Emergence of disconnected clusters in heterogeneous complex systems

The study of correlated clusters in complex systems is crucial across various fields, from magnetism to out-of-equilibrium systems. Percolation theory suggests that highly correlated sites form connected clusters, a picture that holds well at small scales away from criticality. However, the authors posit that this intuition breaks down at criticality where correlation length diverges. Traditional intuition, based on percolation processes, suggests that highly correlated critical clusters, while varying in size, remain connected. This research challenges this assumption, proposing an alternative scenario where highly correlated clusters fragment into disconnected regions at the critical point, in both classical and quantum systems. The disordered contact process (DCP), a simple infection-spreading model, serves as a proof-of-concept for demonstrating this phenomenon. The DCP's critical behavior is well understood statistically, but the detailed dynamics in individual samples are difficult to analyze due to large fluctuations and slow dynamics near criticality. The study aims to understand this complex behavior through a combination of simulations and the SDRG technique, addressing the counter-intuitive organization of dynamical correlations where functional similarity decouples from physical connectivity.

Existing literature extensively covers correlated clusters in various systems. Fortuin and Kasteleyn's work on the random-cluster model (1972) and studies on magnetic domains (Carey & Isaac, 1966; Mansfield & Douglas, 2013) provide a basis for understanding correlated regions. The disordered contact process (DCP) has also received attention, with studies exploring its critical dynamics in the presence of disorder (Moreira & Dickman, 1996; Vojta et al., 2009; Vojta & Dickison, 2005; Vojta, 2012; Vojta, 2006). Previous research using the strong disorder renormalization group (SDRG) method (Hooyberghs et al., 2003, 2004; Iglói & Monthus, 2005, 2018) has provided insights into the DCP's critical behavior, but direct observation of the spatially disconnected clusters has been elusive due to slow dynamics and large fluctuations. Studies on the contact process on networks with long-range connections (Juhász & Kovács, 2013; Muñoz et al., 2010; Juhász et al., 2012) have further contributed to understanding the effect of network topology on critical behavior. The work on the infinite-randomness fixed-point (IRFP) of SDRG transformations (Fisher, 1992, 1995; Neugebauer et al., 2006; Fallert & Taraskin, 2009; Hoyos, 2008; Kovács & Iglói, 2010, 2011) is also relevant, showing how this fixed point governs the long-time behavior and the critical exponents of the DCP.

The study employs a two-pronged approach combining numerical simulations and an analytical technique: 1. **Disordered Contact Process (DCP) Simulation:** The DCP, a model for infection spreading, is defined on a network with binary variables (nᵢ = 0, 1) representing healthy (0) or infected (1) sites. The model includes infection and healing rates (λᵢⱼ and μᵢ, respectively), which can vary depending on site or link properties to introduce disorder. Two simulation implementations are described; one for regular lattices and one for non-regular networks. A quasi-stationary simulation with reflecting boundary conditions is used to avoid the absorbing state (all sites healthy), allowing observation of long-time dynamics. The pseudo-critical point for each sample is determined using susceptibility, maximizing it over the healing rate (μ). Simulations are run for an extended time (10³⁰) to obtain accurate local density measurements, averaged over multiple independent runs to enhance statistical robustness. Infection rates are drawn from a power-law distribution P(λ) = λ⁻¹⁻α, where α controls the disorder strength. 2. **Strong Disorder Renormalization Group (SDRG):** The SDRG method is used to efficiently determine the long-time behavior of the DCP. The method iteratively removes or merges sites/clusters based on the largest rate (infection or healing) within the system, simplifying the model while retaining essential dynamical features. Specifically, if the largest rate is an infection rate (λᵢⱼ), sites *i* and *j* are merged into a cluster with an effective healing rate calculated using equation (1). If the largest rate is a healing rate (μᵢ), site *i* is removed and effective infection rates between its neighbors are calculated (equation 2). The process continues until only one cluster remains, which dominates the long-time dynamics. The SDRG method, particularly its application to the IRFP, enables efficient computation of the long-time dynamics and allows for the characterization of the critical behavior of the DCP. The study compares the spatial structure of the final SDRG cluster with the simulated local occupancy maps. To address inconsistencies arising from the approximative nature of SDRG, the authors match the critical point in both simulation and SDRG calculations using the healing rates as the control parameter. While the infection rates are kept identical for comparison, the healing rate serves as a tuning parameter to align the critical behavior of the two approaches. The agreement between the two methods is then assessed by comparing the spatial distribution of activities in simulations to the spatial extent and probability distribution of the last remaining cluster in the SDRG calculations.

The study's key findings demonstrate a significant departure from traditional understanding of critical behavior in disordered systems: 1. **Disconnected Correlated Clusters:** The simulations and SDRG analysis consistently show that at criticality, the DCP's long-time dynamics are dominated by a single, large cluster of highly correlated sites. Crucially, this cluster is not spatially connected; it contains significant gaps in spatial density of active sites. This is visually confirmed in Figures 3, 4, 5, and 6 for 1D, 2D, 3D lattices, and 2D lattices with long-range interactions. These figures show remarkable agreement between the SDRG clusters and the simulated activity patterns, where the most active sites largely coincide with the sites included in the final SDRG cluster. 2. **Fractal Nature of Clusters:** The clusters identified have a fractal dimension (d_f), which varies with dimensionality. The authors indicate the approximate values for different dimensions: d_f ≈ 0.819 (1D), d_f = 1.02(2) (2D), and d_f = 1.16(2) (3D). In 2D systems with long-range interactions (Figure 6), the fractal dimension is formally zero, emphasizing the highly disconnected nature of the dominant cluster. 3. **IRFP Dominance:** The observed behavior strongly supports the relevance of the infinite-randomness fixed-point (IRFP) in governing the critical behavior of the DCP. The SDRG method's ability to accurately predict the spatial distribution of activity strongly implies that the dynamics are governed by this IRFP and, consequently, the existence of large, disconnected, highly correlated clusters is a hallmark of this critical behavior. 4. **Relevance to Other Systems:** The authors argue that the similarity of the SDRG equations suggests these findings extend to other systems, particularly the disordered quantum Ising model, indicating that quantum correlated magnetic domains can also exhibit spatial disconnectivity.

The findings significantly challenge the conventional understanding of criticality in complex systems. The emergence of disconnected yet strongly correlated clusters is a counter-intuitive result, implying that functional connections can exist without direct physical links. This has profound implications for understanding information processing in complex systems. In systems like the brain, for instance, this indicates that strongly correlated brain regions are not necessarily directly physically connected. Functional connectivity may thus significantly differ from anatomical connectivity, requiring a reassessment of brain network analysis. The use of quasi-stationary simulations and the SDRG method provides an efficient way to probe these intricate dynamics, revealing the structure of these disconnected yet correlated clusters which had not been previously observed. The good agreement between simulation and SDRG results demonstrates the efficacy of the employed methodology. The findings highlight the importance of considering disorder and its impact on the emergence of collective behavior, challenging the reliance on percolation theory alone to understand correlation structures in complex systems. Future work could extend this research to more complex network topologies, including those observed in real-world systems such as brain networks.

This study reveals the surprising emergence of disconnected yet highly correlated clusters in the critical dynamics of disordered complex systems. Using the disordered contact process as a model, simulations and SDRG analysis demonstrate that a single, large but spatially fragmented cluster dictates the long-time behavior at criticality. This contradicts traditional intuition based on percolation theory. The authors show that this behavior is likely relevant to other systems displaying infinite randomness criticality, such as disordered quantum Ising magnets. This necessitates a reevaluation of how we understand correlations in complex systems, especially in biological contexts where functional connectivity may not directly correspond to physical connections. Future research should focus on applying this methodology to experimentally obtained datasets, especially in neuroscience, where understanding functional connectivity is crucial.

While the study provides strong evidence for the existence of disconnected correlated clusters, some limitations exist. The SDRG method, while efficient, relies on approximations, particularly at early stages of the renormalization process. The quasi-stationary simulations, though helpful in accessing long-time dynamics, still involve finite-size effects. The accuracy of the observed fractal dimensions could be improved with larger systems. Furthermore, the extension of the findings to other systems is based on theoretical similarity of SDRG equations, demanding further investigations in specific models. Finally, the study focuses on specific types of disorder; testing other types of disorder could provide further insights into the robustness of these findings.