rho(e)^2\\
\text{subject to}\quad&\rho\ge 0,\\
& \gamma^T\rho\ge 1\quad\forall\;\gamma\in\Gamma
\end{split}
\end{equation*}
\rho\in\text{Adm}(\Gamma)
Fulkerson Duality
\mathbb{R}^E
Fulkerson Duality
\mathbb{R}^E
Fulkerson Duality
\mathbb{R}^E
\text{Adm}(\Gamma)
Fulkerson Duality
\hat{\Gamma} = \text{ext}(\text{Adm}(\Gamma))
\mathbb{R}^E
\text{Adm}(\Gamma)
Fulkerson Duality
\text{Mod}_{p,\sigma}(\Gamma)^{\frac{1}{p}}\text{Mod}_{q,\hat{\sigma}}(\hat{\Gamma})^{\frac{1}{q}} = 1
\hat{\sigma} = \sigma^{-\frac{q}{p}}
Key Example:
\(\Gamma\) = \(ab\)-paths
\(\hat{\Gamma}\) = \(ab\)-cuts
\begin{equation*}
\begin{split}
\underset{\eta\in\mathbb{R}^E_{\ge 0}}{\text{minimize}}\quad&\sum_{e\in E}\eta(e)^2\\
\text{subject to}\quad&\eta\ge 0,\\
& \hat{\gamma}^T\eta\ge 1\quad\forall\;\hat{\gamma}\in\hat{\Gamma}
\end{split}
\end{equation*}
\(1-\alpha\)
\(1-\alpha\)
\(1\)
\(\alpha\)
\mathcal{E}(\eta) = 1 + \alpha^2 + 2(1-\alpha)^2
\alpha=\frac{2}{3}\quad\implies\mathcal{E}(\eta) = \frac{5}{3}=\text{Mod}(\hat{\Gamma})
\(\frac{1}{3}\)
\(\frac{1}{3}\)
\(1\)
\(\frac{2}{3}\)
\(\frac{3}{5}\)
\(\frac{2}{5}\)
\(\frac{1}{5}\)
\(\frac{1}{5}\)
\text{Mod}(\Gamma)=\frac{3}{5}
\text{Mod}(\hat{\Gamma})=\frac{5}{3}
\eta^* = \frac{\rho^*}{\text{Mod}(\Gamma)}
\(\frac{1}{3}\)
\(\frac{1}{3}\)
\(1\)
\(\frac{2}{3}\)
\(\mu(\gamma_2)=\frac{2}{3}\)
\mathbb{E}_{\mu^*}|\underline{\gamma}\cap\underline{\gamma'}|=\frac{5}{3}
\text{Mod}(\hat{\Gamma})=\frac{5}{3}
\eta^*(e) = \mathbb{E}_{\mu^*}(\underline{\gamma}(e))
\(\mu(\gamma_1)=\frac{1}{3}\)
Families and Structural Properties
| family | property |
|---|---|
| connecting paths | metrics |
| spanning trees | hierarchical structure |
| cycles | communities |
| center-to-shell paths | centrality |
| via walks | betweenness |
Modulus Doesn't Need Graphs
What do we need?
- "ground set" \(E\)
- "objects" \(\Gamma\in\mathbb{R}^E_{\ge 0}\)
For example,
- hypergraphs
- matroids
- permutations
- ...
Play With the Code
Thank You
these slides
Introduction to Discrete Modulus
By nathan_albin
