time for the random walk on the graph and $N$ is the number of nodes.

This procedure is equivalent to evolving the discretized heat equation on the graph and can be demonstrated as follows. The heat equation is given by, $$\begin{aligned}
    \frac{\partial u}{\partial t} = \Delta u\,,\end{aligned}$$ where $u$ is a function of time and space, $\partial u/\partial
t$ is the partial derivative of $u$ with respect to time, and $\Delta$ is the Laplace operator [@Cit:cluster2].

When the above equation is discretized on a graph $\mathcal{G}=(V,E)$ one gets the following equation: $$\begin{aligned}
    \u_{i}(t+1) = \u_{i}(t) - \displaystyle\sum_{j\in\mathcal{N}(i)}\L_{ij}\u_{j}(t)\,,\end{aligned}$$ for $i,j \in V$. Here $\u_{i}(t)$ is the scalar value of $u$ on node $i$ at time $t$ and $\mathcal{N}(i)$ are the neighbors of node $i$ in the graph. The graph Laplacian $\L=[\L_{ij}]$ is the discrete counterpart of the $\Delta$ operator. The above iteration can be re-written, in matrix form, $\u(t+1) = (\I-\L)\,\u(t)$ where $\u(t) =
(\u_{1}(t),\dots,\u_{N}(t))^{T}$. The solution of this iteration is, $$\begin{aligned}
    \u(t) = C_0\1 + C_1 (1-\lambda_2)^t \v^{(2)} + \dots
    +C_N (1-\lambda_N)^t
    \v^{(N)}\,,
    \label{eq:HeatSol}\end{aligned}$$ where constants $C_{j}$ depend on the initial condition $\u(0)$. It is interesting to note that in Eqn. \[eq:HeatSol\], the dependence of the solution on higher eigenvectors 