How universal is the mean-field universality class for percolation in complex networks?

TL;DR

This work investigates the universality of mean-field percolation in complex networks by introducing Static Triadic Closure (STC) graphs, which host strong clustering, overlapping short loops, and degree correlations. It derives an exact solution for site percolation on STC graphs and leverages an equivalence with extended-range percolation (R=2) on treelike backbones to obtain the full critical behavior. The key finding is that degree heterogeneity alone does not fix the universality class: clustering and correlations can modify the percolation threshold and, in some regimes, the critical exponents, though the backbone topology often preserves homogeneous MF exponents. This highlights a nontrivial interplay between local loop structure and degree heterogeneity and suggests that a treelike backbone may largely determine critical properties even in highly loopy, correlated networks, with potential implications for inferring backbone structure in real networks.

Abstract

Clustering and degree correlations are ubiquitous in real-world complex networks. Yet, understanding their role in critical phenomena remains a challenge for theoretical studies. Here, we provide the exact solution of site percolation in a model for strongly clustered random graphs, with many overlapping loops and heterogeneous degree distribution. We systematically compare the exact solution with heterogeneous mean-field predictions obtained from a treelike random rewiring of the network, which preserves only the degree sequence. Our results demonstrate a nontrivial interplay between degree heterogeneity, correlations and network topology, which can significantly alter both the percolation threshold and the critical exponents predicted by the heterogeneous mean-field. These findings reveal limitations of heterogeneous mean-field theory, demonstrating that the degree distribution alone is insufficient to determine universality classes in complex networks with realistic structural features.

Figures (3)

• Figure 1: (a) Construction of the STC random graph via triadic closure with $f=1$, from a backbone network $\mathcal{G}_0$ (left), to the clustered graph $\mathcal{G}_1$ (right). Note that the degree of a node with original degree $k$ becomes $k+\sum_{i=1}^k r_i$ (for the node indicated by the arrow, $k=3$, $r_1=r_2=r_3=2$, and $K=9$). (b) Equivalence between extended-range percolation with $R=2$ (left) and standard percolation in the STC graph (right). Some nodes are inactive (empty circles). On the left, the dashed lines denote the extended-range connected component with $R=2$ in the backbone $\mathcal{G}_0$. The standard connected components ($R=1$) are also represented with continuous lines for comparison. On the right, continuous lines denote the standard connected component in $\mathcal{G}_1$. Note its equivalence with the $R=2$-connected component on the left.
• Figure 2: Numerical simulations of site percolation in STC random graphs (full symbols) and in a random rewiring of these networks (UCM) (empty symbols), keeping the same degree sequence but with no correlations nor the short-loop structure. The synthetic networks considered have $N=10^6$ nodes, $k_{\text{min}}=2$ and (a) $\widetilde{\gamma}_d=4.5$, (b) $\widetilde{\gamma}_d=3.5$, (c) $\widetilde{\gamma}_d=2.5$. Results are averaged over $100$ independent realizations. Continuous lines are the exact solution for site percolation in STC, Eqs. \ref{['eq:order_parameter']}, and for site percolation in the UCM counterpart, Eq. \ref{['eq:order_parameter_UCM']}.
• Figure 3: (a) Modified branching process for percolation in STC graphs. Here, the edges of the backbone are represented by solid lines, and the edges created by the STC mechanism are represented by dashed lines. The first generation of offsprings emanating from $i$ (green shaded region) is formed by the active nodes that are $i$'s nearest neighbors in $\mathcal{G}_0$ (node $j$), and the active nodes that are $i$'s second neighbors in $\mathcal{G}_0$ and are reachable in $\mathcal{G}_1$ only by dashed lines (node $l$). Note that this modified branching process avoids overcounting effects caused by the presence of loops: node $k$, which is a common neighbor of both $i$ and $j$ (red triangle) belongs to the second generation of offsprings. (b) The standard branching process approach in a random rewiring of the same network, in which the loop structure has been destroyed. The first generation of offsprings emanating from node $i$ are nodes $j,k,l$. This difference with the modified branching process in panel (a) is crucial.