Hierarchical Percolation Model

• Hierarchical percolation is a network model defined on recursively structured substrates where connection probabilities decay with ultrametric distance and nested modularity is explicit.
• The model leverages exact recursion and renormalization methodologies to determine precise critical thresholds, fractal exponents, and scaling relations across diverse classes.
• It finds applications in urban morphology, infrastructure vulnerability, and branching transport, demonstrating both discontinuous and continuous phase transitions under varied conditions.

A hierarchical percolation model is a percolation system defined on a recursively organized substrate, or on a network whose connectivity is itself stratified by scale, module, or generation. In this setting, occupation variables are coupled to explicit hierarchy: edges may be replaced by motifs, vertices may belong to nested modules, or connection probabilities may decay with an ultrametric distance. The central consequence is that percolation can often be analyzed by exact recursion, renormalization, or multiscale arguments, while the resulting phase structure can differ sharply from that of Euclidean lattices or locally tree-like random graphs (Hasegawa et al., 2013, Hutchcroft, 2021, Alves et al., 9 Jun 2026).

1. Recursive substrates and model classes

Hierarchical percolation appears in several mathematically distinct, but structurally related, forms. One class uses deterministic recursive graphs. The Dorogovtsev-Goltsev-Mendes network starts from a triangle and replaces each edge by adding a new node connected to both endpoints; equivalently, each generation is formed by joining three copies of the previous generation at their roots. The number of nodes is $N_n=(3^n+3)/2$, and the degree distribution is $p(k)\propto k^{-\gamma}$ with $\gamma=1+\ln 3/\ln 2 \approx 2.585$. The $(u,v)$-flower replaces each link by two linear chains of length $u$ and $v$, so each generation is $w=u+v$ copies of the previous one. Diamond hierarchical lattices and Wheatstone hierarchical lattices are built by replacing each bond with a fixed motif, which makes the graph exactly self-similar (Hasegawa et al., 2013, Lee et al., 2014, Levitan, 2022, Andrade et al., 2013).

A second class uses ultrametric hierarchical lattices. In the $d$-dimensional hierarchical lattice $\mathbb{H}_L^d$, vertices are finitely supported sequences over $\mathbb{T}_L^d$, and the distance is ultrametric: $p(k)\propto k^{-\gamma}$0, where $p(k)\propto k^{-\gamma}$1 is the largest index at which the coordinates differ. Balls of radius $p(k)\propto k^{-\gamma}$2 contain about $p(k)\propto k^{-\gamma}$3 points. Closely related models are defined on the hierarchical lattice of order $p(k)\propto k^{-\gamma}$4, whose vertices are finite sequences over $p(k)\propto k^{-\gamma}$5 with ultrametric distance given by the largest differing coordinate (Hutchcroft, 2021, Dawson et al., 2010, Koval et al., 2010).

A third class uses explicit modular hierarchy. In a stochastic block formulation, the hierarchy is encoded by $p(k)\propto k^{-\gamma}$6, the numbers of modules at each level, and $p(k)\propto k^{-\gamma}$7, the average degrees at each level, with $p(k)\propto k^{-\gamma}$8. Nodes carrying the highest-level interconnections are attacked first, then those at the next level, and so on. This produces percolation behavior that is tied directly to organizational scale rather than only to degree or distance (Shekhtman et al., 2018).

Model family	Hierarchical construction	Percolation feature
DGM network	Each edge generates a new node; also three-copy root gluing	Site threshold $p(k)\propto k^{-\gamma}$9
$\gamma=1+\ln 3/\ln 2 \approx 2.585$0-flower	Each link becomes two chains of lengths $\gamma=1+\ln 3/\ln 2 \approx 2.585$1 and $\gamma=1+\ln 3/\ln 2 \approx 2.585$2	Degree-ordered percolation recursions
Diamond hierarchical lattice	Bond replacement by a diamond motif	Exact RG and exact exponents
Ultrametric hierarchical lattice	Nested balls with ultrametric distance	Long-range critical scaling
Stochastic block hierarchy	Nested modules with level-dependent degrees	Multiple jumps under attack

This taxonomy suggests a useful working definition: a hierarchical percolation model is less a single model than a class of percolation problems in which the substrate possesses an explicit recursive or multilevel geometry.

2. Exact recursions and renormalization

The main technical advantage of hierarchy is closure under coarse-graining. In site percolation on the DGM network, with node occupation probability $\gamma=1+\ln 3/\ln 2 \approx 2.585$3 and $\gamma=1+\ln 3/\ln 2 \approx 2.585$4, the generating functions $\gamma=1+\ln 3/\ln 2 \approx 2.585$5 and $\gamma=1+\ln 3/\ln 2 \approx 2.585$6 obey

$\gamma=1+\ln 3/\ln 2 \approx 2.585$7

$\gamma=1+\ln 3/\ln 2 \approx 2.585$8

From these recursions one obtains the mean root-cluster size and the fractal exponent

$\gamma=1+\ln 3/\ln 2 \approx 2.585$9

with

$(u,v)$0

The formalism distinguishes a critical phase with nontrivial cluster scaling from a genuinely percolating phase with a unique giant component (Hasegawa et al., 2013).

For degree-ordered percolation on the $(u,v)$1-flower, the probability $(u,v)$2 that the two roots are connected satisfies

$(u,v)$3

with initial condition

$(u,v)$4

The threshold is determined by fixed-point analysis: $(u,v)$5 The order parameter recursion closes on a $(u,v)$6 matrix $(u,v)$7, and the finite-size scaling exponents follow from $(u,v)$8 and the largest eigenvalue $(u,v)$9 (Lee et al., 2014).

On the diamond hierarchical lattice, decimation gives an exact renormalization map for bond occupation,

$u$0

The nontrivial fixed point solves $u$1, giving

$u$2

Linearization at $u$3 yields the correlation-length exponent $u$4, while a $u$5 linear system for the masses of connected and dangling pieces yields the fractal dimension $u$6. The same recursive structure can be extended to the full conductance distribution, which is essential because the conductivity RG is not closed on means (Levitan, 2022).

A more abstract formulation replaces each edge of a seed graph $u$7 by a copy of $u$8 itself. If $u$9 denotes the crossing probability between the two distinguished vertices of $v$0, then

$v$1

Under the stated nontriviality hypotheses, $v$2 has a unique fixed point $v$3, and this fixed point governs the phase transition and the scaling window (Alves et al., 9 Jun 2026).

These constructions make hierarchical percolation a rare setting in which recursive probability, real-space renormalization, and dynamical-systems methods are simultaneously exact rather than heuristic.

3. Thresholds, fragility, and phase structure

Hierarchy does not imply robustness. In site percolation on the DGM hierarchical scale-free network, the percolation threshold is $v$4: for any $v$5, the fraction of nodes in the largest cluster vanishes in the thermodynamic limit. The same network has lower and upper critical points $v$6 and $v$7, so the entire interval $v$8 is a critical phase with no giant component but nontrivial scaling. By contrast, bond percolation on the same network has threshold $v$9, and intentional attacks on high-degree nodes again give threshold $w=u+v$0. The physical interpretation given in the source is that the hierarchy creates a few high-degree hubs essential for global connectivity, so random node loss or hub removal fragments the graph into small pieces (Hasegawa et al., 2013).

Degree-ordered percolation on $w=u+v$1-flowers shows a different hierarchy effect. When $w=u+v$2, the threshold is finite and nonzero; for the $w=u+v$3-flower, $w=u+v$4. When $w=u+v$5, the threshold vanishes, $w=u+v$6, but the critical behavior differs from bond percolation: the degree-ordered model has $w=u+v$7, whereas bond percolation has an infinite-order transition with $w=u+v$8. For $w=u+v$9, degree-ordered percolation and bond percolation share critical exponents but differ by a shifted threshold (Lee et al., 2014).

In hierarchical modular networks, attacks that remove higher-level interconnecting nodes first can generate multiple discontinuous transitions in the giant component. The cutoff at level $d$0 is

$d$1

and a discontinuous jump occurs when

$d$2

Each jump corresponds to fragmentation at a distinct organizational scale, such as neighborhoods, cities, or larger regions. In interdependent hierarchical networks, the multiplicative dependency terms make all transitions discontinuous in the treelike case, while in random-regular networks of networks the final transition can be continuous or discontinuous depending on $d$3 (Shekhtman et al., 2018).

A separate line of work on load-bearing branching hierarchical networks identifies a singular configuration, the V lattice, as a critical case. Typical realizations exhibit a second-order directed-percolation transition, while the V lattice exhibits a first-order, explosive transition in the order parameters $d$4 and $d$5. Small perturbations destroy both the power-law avalanche statistics and the explosive character: for perturbed V lattices,

$d$6

with $d$7, so $d$8 as $d$9 (Gupte et al., 2014).

Two recurrent misconceptions are therefore excluded by the literature. First, a scale-free degree sequence does not by itself determine robustness; the DGM example shows that hierarchy can override the usual $\mathbb{H}_L^d$0 intuition. Second, explosive percolation is not a generic outcome of hierarchy; in the V lattice it is a finely structured special case, unstable under perturbation.

4. Ultrametric long-range models

On ultrametric hierarchical lattices, percolation is usually defined through distance-dependent independent edges. In one formulation, any pair of vertices at distance $\mathbb{H}_L^d$1 is connected with probability

$\mathbb{H}_L^d$2

For fixed $\mathbb{H}_L^d$3, the critical value $\mathbb{H}_L^d$4 is nontrivial if and only if $\mathbb{H}_L^d$5. More precisely, $\mathbb{H}_L^d$6 for $\mathbb{H}_L^d$7, $\mathbb{H}_L^d$8 for $\mathbb{H}_L^d$9, and $\mathbb{T}_L^d$0 for $\mathbb{T}_L^d$1. The infinite component, when it exists, is unique, and both the percolation probability and $\mathbb{T}_L^d$2 are continuous in the stated senses (Koval et al., 2010).

A related model on the hierarchical lattice of order $\mathbb{T}_L^d$3 connects points at distance $\mathbb{T}_L^d$4 with probability

$\mathbb{T}_L^d$5

Here the qualitative regimes are controlled by $\mathbb{T}_L^d$6. When $\mathbb{T}_L^d$7, percolation occurs if the $\mathbb{T}_L^d$8 are sufficiently large; when $\mathbb{T}_L^d$9, percolation does not occur for bounded $p(k)\propto k^{-\gamma}$00; and when $p(k)\propto k^{-\gamma}$01, the model is critical in a more delicate sense. In that borderline regime, renormalization-group arguments and Erdős–Rényi connectivity estimates yield both sufficient conditions for percolation and sufficient conditions for non-percolation, together with the intermediate notion of pre-percolation (Dawson et al., 2010).

For long-range Bernoulli bond percolation on $p(k)\propto k^{-\gamma}$02, each pair of points $p(k)\propto k^{-\gamma}$03 is joined independently with probability

$p(k)\propto k^{-\gamma}$04

At the critical point $p(k)\propto k^{-\gamma}$05, the two-point function obeys

$p(k)\propto k^{-\gamma}$06

The model has mean-field critical behavior when $p(k)\propto k^{-\gamma}$07 and does not have mean-field critical behavior when $p(k)\propto k^{-\gamma}$08. The triangle condition holds if and only if $p(k)\propto k^{-\gamma}$09, and this identifies the threshold for mean-field behavior (Hutchcroft, 2021).

The same long-range hierarchical setting also permits precise control of critical cluster volumes. At $p(k)\propto k^{-\gamma}$10,

$p(k)\propto k^{-\gamma}$11

Thus the critical exponent is $p(k)\propto k^{-\gamma}$12 below the upper-critical dimension $p(k)\propto k^{-\gamma}$13, while the upper-critical case exhibits a $p(k)\propto k^{-\gamma}$14 correction. The source explicitly notes that these polylogarithmic corrections differ from those predicted for nearest-neighbor percolation on $p(k)\propto k^{-\gamma}$15 (Hutchcroft, 2022).

A finite-volume critical-window theory further shows that, for $p(k)\propto k^{-\gamma}$16, maximal components in balls $p(k)\propto k^{-\gamma}$17 of the hierarchical lattice have Brownian metric scaling limits and fall into the Erdős–Rényi universality class. In this regime, critical component sizes scale as $p(k)\propto k^{-\gamma}$18 and diameters as $p(k)\propto k^{-\gamma}$19. When $p(k)\propto k^{-\gamma}$20, the girth of each maximal component in the critical window is $p(k)\propto k^{-\gamma}$21, whereas for $p(k)\propto k^{-\gamma}$22 the girth equals $p(k)\propto k^{-\gamma}$23 (Sen, 11 Sep 2025).

5. Critical exponents, universality, and exact solvability

Hierarchical lattices provide exact exponent calculations that are rarely available elsewhere. On the hierarchical diamond lattice of effective dimension $p(k)\propto k^{-\gamma}$24, decimation yields

$p(k)\propto k^{-\gamma}$25

and the mass renormalization matrix gives

$p(k)\propto k^{-\gamma}$26

With $p(k)\propto k^{-\gamma}$27, the scaling relations then produce

$p(k)\propto k^{-\gamma}$28

A separate RG for the conductance distribution gives $p(k)\propto k^{-\gamma}$29 and hence the conductivity exponent

$p(k)\propto k^{-\gamma}$30

The simulations reported in the source confirm the geometric exponents and the conductivity exponent (Levitan, 2022).

Exact solvability does not imply fixed universality. On the diamond hierarchical lattice, a modified model introduces erasing probabilities $p(k)\propto k^{-\gamma}$31 into the recursion

$p(k)\propto k^{-\gamma}$32

The critical exponents $p(k)\propto k^{-\gamma}$33 and $p(k)\propto k^{-\gamma}$34 vary continuously with the erasing probability. As $p(k)\propto k^{-\gamma}$35, one obtains $p(k)\propto k^{-\gamma}$36, while the transition remains continuous with $p(k)\propto k^{-\gamma}$37, and $p(k)\propto k^{-\gamma}$38 can be made as small as desired. For $p(k)\propto k^{-\gamma}$39, the exact value reported is $p(k)\propto k^{-\gamma}$40. The model is equivalent to the $p(k)\propto k^{-\gamma}$41 limit of a Potts model with specific long-range interactions between root nodes (Andrade et al., 2015).

Recursive self-similarity also controls geometric observables other than the order parameter. On the Wheatstone hierarchical lattice, the average cutting-path length satisfies

$p(k)\propto k^{-\gamma}$42

For $p(k)\propto k^{-\gamma}$43, $p(k)\propto k^{-\gamma}$44, so

$p(k)\propto k^{-\gamma}$45

For $p(k)\propto k^{-\gamma}$46, $p(k)\propto k^{-\gamma}$47, giving

$p(k)\propto k^{-\gamma}$48

The latter value is reported to be very close to that found for cutting paths in standard $p(k)\propto k^{-\gamma}$49-D square lattices and watersheds (Andrade et al., 2013).

A recent general theory of percolation on hierarchical lattices built from an arbitrary seed graph $p(k)\propto k^{-\gamma}$50 establishes, under sharp hypotheses, a unique phase transition, existence of the critical exponents $p(k)\propto k^{-\gamma}$51, $p(k)\propto k^{-\gamma}$52, $p(k)\propto k^{-\gamma}$53, and $p(k)\propto k^{-\gamma}$54, and scaling relations on the Benjamini–Schramm limit $p(k)\propto k^{-\gamma}$55. If $p(k)\propto k^{-\gamma}$56 is the unique nontrivial fixed point and $p(k)\propto k^{-\gamma}$57, then

$p(k)\propto k^{-\gamma}$58

and

$p(k)\propto k^{-\gamma}$59

where $p(k)\propto k^{-\gamma}$60 is the Perron–Frobenius eigenvalue of an explicit matrix $p(k)\propto k^{-\gamma}$61. The same framework proves uniqueness of the infinite cluster, continuity of $p(k)\propto k^{-\gamma}$62, a scaling window of size $p(k)\propto k^{-\gamma}$63, and sharp noise sensitivity for crossing functions (Alves et al., 9 Jun 2026).

Taken together, these results show that hierarchy can support both classical exponent relations and explicit violations of fixed universality, including continuously varying exponents and model-dependent logarithmic corrections.

6. Applications, extensions, and broader interpretations

Hierarchical percolation has been used as a model of vulnerability in infrastructure-like systems. In modular hierarchical networks, multiple jumps in the giant component track separation at different levels of organization, and the same formalism extends to interdependent networks of networks, where cascading failures make the system more fragile. The source treats neighborhoods, cities, and larger territorial scales as canonical examples of such levels (Shekhtman et al., 2018).

In urban morphology, hierarchical percolation on Britain’s street network and its intersections uncovers nested spatial units ranging from city cores to regional fractures. On intersections, the fractal dimension reaches a maximum at $p(k)\propto k^{-\gamma}$64; on the street network, the maximum occurs around $p(k)\propto k^{-\gamma}$65. At this “urban” threshold, the percolation clusters show high correspondence with urban boundaries recovered from satellite images and population-density methods, with grid-based correlation $p(k)\propto k^{-\gamma}$66 (Arcaute et al., 2015).

Branching hierarchical transport networks provide another application. Packet transmission on weight-bearing hierarchical $p(k)\propto k^{-\gamma}$67-D lattices is mapped to a site-percolation problem in which occupied and unoccupied subnetworks compete. The stationary occupation numbers follow Maxwell–Boltzmann statistics, and the V lattice again appears as a critical realization with explosive percolation, while the original lattice and its reconnection variants show continuous transitions and finite-size scaling (Kachhvah et al., 2011).

A related but distinct hierarchical deposition model links geometry and percolation at the same threshold. Blocks of size $p(k)\propto k^{-\gamma}$68 are deposited generation by generation. The number of coastal points or coastlines is Euclidean when $p(k)\propto k^{-\gamma}$69, logarithmic fractal when $p(k)\propto k^{-\gamma}$70, and fractal when $p(k)\propto k^{-\gamma}$71, with

$p(k)\propto k^{-\gamma}$72

For $p(k)\propto k^{-\gamma}$73, the percolation threshold is

$p(k)\propto k^{-\gamma}$74

and the source states that this coincides exactly with the onset of logarithmic fractality in the coastline geometry (Berx et al., 2021).

In nonequilibrium statistical mechanics, a one-dimensional kinetic replication model with parallel update exhibits a “tower of percolation patterns” inside the active phase. Five phases with distinct patterns of percolation are detected, and all transitions belong to the directed percolation universality class. The work proposes an extension of the Janssen–Grassberger conjecture to accommodate multiple active phases with distinct absorbing states (Ovchinnikov et al., 2024).

Finally, a recent arithmetic formulation relates long-range percolation on lattices and on hierarchical lattices through adelic constructions. In that framework, a power-mean deformation interpolates between Euclidean and toric kernels, while adelic product formulas connect lattice, toric, and hierarchical percolation models. This suggests a broader mathematical interpretation of hierarchical percolation as one member of a larger family of geometrically and arithmetically related percolation theories (Marcolli, 11 Aug 2025).