Pseudo-Percolation Transitions

• Pseudo-percolation transitions are percolation-like phenomena exhibiting nonstandard threshold behaviors, including explosive, finite-size, and hybrid regimes.
• They reveal how deviations from conventional independent occupation rules lead to apparent discontinuities and nuanced finite-size scaling in large complex networks.
• These transitions illustrate the influence of geometric constraints, correlated occupation rules, and substrate structure on the onset of macroscopic connectivity.

Pseudo-percolation transitions are percolation-like phenomena in which the observed threshold behavior departs from the standard scenario of independent occupation and a single continuous emergence of a macroscopic giant cluster. In the current literature, the label covers several distinct cases: apparently discontinuous but asymptotically continuous “explosive” transitions, finite-size pseudocritical phenomena, zero-threshold transitions with exponentially suppressed order parameters, quasi-critical phases with subextensive largest clusters, and geometry-driven crossovers in systems whose connectivity is not generated by ordinary random occupation (Costa et al., 2010, Shi et al., 3 Feb 2025, Sun et al., 2020, Shim et al., 2012). The terminology is not uniform: closely related work is also framed in terms of explosive, discontinuous, weakly discontinuous, or hybrid percolation transitions rather than “pseudo-percolation” proper (Lee et al., 2016).

1. Conceptual scope and relation to standard percolation

Ordinary percolation provides the baseline. In the review of diverse transition types, the order parameter is the giant-cluster fraction, which in the continuous case grows as $m(t)\sim (t-t_c)^\beta$ above threshold, while the finite-cluster distribution obeys $n_s(z)\sim s^{-\tau}e^{-s/s^*}$ and becomes scale free at criticality, $n_s(z)\sim s^{-\tau}$ (Lee et al., 2016). Pseudo-percolation phenomena are defined relative to this baseline: the transition may remain continuous but become visually jump-like, may occur under a nonstandard control parameter, may be shifted from the true critical point, or may produce mesoscopic rather than extensive connectivity.

One important usage arises in canonical continuum systems. For fixed point sets in Euclidean space, discs or spheres are grown around a fixed number $N$ of points, and the control parameter is the radius $r$, or more precisely $\hat r=r/\mathrm{MNN}$. This is called a “pseudo-percolation” transition because it is formulated in the canonical ensemble rather than the usual grand-canonical one, even though for homogeneous Poisson point processes the measured exponents match standard isotropic percolation and the upper critical dimension is $d_c=6$ (Villegas et al., 2022). A different usage appears in finite two-dimensional percolation, where the pseudocritical point $p_L$ lies inside the critical finite-size scaling window but is not the true $p_c$, so the system shows scaling without full critical universality (Shi et al., 3 Feb 2025).

A further usage is structural rather than finite-size based. On pseudo-fractal simplicial and cell complexes, the percolation threshold is continuous and located at $p_c=0$, with order parameter

$n_s(z)\sim s^{-\tau}e^{-s/s^*}$0

so the transition is percolation-like but not of the standard nonzero-threshold form (Sun et al., 2020). In nonlocal constrained network growth, the system can pass from a non-percolating regime to a quasi-critical phase with $n_s(z)\sim s^{-\tau}e^{-s/s^*}$1 and no ordinary extensive giant cluster, again motivating the “pseudo” qualifier (Shim et al., 2012).

2. Pseudo-discontinuity, explosive growth, and correlated occupation rules

The canonical example of pseudo-discontinuity is explosive percolation. In the analytically tractable model studied by da Costa and collaborators, cluster growth is more restrictive than in the original Achlioptas product rule: at each step two pairs of nodes are sampled, the node in the smaller cluster is chosen from each pair, and those selected nodes are linked. The infinite-system dynamics obey a Smoluchowski-type equation,

$n_s(z)\sim s^{-\tau}e^{-s/s^*}$2

with $n_s(z)\sim s^{-\tau}e^{-s/s^*}$3 above threshold. The transition is continuous, with

$n_s(z)\sim s^{-\tau}e^{-s/s^*}$4

and the apparent jump is attributed to the exceptionally small order-parameter exponent. At criticality, $n_s(z)\sim s^{-\tau}e^{-s/s^*}$5 with $n_s(z)\sim s^{-\tau}e^{-s/s^*}$6, and the scaling functions are roughly Gaussian-shaped rather than the standard exponential form (Costa et al., 2010). The paper’s $n_s(z)\sim s^{-\tau}e^{-s/s^*}$7 illustration, where a single step above threshold yields $n_s(z)\sim s^{-\tau}e^{-s/s^*}$8, shows why finite-size data can mimic a discontinuity.

The location of the largest jump is a key diagnostic. In Erdős–Rényi and other globally continuous “explosive” models, the largest jump in $n_s(z)\sim s^{-\tau}e^{-s/s^*}$9 asymptotically converges to the true percolation threshold. By contrast, in genuinely discontinuous staircase models such as the Devil’s staircase, Nagler–Gutch, and modified ER models, the largest jump can occur well into the supercritical regime. In the generalized Bohman–Frieze–Wormald model, stable regimes have the first macroscopic jump at threshold, whereas unstable regimes exhibit additional supercritical discontinuous mergers of giant components (Chen et al., 2013).

This distinction is sharpened by the classification of discontinuous cluster-merging processes into two types. Type-I discontinuous transitions reach $n_s(z)\sim s^{-\tau}$0 at $n_s(z)\sim s^{-\tau}$1 and are preceded by a homogeneous mesoscopic cluster population. Type-II transitions have $n_s(z)\sim s^{-\tau}$2 at $n_s(z)\sim s^{-\tau}$3 and are preceded by a heterogeneous “powder keg” in which a finite fraction $n_s(z)\sim s^{-\tau}$4 of the mass is already stored in large clusters. The necessary conditions are expressed directly in terms of $n_s(z)\sim s^{-\tau}$5, making the pre-jump cluster-size distribution, rather than the abruptness of $n_s(z)\sim s^{-\tau}$6 alone, the decisive object (Cho et al., 2014).

Correlated percolation provides the broader context. After the Riordan–Warnke result on Achlioptas-type models, correlated rules are no longer taken as sufficient evidence for true discontinuity. Genuine discontinuous transitions do occur in models such as random-graph $n_s(z)\sim s^{-\tau}$7-core percolation for $n_s(z)\sim s^{-\tau}$8, the spiral model, and the counter-balance model, and a tricritical point appears in the mixed $n_s(z)\sim s^{-\tau}$9-core model at $N$0 with $N$1 (Cao et al., 2012). Earlier suppression-rule models made the underlying mechanism explicit: in the largest-cluster and Gaussian models, it is sufficient to suppress clusters that differ strongly from the average size, or merely to suppress the largest cluster, to obtain a discontinuous transition with compact clusters, a Gaussian-like cluster-size distribution, and fractal interfaces of dimension $N$2 or $N$3 (Herrmann et al., 2011). This suggests that pseudo-percolation behavior is often a question of whether the occupation rule delays spanning without eliminating critical scaling.

3. Constraint-induced mesoscopic, quasi-critical, and hybrid regimes

A direct realization of a pseudo-percolation phase structure appears in the pair-exclusion model with a nonlocal constraint. Each node has an exclusive partner, the $N$4-neighbor set $N$5 has size $N$6, and a link between $N$7 and $N$8 is accepted only if $N$9 and $r$0 share no exclusive pair. Because the acceptance probability satisfies $r$1, the regimes $r$2, $r$3, and $r$4 are sharply separated. For $r$5, the threshold remains $r$6 and the transition is mean-field with $r$7 and $r$8. For $r$9, the transition is non-mean-field, with $\hat r=r/\mathrm{MNN}$0, $\hat r=r/\mathrm{MNN}$1, and $\hat r=r/\mathrm{MNN}$2. For $\hat r=r/\mathrm{MNN}$3, there is no ordinary giant component; instead the system enters a quasi-critical phase where

$\hat r=r/\mathrm{MNN}$4

in the abstract and $\hat r=r/\mathrm{MNN}$5 in the numerical summary (Shim et al., 2012).

A second mechanism is kinetic rather than combinatorial. In diffusion-limited cluster aggregation, Brownian motion gives a cluster velocity $\hat r=r/\mathrm{MNN}$6 with $\hat r=r/\mathrm{MNN}$7, so large clusters move slowly and their growth is suppressed. The giant cluster $\hat r=r/\mathrm{MNN}$8 grows smoothly in physical time, but when aggregation events are counted by

$\hat r=r/\mathrm{MNN}$9

the same process shows an abrupt jump near the end because $d_c=6$0 is highly nonlinear in $d_c=6$1 as $d_c=6$2. Finite-size scaling gives

$d_c=6$3

and the generalized model $d_c=6$4 has a tricritical point at $d_c=6$5 separating discontinuous from continuous behavior (Cho et al., 2011). Here the pseudo-percolation aspect is representation-dependent: the same aggregation dynamics can appear smooth or explosive depending on the control parameter.

Hybrid percolation transitions occupy an intermediate position between pseudo-discontinuity and true first-order behavior. In the review of diverse types, the defining form is

$d_c=6$6

with $d_c=6$7. This combines a jump in the order parameter with critical scaling above threshold and appears both in pruning processes, such as cascading failure and $d_c=6$8-core percolation, and in cluster-merging processes such as the restricted ER model (Lee et al., 2016). A plausible implication is that many systems called pseudo-percolative are better described as hybrids, because abrupt onset alone does not determine the order of the transition.

4. Pseudocriticality, finite-size scaling, and precursor structure

Finite systems introduce a distinct notion of pseudo-percolation. In two-dimensional bond and site percolation, the dynamic pseudocritical point is defined by the largest one-step gap in the largest cluster,

$d_c=6$9

with

$p_L$0

Because $p_L$1, it lies inside the critical finite-size scaling window, so many observables still obey conventional scaling forms near $p_L$2. However, the largest-cluster distribution differs from that at $p_L$3: it approaches a Gumbel form in the subcritical phase, a Gaussian form in the supercritical phase, and a family of crossover distributions inside the critical window. At $p_L$4, strict universality breaks down, but a quasi-universal pattern survives in which bond and site percolation on the same lattice agree within error bars while square and triangular lattices differ (Shi et al., 3 Feb 2025).

The dimensionless observables at $p_L$5 make this breakdown explicit. The critical polynomial $p_L$6, universally zero at $p_L$7, becomes nonzero at $p_L$8, with fitted asymptotic values near $p_L$9 for square bond percolation and $p_c$0 for triangular bond percolation. Wrapping probabilities and Binder cumulants behave similarly: for example, on the square lattice $p_c$1, while square-lattice $p_c$2 and $p_c$3 are approximately $p_c$4 and $p_c$5 for bond percolation at $p_c$6 (Shi et al., 3 Feb 2025). Pseudocriticality therefore preserves the scaling window but not the full fixed-point structure of criticality.

Precursor structure can be more discrete. In continuous and discontinuous models alike, the largest cluster exhibits a hierarchy of micro-transitions before global connectivity. In the generalized BFW model, the micro-transition positions satisfy

$p_c$7

with $p_c$8 close to $p_c$9, and the accumulation point $p_c=0$0 coincides with the macroscopic threshold. In globally competitive percolation, the exact relation is

$p_c=0$1

so the micro-transition cascade has exact discrete scale invariance. In continuous percolation, the predicted subcritical positions are

$p_c=0$2

which the paper checks for ER and two-dimensional site percolation (Chen et al., 2014). These are not percolation transitions in the thermodynamic sense, but they are structured precursors of the final one.

A related precursor phenomenon occurs in functional networks built from spatial correlations. Correlations are computed as

$p_c=0$3

and nodes are linked when $p_c=0$4. The resulting network can undergo a percolation transition before the underlying extended system reaches its true bifurcation, and the cluster-size probabilities $p_c=0$5, especially $p_c=0$6, peak even earlier than the giant component or susceptibility. This behavior was demonstrated for a lake eutrophication model, the Ginzburg–Landau equation, Lorenz’96, and Sea Surface Temperature data associated with El Niño (Rodriguez-Mendez et al., 2016). This suggests that pseudo-percolation may also describe mesoscopic warning structure rather than a change of thermodynamic phase.

5. Substrates, ensembles, and geometry beyond ordinary random graphs

Pseudo-percolation can also refer to transitions produced by atypical substrates rather than atypical occupation rules. In site percolation on pseudo-random $p_c=0$7-regular graphs, the retained subgraph $p_c=0$8 of an $p_c=0$9-graph has a sharp threshold at $n_s(z)\sim s^{-\tau}e^{-s/s^*}$00: when $n_s(z)\sim s^{-\tau}e^{-s/s^*}$01, all components are logarithmic, whereas for $n_s(z)\sim s^{-\tau}e^{-s/s^*}$02 a unique giant of order $n_s(z)\sim s^{-\tau}e^{-s/s^*}$03 appears. Its asymptotic size is

$n_s(z)\sim s^{-\tau}e^{-s/s^*}$04

and the giant has a predictable edge count, a cycle of length at least $n_s(z)\sim s^{-\tau}e^{-s/s^*}$05, and expansion on subsets of size between $n_s(z)\sim s^{-\tau}e^{-s/s^*}$06 and $n_s(z)\sim s^{-\tau}e^{-s/s^*}$07 (Diskin et al., 2021). The transition occurs on a deterministic graph but is governed by spectral pseudorandomness, so the percolation is random-like without the substrate itself being random.

Hierarchical complexes generate an even more unusual geometry. In pseudo-fractal simplicial and cell complexes built by gluing $n_s(z)\sim s^{-\tau}e^{-s/s^*}$08-gons, renormalization gives the recursion

$n_s(z)\sim s^{-\tau}e^{-s/s^*}$09

with fixed points only at $n_s(z)\sim s^{-\tau}e^{-s/s^*}$10 and $n_s(z)\sim s^{-\tau}e^{-s/s^*}$11. For any $n_s(z)\sim s^{-\tau}e^{-s/s^*}$12, the flow reaches $n_s(z)\sim s^{-\tau}e^{-s/s^*}$13, so $n_s(z)\sim s^{-\tau}e^{-s/s^*}$14. Yet the giant component is strongly suppressed: $n_s(z)\sim s^{-\tau}e^{-s/s^*}$15 or, for random polygon mixtures, the same form with $n_s(z)\sim s^{-\tau}e^{-s/s^*}$16 replaced by the smallest polygon size $n_s(z)\sim s^{-\tau}e^{-s/s^*}$17 with $n_s(z)\sim s^{-\tau}e^{-s/s^*}$18 (Sun et al., 2020). The transition is continuous, but its scaling is non-power-law and pinned to zero.

Canonical continuum percolation of fixed point sets provides a different departure from the standard ensemble. Two points are connected when $n_s(z)\sim s^{-\tau}e^{-s/s^*}$19, the order parameter is $n_s(z)\sim s^{-\tau}e^{-s/s^*}$20, and the susceptibility is

$n_s(z)\sim s^{-\tau}e^{-s/s^*}$21

For homogeneous Poisson point processes, the critical radius in two dimensions is $n_s(z)\sim s^{-\tau}e^{-s/s^*}$22, and the measured exponents $n_s(z)\sim s^{-\tau}e^{-s/s^*}$23, $n_s(z)\sim s^{-\tau}e^{-s/s^*}$24, $n_s(z)\sim s^{-\tau}e^{-s/s^*}$25, and $n_s(z)\sim s^{-\tau}e^{-s/s^*}$26 are close to isotropic percolation. By contrast, inhomogeneous Poisson processes with gradients give $n_s(z)\sim s^{-\tau}e^{-s/s^*}$27, and clustered Thomas processes give $n_s(z)\sim s^{-\tau}e^{-s/s^*}$28 (Villegas et al., 2022). Thus the “pseudo” character lies in the ensemble and control parameter for homogeneous systems, but in the critical behavior itself for heterogeneous and clustered ones.

Random geometry can also decide whether any threshold exists. On a randomly stretched square lattice with horizontal edge lengths $n_s(z)\sim s^{-\tau}e^{-s/s^*}$29 and open probabilities $n_s(z)\sim s^{-\tau}e^{-s/s^*}$30, a nontrivial phase transition exists if $n_s(z)\sim s^{-\tau}e^{-s/s^*}$31 for some $n_s(z)\sim s^{-\tau}e^{-s/s^*}$32, whereas no transition occurs for any $n_s(z)\sim s^{-\tau}e^{-s/s^*}$33 if $n_s(z)\sim s^{-\tau}e^{-s/s^*}$34 for some $n_s(z)\sim s^{-\tau}e^{-s/s^*}$35 (Hilario et al., 2019). This is not labeled pseudo-percolation in the paper, but it exhibits the same principle: critical behavior can be created or destroyed by the geometry of the substrate.

6. Rare-event, hydrological, and topological extensions

In sparse networks, pseudo-percolation can be formulated as a large-deviation phenomenon. Standard node percolation studies the typical giant-component size under the product measure $n_s(z)\sim s^{-\tau}e^{-s/s^*}$36. The large-deviation extension introduces

$n_s(z)\sim s^{-\tau}e^{-s/s^*}$37

so $n_s(z)\sim s^{-\tau}e^{-s/s^*}$38 biases the measure toward buffering ($n_s(z)\sim s^{-\tau}e^{-s/s^*}$39) or aggravating ($n_s(z)\sim s^{-\tau}e^{-s/s^*}$40) initial damage configurations. At $n_s(z)\sim s^{-\tau}e^{-s/s^*}$41, the standard continuous transition is recovered. For $n_s(z)\sim s^{-\tau}e^{-s/s^*}$42, the rate function can become nonconvex, the free energy non-differentiable, and the giant component can jump discontinuously from $n_s(z)\sim s^{-\tau}e^{-s/s^*}$43 to $n_s(z)\sim s^{-\tau}e^{-s/s^*}$44 (Bianconi, 2017). In this formulation, the pseudo-percolation transition is not the usual threshold of the typical ensemble but a singularity in the rare-event landscape.

A physically distinct extension appears in terrestrial hydrology. Water occupancy is defined by the Topographic Wetness Index,

$n_s(z)\sim s^{-\tau}e^{-s/s^*}$45

and cells with $n_s(z)\sim s^{-\tau}e^{-s/s^*}$46 are occupied. The largest wetted cluster shows an abrupt jump at

$n_s(z)\sim s^{-\tau}e^{-s/s^*}$47

reported as universal across 14 regions, 11 hydro-climatic zones in China, regional to global scales, and both 30 m and 1 km resolution. The maximum gap statistic $n_s(z)\sim s^{-\tau}e^{-s/s^*}$48 is nonzero but small, so the transition is classified as weakly discontinuous, and a second threshold

$n_s(z)\sim s^{-\tau}e^{-s/s^*}$49

bounds a Griffiths-like interval in which scale-free statistics and lake formation persist (Hu et al., 2023). The proposed mechanism is the combination of long-range correlation from upslope contributing area and directionality from downhill flow.

Topological systems provide yet another extension. In bond-percolated SSH chains, the short-range model has geometric threshold $n_s(z)\sim s^{-\tau}e^{-s/s^*}$50, so any dilution destroys global polarization $n_s(z)\sim s^{-\tau}e^{-s/s^*}$51 in the thermodynamic limit, but the zero-mode count $n_s(z)\sim s^{-\tau}e^{-s/s^*}$52 remains nonzero in the topological energetic regime $n_s(z)\sim s^{-\tau}e^{-s/s^*}$53. In the long-range model, three scales separate: the geometric threshold $n_s(z)\sim s^{-\tau}e^{-s/s^*}$54, the mean-field topological scale $n_s(z)\sim s^{-\tau}e^{-s/s^*}$55, and a distinct scale $n_s(z)\sim s^{-\tau}e^{-s/s^*}$56 for global polarization. The resulting fractured topological region has

$n_s(z)\sim s^{-\tau}e^{-s/s^*}$57

meaning that the system is globally trivial in polarization while still containing a macroscopic number of topological clusters with localized zero modes (Mondal et al., 2023). This suggests that pseudo-percolation can describe fragmentation of global response rather than loss of all local order.

Taken together, these results indicate that pseudo-percolation transitions are not a single universality class. They are a family of mechanisms by which percolation observables become sharp, shifted, fragmented, or ensemble-dependent: very small but nonzero critical exponents can mimic jumps; nonlocal constraints can stabilize quasi-critical phases; pseudocritical points can preserve scaling windows while breaking strict universality; hierarchical or deterministic geometries can move the threshold to $n_s(z)\sim s^{-\tau}e^{-s/s^*}$58 or replace randomness by pseudorandomness; and rare-event or topological formulations can relocate the transition from ordinary connectivity to a large-deviation or fragmented-order setting (Costa et al., 2010, Shim et al., 2012, Shi et al., 3 Feb 2025, Sun et al., 2020, Bianconi, 2017, Mondal et al., 2023).