Editing Discrete Laplace operator (section)

=== Equilibrium behavior ===
To understand <math display="inline">\lim_{t \to \infty}\phi(t)</math>, the only terms <math display="inline"> c_i(t) = c_i(0) e^{-k \lambda_i t}</math> that remain are those where <math display="inline">\lambda_i = 0</math>, since
: <math>\lim_{t\to\infty} e^{-k \lambda_i t} = \begin{cases}
  0, & \text{if} & \lambda_i > 0 \\
  1, & \text{if} & \lambda_i = 0
\end{cases}</math>

In other words, the equilibrium state of the system is determined completely by the [[Kernel (linear algebra)|kernel]] of <math display="inline">L</math>.

Since by definition, <math display="inline">\sum_{j}L_{ij} = 0</math>, the vector <math display="inline">\mathbf{v}^1</math> of all ones is in the kernel. If there are <math display="inline">k</math> disjoint [[Connected component (graph theory)|connected components]] in the graph, then this vector of all ones can be split into the sum of <math display="inline">k</math> independent <math display="inline">\lambda = 0</math> eigenvectors of ones and zeros, where each connected component corresponds to an eigenvector with ones at the elements in the connected component and zeros elsewhere.

The consequence of this is that for a given initial condition <math display="inline">\phi(0)</math> for a graph with <math display="inline">N</math> vertices
: <math>\lim_{t\to\infty}\phi(t) = \left\langle \phi(0), \mathbf{v^1} \right\rangle \mathbf{v^1}</math>

where
: <math>\mathbf{v^1} = \frac{1}{\sqrt{N}} [1, 1, \ldots, 1] </math>

For each element <math display="inline">\phi_j</math> of <math display="inline">\phi</math>, i.e. for each vertex <math display="inline">j</math> in the graph, it can be rewritten as
: <math>\lim_{t\to\infty}\phi_j(t) = \frac{1}{N} \sum_{i = 1}^N \phi_i(0) </math>.

In other words, at steady state, the value of <math display="inline">\phi</math> converges to the same value at each of the vertices of the graph, which is the average of the initial values at all of the vertices.  Since this is the solution to the heat diffusion equation, this makes perfect sense intuitively.  We expect that neighboring elements in the graph will exchange energy until that energy is spread out evenly throughout all of the elements that are connected to each other.