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

Abstract

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 the 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 z = 4.

The Kuramoto model in complex networks

“…These results suggest that the presence of triangles in the topology poorly affects the network dynamics, since the dependence on r can be described by MFAs developed for local tree-like networks [108]. It is noteworthy that similar findings were reported on the performance of other dynamical processes in clustered networks, such as bond percolation, k-core size percolations, and epidemic spreading [94,95]. As demonstrated in [94,95], mean-field theories for local tree-like networks yield remarkably accurate results even for networks with high values of clustering coefficient if the average shortest path is sufficiently small.…”

“…It is important to emphasize that most of analytical approaches are based on MFAs that are only valid for uncorrelated networks in the limit of large populations of oscillators and sufficiently high average degree. Obviously this imposes a constraint to the thorough comprehension of synchronization of real networks, since they are finite and often exhibit sparsity, degree-degree correlations, presence of loops, community structure, and other properties that make the mean-field calculations no longer valid [25,94,95]. While there is still an ongoing effort to generalize MFAs for more sophisticated topologies [96], many numerical studies have extensively investigated synchronization of Kuramoto oscillators in networks with properties observed in real structures.…”

Competition-Induced Criticality in a Model of Meme Popularity

“…The theory matches the simulation results rather well, despite the fact that this network is not treelike-indeed, 44% of links are reciprocal links-and does not have a homogeneous in-degree distribution, as assumed in the derivation of the theory. The accuracy of results from tree-based theories applied to real-world networks has been noted previously [42] and is examined further for this model in Sec. S4 of [33].…”

Epidemic fronts in complex networks with metapopulation structure

“…where q * satisfies (15), and v * is given by the basic reproductive number multiplied by the average product of relative position and incidence, −∆xe −∆xq * , on the end of a randomly selected edge. Fig.…”

“…3 that the epidemic front is broader in the scale-free network than in the Poisson. This difference comes from the much larger front speed of the former, which had average excess degrees an order of magnitude larger than the latter, (12) and (15), and the relatively similar relaxation times (21) for the two classes of networks (implying that the time scale over which a site is infectious in each network is roughly the same). In the more homogeneous Poisson networks, the front is more narrow and propagates through the lattice on the same time scales as the local infection dynamics; whereas in the scale-free case, the leading edge of the front propagates quickly through the lattice, followed by a slower relaxation to the stable equilibrium state behind the front.…”