Finite weighted network diffusion data

Let $G=(V,E)$ be a finite simple graph with ordered vertex and edge sets. Choose an orientation of each edge and let $B\in {\mathbb{R}}^{V\times E}$ be the resulting incidence matrix of An oriented incidence matrix of a finite simple graph: a column has $-1$ at its tail, $+1$ at its head, and zero elsewhere. Let $D\subseteq V$ be the vertices with prescribed Dirichlet values and let $I=V\setminus D$ be the free vertices. Assign a mass $m_i>0$ to each $i\in I$ and a conductance $c_e\ge 0$ to each $e\in E$. Put

$M=\mathrm{diag}(m_i{)}_{i\in I},C=\mathrm{diag}(c_e{)}_{e\in E},L_C=BC{B}^{\mathsf{T}}\mathrm{.}$

The matrix $L_C$ is the conductance-weighted graph stiffness. Its free Dirichlet restriction $L_D$ is the principal submatrix $(L_C{)}_{I,I}$. The finite weighted network diffusion equation is

$M{u}^{\mathrm{\prime }}(t)+L_{D}u(t)=f(t),$

where $u(t)\in {\mathbb{R}}^I$ and $f(t)\in {\mathbb{R}}^I$. If $D=\mathrm{\varnothing }$, this is the closed Neumann network equation. If the boundary values are instead a nonzero vector $g(t)$, restriction of $L_C(u,g)$ to $I$ gives $M{u}^{\mathrm{\prime }}+L_{D}u=f-(L_C{)}_{I,D}g$.

When every $c_e=1$, $L_C=B{B}^{\mathsf{T}}$ agrees with the ordinary graph Laplacian by The Laplacian equals $B{B}^{\mathsf{T}}$ for every oriented incidence matrix $B$. Zero conductances are allowed and remove those edges from the diffusion coupling; they do not remove the vertices or masses.