Emergence of disconnected clusters in heterogeneous complex systems

The paper challenges the traditional, percolation-inspired intuition that highly correlated critical clusters are spatially connected. It proposes an alternative scenario in which, at the critical point, highly correlated sites can form spatially disconnected regions. The authors focus on the disordered contact process (DCP) as a proof-of-concept, where quenched disorder is known to induce ultra-slow dynamics and infinite-randomness criticality. While ensemble-averaged critical properties are well understood, simulating individual disordered samples near criticality is difficult due to extremely slow dynamics and large fluctuations. To overcome this, the study combines an efficient strong-disorder renormalization group (SDRG) approach with quasi-stationary simulations to probe the organization of correlations and identify whether the most highly correlated sites are connected or disconnected in space.

Background is provided on the contact process and its absorbing-state phase transition, which for regular lattices with uniform rates belongs to the directed percolation universality class. In the presence of quenched disorder, extensive Monte Carlo studies in 1D–3D have established ultra-slow critical dynamics and infinite-randomness behavior consistent with SDRG predictions. Prior work on spatial networks with long-range links (probability decaying with distance) showed similar ultra-slow scaling, variable critical exponents depending on long-range parameters, and irrelevance of additional rate disorder at s=2d. SDRG has been developed and applied to the DCP and related random quantum Ising models, establishing infinite-randomness fixed points and asymptotically exact decimation rules below four dimensions. These studies motivate using SDRG to understand correlation structures and guide simulations in heterogeneous media.

Model: The disordered contact process (DCP) is defined on a network with adjacency A_ij. Each site i has a binary state n_i∈{0,1} (infected/healthy). Infected sites heal with (possibly site-dependent) rate μ_i. Infection attempts occur along undirected links with symmetric rates λ_ij=λ_ji; an infected i infects neighbor j at rate λ_ij if j is healthy. This is the SIS model variant. The system exhibits a transition between an absorbing (all healthy) phase and an active phase with nonzero infected density.

Strong-disorder renormalization group (SDRG): Iteratively eliminate the largest rate Ω=max{μ_i,λ_ij} via two decimations:

• If Ω=λ_ij (strong link), merge sites i and j into a cluster with effective healing rate ln μ̃ = ln μ_i + ln μ_j − ln λ_ij + ln 2 (the ln 2 term neglected in practice near the IRFP). Valid when μ_i,μ_j≪λ_ij.
• If Ω=μ_i (strong healing), delete site i and generate effective infection between each neighbor pair j,k via ln λ̃_jk = ln λ_ji + ln λ_ik − ln μ_i, valid when λ_ji,λ_ik≪μ_i. A maximum rule keeps only the larger of multiple parallel links that may arise. Repeating these steps lowers Ω and the number of degrees of freedom. Approaching the infinite-randomness fixed point (IRFP), the distribution of logarithmic rates broadens without bound, making the steps asymptotically exact and justifying the approximations (maximum rule, dropping ln 2). Implementations exploit these for computational efficiency. In finite systems, clusters are removed until none remain; the removal scales Ω_n (with Ω_1 the last) set widely separated lifetimes τ_n≈Ω_n^−1. For times τ_2≪t≪τ_1, dynamics is dominated by the last remaining cluster. The last-cluster structure is also used to define a pseudocritical point by checking whether it changes upon doubling the system.

Quasi-stationary simulations: For regular lattices, an occupied site i is chosen at random; with probability μ/(μ+1) it heals; otherwise a neighbor j is chosen uniformly among n neighbors and infected with probability λ_ij if healthy. The time increment is Δt=1/N_i, where N_i is the number of infected sites. For non-regular networks, maintain a list of active directed links with infected sources; let N_e be its size. With probability μ N_i/(μ N_i+N_e) a healing event occurs on a randomly chosen infected site; otherwise select an active link (i→j) uniformly and attempt infection of j with probability λ_ij if healthy. The time increment is Δt=1/(μ N_i+N_e). A quasi-stationary (QS) reflecting boundary condition is imposed: when only a single active site remains, the healing event is rejected to prevent absorption. Stationarity is verified by observing invariant mean density and variance under increased relaxation. Typical total simulation times were ~10^28 steps (half for relaxation), and for detailed measurements ~10^30 steps. Pseudocritical points in each disorder realization are determined by maximizing the susceptibility X = N(⟨p^2⟩−⟨p⟩^2)/⟨p⟩, where p is the global active-site density, evaluated in the QS state.

Disorder specification: Infection rates λ are drawn from a power-law distribution P(λ)=λ^{−1−α} with α>0 to ensure sufficiently strong disorder and broad distributions; healing rates are kept uniform to serve as a tunable control parameter. For fair comparison, the same random infection rates (random environment) are used in SDRG and simulations, while the uniform healing rate is tuned in each method to its internal indicator of criticality to compensate for early-stage SDRG shifts.

Comparison protocol: SDRG is run to the stage with a single remaining cluster (capturing dynamics for τ_2≪t≪τ_1). QS simulations at the pseudocritical point yield spatial maps of local occupation probabilities. These maps are compared to the SDRG’s largest clusters (and subleading ones for moderate sizes) for 1D, 2D, 3D lattices and 2D lattices with long-range links (algebraically decaying connection probability, exponent s=4, number of shortcuts N/16).

• Across 1D, 2D, and 3D lattices and 2D lattices with long-range connections, the critical dynamics is governed by one dominant, highly correlated cluster that is spatially disconnected: simultaneously active (infected) sites typically are not directly connected by edges.
• Strong agreement is observed between quasi-stationary simulation density profiles and SDRG-predicted dominant clusters, validating SDRG’s asymptotic description of correlation structure even for moderate system sizes.
• Finite-size samples display widely separated cluster lifetimes; during τ_2≪t≪τ_1, activity is overwhelmingly concentrated on the last SDRG cluster, which captures most of the probability weight.
• The dominant and secondary clusters exhibit deep internal gaps in the local activity profile, signaling an asymptotically disconnected, fractal-like structure of correlated activity.
• Reported fractal dimensions of SDRG clusters at criticality: in 2D, d_f≈1.02(2); in 3D, d_f≈1.16(2). In 2D with long-range connections (s=4), critical clusters are even more disconnected with formally zero fractal dimension.
• Because the SDRG recursion for the DCP maps to that of the random quantum Ising model, the results imply quantum-correlated but spatially disconnected magnetic domains at criticality in that class as well.

The findings demonstrate that at the infinite-randomness critical point of heterogeneous spreading dynamics, functional correlations decouple from physical connectivity: the most strongly correlated sites need not be directly connected by network edges. This resolves the apparent contradiction with percolation-based intuition by showing that ultra-slow, rare-event dominated dynamics organizes activity into sparse, spatially disconnected clusters sustained by repeated, indirect infection paths and positive feedback through the network. The SDRG framework accounts for this by generating renormalized, effective interactions along indirect paths and predicting dominance by one late-time cluster. The validation via quasi-stationary simulations confirms SDRG’s asymptotic picture for individual realizations. The implication extends to disordered quantum magnets (random quantum Ising model), where the same SDRG structure indicates quantum-correlated domains that are spatially disconnected at criticality. Beyond physics, the results suggest that in complex systems such as brain networks, strongly correlated functional regions need not be directly connected structurally, highlighting a qualitative difference between functional and anatomical connectivity, particularly near critical-like operating regimes.

By combining quasi-stationary simulations with a computationally efficient SDRG approach, the study shows that the critical behavior of the disordered contact process is dominated by a single, highly correlated yet spatially disconnected cluster across 1D–3D lattices and in 2D with long-range links. The agreement between simulations and SDRG at the level of individual samples substantiates the infinite-randomness, ultra-slow critical scenario and reveals the counter-intuitive organization of correlations decoupled from direct connectivity. The similarity of SDRG recursions implies analogous behavior in the disordered quantum Ising model, predicting quantum-correlated but disconnected domains. Future work should apply these methods to empirical connectomes and other real-world heterogeneous networks to test how functional clusters relate to structural connectivity, and to quantify finite-size and disorder-strength effects on cluster geometry and dynamics.

• Early-stage SDRG approximations (maximum rule, neglecting ln 2) and the finite breadth of disorder can shift the apparent critical point; thus, per-sample tuning via internal indicators is required to ensure criticality.
• The asymptotic exactness of SDRG holds below four dimensions and for sufficiently strong disorder; weaker disorder or higher dimensions may exhibit larger corrections.
• Individual-sample simulations near criticality suffer from extremely slow dynamics and large fluctuations, necessitating very long quasi-stationary runs and reflecting-boundary conditions that may introduce finite-size biases.
• Finite system sizes mean that only the last few clusters and their lifetimes are accessible; scale separation improves with size but remains a limitation for moderate N.
• Reported fractal dimensions for clusters rely on SDRG and prior studies; precise numerical estimation in the present simulations is limited by system size and computational cost.