Double percolation phase transition in clustered complex networks

The internal organization of complex networks often has striking consequences on either their response to external perturbations or on their dynamical properties. In addition to small-world and scale-free properties, clustering is the most common topological characteristic observed in many real networked systems. In this paper, we report an extensive numerical study on the effects of clustering on the structural properties of complex networks. Strong clustering in heterogeneous networks induces the emergence of a core-periphery organization that has a critical effect on the percolation properties of the networks. We observe a novel double phase transition with an intermediate phase in which only the core of the network is percolated and a final phase in which the periphery percolates regardless of the core. This result implies breaking of the same symmetry at two different values of the control parameter, in stark contrast to the modern theory of continuous phase transitions. Inspired by this core-periphery organization, we introduce a simple model that allows us to analytically prove that such an anomalous phase transition is in fact possible.

💡 Research Summary

The paper investigates how clustering—a ubiquitous feature of real-world networks—affects the structural and percolation properties of heterogeneous complex networks. While previous studies have reported conflicting results on whether clustering raises or lowers the percolation threshold, this work shows that strong clustering fundamentally reshapes the network into a core‑periphery architecture, leading to a novel “double percolation” phenomenon.

Using synthetic scale‑free networks with degree exponent (\gamma = 3.1) and a tunable clustering spectrum (\bar c(k)), the authors perform extensive bond‑percolation simulations (10⁴ realizations per network) across a wide range of system sizes ((N = 5\times10^3) to (5\times10^5)). For low clustering the susceptibility (\chi) exhibits a single sharp peak, indicating the usual continuous percolation transition. When clustering is increased (e.g., (\bar c(k)=0.25)), (\chi) develops two distinct peaks. Finite‑size scaling reveals that both peaks diverge as power laws (\chi_{\max}\sim N^{\gamma/\nu}), confirming that each corresponds to a genuine continuous phase transition rather than a smeared crossover. The first peak moves toward (p=0) as (N) grows, implying that the network is always percolated in the thermodynamic limit; the second peak remains at a finite occupation probability, signalling a second, independent transition.

To uncover the structural origin of the two transitions, the authors apply the m‑core decomposition, which iteratively removes edges that belong to fewer than (m) triangles. High clustering yields a hierarchy of nested m‑cores: the innermost core (high‑m) is a dense, well‑connected subgraph, while the outer layers consist of many small, loosely connected components—the periphery. Visualizations (Fig. 3) demonstrate that the core remains intact when edges of multiplicity zero are stripped away, whereas the periphery fragments. This core‑periphery split suggests that the first susceptibility peak corresponds to the percolation of the dense core, while the second peak marks the percolation of the peripheral subgraph.

To prove that such a double transition can arise purely from topology, the authors construct an analytically tractable model: two Erdős‑Rényi graphs (core and periphery) of sizes (N_c) and (N_p) ((r=N_c/N_p<1)), with average internal degrees (\bar k_c>\bar k_p) and inter‑layer average degrees (\bar k_{cp},\bar k_{pc}). When the number of inter‑layer links scales sub‑linearly with system size ((\sim N^{\alpha}), (0<\alpha<1)), the inter‑layer average degrees vanish in the thermodynamic limit, effectively decoupling the two layers. Consequently, the combined network exhibits two distinct percolation thresholds: (p_c^{(c)}=\bar k_c^{-1}) for the core and (p_c^{(p)}=\bar k_p^{-1}) for the periphery. Simulations confirm that for (\alpha=0.5) the order parameter (g(p)) shows a clear discontinuity in its derivative at the second threshold, matching the double‑peak susceptibility observed in clustered networks. When (\alpha=1) (macroscopic coupling), the two transitions merge into a crossover, reproducing the behavior seen in earlier studies that reported a single shifted threshold.

The authors thus demonstrate that strong clustering can spontaneously generate a core‑periphery organization that produces two separate continuous percolation transitions. This finding challenges the conventional wisdom that a connected system can break the same symmetry only once as a control parameter varies. It also has practical implications: in epidemic spreading, for example, the core may become infected at a lower transmissibility, while the peripheral population remains largely protected until a higher transmissibility is reached, potentially informing targeted vaccination strategies.

In summary, the paper provides (i) comprehensive numerical evidence of a double percolation transition in highly clustered heterogeneous networks, (ii) a structural explanation via m‑core decomposition, and (iii) an analytically solvable core‑periphery random‑graph model that reproduces the phenomenon. The work opens new avenues for studying how higher‑order structures (triangles, cliques) shape critical phenomena on complex networks.