TITLE: What does the value of eigenvectors of a graph Laplacian matrix mean?
QUESTION [5 upvotes]: I know that the eigenvectors of a Laplacian matrix of a graph are so important. They show the locality over the graph (as I know). But whatever I've read about an eigenvector of Laplacian graph is about the smoothness of the eigenvector and the relative value of neighbor nodes in an eigenvector (how close the value of two neighbor nodes are in an eigenvector). Is there any intuition behind the exact values in an eigenvector?
For example, if $u_i$ is an eigenvector and $u_{ij}$ shows the $j$-th value of $u_i$, and $u_{ij}$ is the maximum value in $u_{i}$, can we say anything about node $j$? Do higher values show anything relative to lower values? Or does the sign of the value show anything?
I want to get an intuition about the values of an eigenvector.

REPLY [2 votes]: As a complementary answer, one can think about random walks on $G$. The ordinary random walk on $G$, where at each time step the process makes a transition uniformly at random, is actually not so intimately connected with the ordinary Laplacian matrix $L$ (as was discussed in the comments on Misha Lavrov's answer). It is instead connected with the symmetric normalized Laplacian matrix $L_{sym}$. The matrix $L_{sym}$ is related to the transition probability matrix of the random walk $P$ by the identity $L_{sym}=I-D^{1/2} P D^{-1/2}$. The small nonzero eigenvalues of $L_{sym}$ correspond to near-$1$ eigenvalues of $P$. Because $p(t)=p(0) P^t$, the left eigenvectors of $P$ with eigenvalues near $1$ correspond to "slow probability currents" between the vertices where the eigenvector is positive and the vertices where the eigenvector is negative, with relatively more probability lost or gained at a vertex where the absolute value of the eigenvector is larger.
Just about everything I said above does in fact apply to the Laplacian, once you change the setting. Specifically, the Laplacian is really connected to a continuous time Markov chain on $G$ where each edge is a possible transition, with all edges having unit rate. Thus in this situation the typical time to stay at a high degree vertex is lower than at a low degree vertex (unlike for the ordinary random walk). The generator matrix, say $A$, of this CTMC is the negative of the Laplacian (its diagonal has negative entries), and the probability distribution evolves as $p'=pA$ so $p(t)=p(0) e^{At}=p(0) e^{-Lt}$. So now in this setting the left eigenvectors of $L$ with eigenvalues close to zero (which are the left eigenvectors of the adjacency matrix with eigenvalues close to $1$) generate the slow probability currents.