Collective relaxation dynamics of small-world networks

Complex networks exhibit a wide range of collective dynamic phenomena, including synchronization, diffusion, relaxation, and coordination processes. Their asymptotic dynamics is generically characterized by the local Jacobian, graph Laplacian, or a similar linear operator. The structure of networks with regular, small-world, and random connectivities are reasonably well understood, but their collective dynamical properties remain largely unknown. Here we present a two-stage mean-field theory to derive analytic expressions for network spectra. A single formula covers the spectrum from regular via small-world to strongly randomized topologies in Watts-Strogatz networks, explaining the simultaneous dependencies on network size $N$, average degree $k$, and topological randomness $q$. We present simplified analytic predictions for the second-largest and smallest eigenvalue, and numerical checks confirm our theoretical predictions for zero, small, and moderate topological randomness $q$, including the entire small-world regime. For large $q$ of the order of one, we apply standard random matrix theory, thereby overarching the full range from regular to randomized network topologies. These results may contribute to our analytic and mechanistic understanding of collective relaxation phenomena of network dynamical systems.

Figure 1

Rewiring—stochastic and mean field. Cartoon for $N=10$ and $k=4$. Instead of taking out (step 1) and putting back (step 2) edges randomly (left column), the corresponding weight is subtracted (step 1) uniformly and added (step 2) in two fractions (right column).

Figure 2

The banded structure of the mean-field graph Laplacian ${\tilde{\mathrm{\Lambda }}}^{\mathrm{mf}}$ given in Eqs. (12) and (19). It has the weights ${\overline{w}}_1=1-q+w_1$ for $c_i|i\in S_1$ and $w_2$ for $c_i|i\in S_2$ [see Eq. (13) for the definition of $S_i$]. For $q=0$, and hence ${\overline{w}}_1=1$ and $w_2=0$, we recover the exact ring Laplacian.

Figure 3

Interpolating the eigenvalues. Equations (29) (blue) and (33) (red) both contain the eigenspectrum ${\tilde{\lambda }}_l^{\mathrm{mf}}$ for $l\in \{2,\cdots ,N\}$ correctly (green circles). While ${\tilde{\lambda }}_l^{\mathrm{mf}}$ in Eq. (29) includes ${\tilde{\lambda }}_1^{\mathrm{mf}}$ as ${lim}_{l\to 1}{\tilde{\lambda }}_l^{\mathrm{mf}}={\tilde{\lambda }}_1^{\mathrm{mf}}=0$ as well, ${\tilde{\lambda }}_l^{\mathrm{mf}}$ for $l\in \{2,...,N\}$ in Eq. (33) does not: To further simplify expressions, we have used Eqs. (30) and (31) only valid for integer $l$, but apparently not for $l=0$.

Figure 4

${\tilde{\lambda }}_2^{\mathrm{mf}}$ always constitutes the second largest eigenvalue. Functions $f\left(x\right)$ [Eq. (36)], the oscillating function $sin\left[\right(k+1\left)x\right]$, and the envelope function $1/sin\left(x\right)$ are plotted vs. $x=\frac{(l-1)\pi }{N}\in (0,\pi )$ for $k=10$. Obviously, a larger $k$ leads to more roots of $f\left(x\right)$, but otherwise the functions show the same characteristics for all $k\le N-1$: $f\left(x\right)$ has a local maximum at $x=0$ and decreases strictly monotonically up to the following minimum. For larger $x$ the envelope function guarantees that all values up to $x=\pi /2$ are smaller than $f(x_{l=2}=\frac{\pi }{N})$.

Figure 5

Important points of the function $f\left(x\right)$. The function $f\left(x\right)$ is plotted for $k=100$. The boundary points [green, Eq. (42)] of the function $f\left(x\right)$ and the envelope function $1/sin\left(x\right)$, roots of $f\left(x\right)$ [purple, Eq. (41)], and the $x$ value $x_{l=2}$ [blue, Eq. (40), $N=1000$] corresponding to the eigenvalue ${\tilde{\lambda }}_2^{\mathrm{mf}}$ are highlighted.

Figure 6

Second-largest eigenvalues from regular to randomized networks. Numerical measurements for undirected ($\times$) and directed ($◯$) networks in comparison with the analytical mean-field predictions [Eq. (50), solid lines] as a function of $q$, for different degrees $k$. The error bars on the numerical measurements are smaller than the data points ($N=1000$, each data point averaged over 100 realizations). Adapted from Ref. [28].

Figure 7

Scaling of the real part of the second-largest eigenvalue for fixed network size. Numerical measurements for undirected ($\times$) and directed ($◯$) networks in comparison with the analytical mean-field predictions [Eq. (50), solid lines]. The analytical approximations obtained from the expansions in Eq. (51) are depicted by the dashed lines. The error bars on the numerical measurements are smaller than the data points, $N$ fixed to 1000.

Figure 8

Second-largest eigenvalues in dependence on edge density and network size. (a) Numerical measurements for directed ($◯$) and undirected ($\times$) networks (error bars smaller than the data points) in comparison with the analytic mean field prediction (53) for $N=2000$. (b) Asymptotic ($N\to \infty$) real parts of the second-largest eigenvalues ${\lambda }_2$ in dependence on the network size $N$ for fixed-edge density $d=k/N=0.1$ [$q$ values and symbols as in panel (a)]. (c) Asymptotic ($N\to \infty$) real parts of the second-largest eigenvalues ${\lambda }_2$ in dependence on the network size $N$ for fixed-edge density $d=k/N=0.5$ [$q$ values and symbols as in (a)]. Panels (a) and (b) are adapted from Ref. [28].

Figure 9

Smallest eigenvalues from regular to randomized networks. Numerical measurements for undirected ($\times$) and directed ($◯$) networks in comparison with the analytical mean-field predictions [Eq. (45), solid lines] as a function of $q$, for different degrees $k$. Dashed lines show the analytic estimations of the smallest eigenvalues [Eq. (48)]. The error bars on the numerical measurements are smaller than the data points ($N=1000$, each data point averaged over 100 realizations).

Figure 10

Analytic prediction of the second-largest eigenvalues close to $q=1$. (a) Numerical measurements for undirected ($\times$) networks in comparison with the analytical predictions ${\lambda }_2^{\mathrm{wsc}}$ via Wigner's semicircle law [Eq. (67), solid lines], for different degrees $k$. (b) Numerical measurements for directed ($◯$) networks in comparison with the analytical predictions ${\lambda }_2^{\mathrm{rmt}}$ from the theory of asymmetric random matrices [Eq. (68), solid lines]. The error bars on the numerical measurements are smaller than the data points ($N=1000$, each data point averaged over 100 realizations). Dashed lines are only a guide to the eye. Taken from Ref. [28].