We study continuous time random walks on a network. We find that the steady state of the random walk on a given node is independent of the nature of the distribution of waiting times $\psi \left(\tau \right)$ and depends only on the local neighborhood of the node. The relaxation toward the steady state is, however, dependent on $\psi \left(\tau \right)$. For $\psi \left(\tau \right)\stackrel{\text{large}\tau }{\sim }{\tau }^{-1-\alpha },0<\alpha <1$, the relaxation behavior depends on $\alpha$. For $\alpha >1,\psi \left(\tau \right)$ possesses a finite mean and the relaxation exhibits a scaling behavior independent of the specific details of $\psi \left(\tau \right)$. We find that the distribution of recurrence times for a given node possesses the same qualitative form as $\psi \left(\tau \right)$. Starting from a node $i$, we study the mean number of distinct nodes visited by the random walk $\langle S_i\left(t\right)\rangle$, and find that analogous to the relaxation behavior, its properties are dependent on the exponent $\alpha$. Interestingly, $\langle S_i\left(t\right)\rangle$ grows algebraically with time; that is, $\langle S_i\left(t\right)\rangle \sim t^{\alpha }$ for $\alpha <1$. On the other hand, for $\alpha >1,\langle S_i\left(t\right)\rangle \sim Nf(t/N\langle \tau \rangle )$, with the scaling function $f$ depending only on the structural details of the network of size $N$. In addition, for exponentially distributed waiting times, $\langle S_i\left(t\right)\rangle$ grows linearly with $t$ at small times for homogeneous networks.

Close

PrevNext

View figure in article

• Network searches
• Random walks
• Transport in networks