Accuracy of mean-field theory for dynamics on real-world networks

Mean-field analysis is an important tool for understanding dynamics on complex networks. However, surprisingly little attention has been paid to the question of whether mean-field predictions are accurate, and this is particularly true for real-world networks with clustering and modular structure. In this paper, we compare mean-field predictions to numerical simulation results for dynamical processes running on 21 real-world networks and demonstrate that the accuracy of such theory depends not only on the mean degree of the networks but also on the mean first-neighbor degree. We show that mean-field theory can give (unexpectedly) accurate results for certain dynamics on disassortative real-world networks even when the mean degree is as low as 4.

Figure 1

Order parameter for synchronization in the Kuramoto phase oscillator model running on (a) the Facebook Oklahoma network [19] and (b) the US Power Grid network [20] as a function of the coupling parameter $K$. The order parameter $r_2$ is defined in Eq. (2). The MF theory of [21] is given by Eq. (3).

Figure 2

Fraction of infected nodes in the steady state of the SIS process on the (a) Autonomous System (AS) Internet [26] and (b) Electronic Circuit (s838) [25] networks as a function of the spreading rate $\beta$. The MF theory is from Refs. [49, 50]. Observe that uncorrelated and correlated MF theories are indistinguishable in panel (b).

Figure 3

Survival probability for voter-model trajectories in the disordered state as a function of time on (a) C. elegans neural network [20] and (b) a synthetic clustered network with $\gamma (3,3)=1$, which is generated as described briefly in the main text (see Ref. [10] for details). Both networks contain $N\approx 300$ nodes. The theory curves are from Refs. [6, 55].

Figure 4

Results for dynamics of Figs. 1, 2, 3 for all six networks used in those figures. Curves and symbols are as in Figs. 1, 2, 3. For the voter model, black squares show the numerical results obtained by rewiring networks in a manner that conserves both degree distribution and degree-degree correlations [60].

Figure 5

Accuracy of cMF theory for SIS dynamics as a function of values of mean degree $z$ and mean nearest-neighbor degree $d$ for many real-world networks. The numbers that label the networks are enumerated in column 1 of Table . The colors indicate the magnitude of the relative error $E_S$, defined in Eq. (8), between cMF theory and numerical simulations for SIS dynamics with $I_{\mathrm{theory}}=0.5$.

Figure 6

Error (diamonds) of cMF theory in predicting steady-state infected fraction for SIS dynamics as a function of Pearson correlation coefficient $r$ in the synthetic networks (described in the text) with $(k_1,k_2)=(3,12)$. Also shown are the errors for the class of degree-3 nodes (crosses) and for the class of degree-12 nodes (triangles): these are calculated similar to Eq. (8), using $i_k\left(\infty \right)$ [from Eq. (4)] for $I_{\mathrm{theory}}$ and the average infected fraction of degree-$k$ nodes in simulations for $I_{\mathrm{numerical}}$. The solid line gives the mean nearest-neighbor degree $d$.

Figure 7

Mean (black squares) and standard deviation (error bars show one standard deviation above and below the mean) of the distribution of $f_i$ values (described in text) on the same networks as in Fig. 6. The values of $I_{\mathrm{theory}}$ are given by cMF theory, using $i_3\left(\infty \right)$ from Eq. (4).

Figure 8

Relative errors $E_{\mathit{SI}}$ (symbols), given by Eq. (10), in the calculation of the fraction of SI edges using MF assumption (iii) on the same networks as in Fig. 6. The solid line gives the mean nearest-neighbor degree $d$.

Figure 9

As in Fig. 5, except that color indicates the magnitude of the relative error, determined by comparing numerical results to (a) MF theory for the Kuramoto order parameter $r_2$ (with $r_{2\text{theory}}=0.6$) from Eq. (12); (b) MF theory for the voter-model survival probability $P_s\left(t\right)$ at time $t$ (with $P_{s\text{theory}}={10}^{-2}$) from Eq. (13); and (c) PA theory for the voter-model survival probability.

Figure 10

Location of real-world networks in the $(q,d)$ parameter plane, where $q$ is the error measure (14), which was shown in Ref. [60] to be correlated with the error of PA theory for bond percolation. We use the same symbols as in Fig. 5.