DERP: Directed Extended-Range Percolation

• Directed Extended-Range Percolation (DERP) is a family of models that extend classic directed percolation using long-range activation kernels or bounded relay pathways.
• It employs nonlocal interactions through power-law kernels and directed paths, leading to anomalous critical behavior with continuously varying exponents.
• Exact threshold equations from message-passing and spectral methods highlight DERP’s sensitivity to degree correlations and non-reciprocal, bounded connectivity.

Searching arXiv for the cited DERP-related papers and closely related terminology. arxiv_search({"^^^^^^^^4query4 extended-range percolation\" OR ^^^^^^^^4all:\4 Directed Percolation on a Dynamic Network using Rydberg Facilitation\" OR ^^^^^^^^4all:\4 percolation in complex networks\"","max_results":^^^^4all:\4query4^^^^,"sort_by":"submittedDate","sort_order":"descending"}) arxiv_search({"^^^^^^^^4query4 extended-range percolation","max_results":^^^^4all:\4query4^^^^}) Directed Extended-Range Percolation (DERP) denotes directed percolation processes in which connectivity or activation extends beyond strictly local contacts. In the recent literature, the term is used in two closely related senses. In non-equilibrium statistical mechanics, DERP refers to absorbing-state spreading with directed time evolution and extended-range spatial activation kernels, often power-law tailed and sometimes accompanied by heavy-tailed waiting times; in cold-atom realizations this emerges from motion-induced facilitation in Rydberg gases. In network science, DERP denotes percolation on directed graphs where trusted nodes communicate through directed paths of bounded length PRESERVED_PLACEHOLDER_^^^^4query4^^^^ via relays, with directionality suppressing immediate backtracking on non-reciprocal edges. The common structure is an extension beyond nearest-neighbor or uninterrupted-path constraints while retaining a privileged direction, either temporal or graph-theoretic (&&&^^^^4query4^^^^&&&, &&&^^^^4all:\4^^^^&&&).

^^^^4all:\4^^^^. Terminological scope and relation to neighboring theories

DERP sits at the intersection of directed percolation, long-range spreading, and extended-range connectivity. In the absorbing-state setting, the relevant baseline is ordinary directed percolation (DP), where activity spreads locally and decays toward an absorbing state. DERP modifies that baseline by replacing short-range spreading with an extended-range kernel, typically of Lévy-flight type, or by generating such a kernel dynamically through motion (&&&^^^^4query4^^^^&&&, &&&^^^^4 OR all:\4^^^^&&&). In the network setting, the baseline is extended-range percolation (ERP), where connectivity between trusted nodes is allowed through untrusted facilitators provided that the path-length or gap constraint is bounded by PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^; DERP adds edge orientation and, in the non-reciprocal case, forbids immediate backtracking (Cirigliano et al., 2023, &&&^^^^4all:\4^^^^&&&).

The surrounding ERP literature is important because much of the directed theory is built from undirected exact results. The paper on extended-range percolation in complex networks develops exact ERP results for infinite random uncorrelated undirected networks and a message-passing formulation for sparse real-world networks, then gives a principled extension to directed networks by replacing undirected excess-degree generating functions with directed out- and in-degree counterparts (Cirigliano et al., 2023). The later general ERP theory gives exact message passing for arbitrary range PRESERVED_PLACEHOLDER_^^^^4 OR all:\4^^^^ on simple and multiplex networks, and its directed extension is presented as an adaptation of that formalism rather than as the central solved model of the original paper (Cirigliano et al., 2024). By contrast, the ^^^^4 OR all:\4query4 OR all:\4^^^^6 DERP paper treats directed non-reciprocal networks as the primary object and derives exact threshold and critical-index results on locally tree-like structures (&&&^^^^4all:\4^^^^&&&).

A persistent source of confusion is that “extended range” does not have a single operational meaning across these subfields. In absorbing-state models it refers to a nonlocal activation kernel such as PRESERVED_PLACEHOLDER_^^^^4 OR all:\4^^^^ or $P(r)\sim r^{-\beta}$. In network-theoretic DERP it refers to bounded-length directed paths through relays, or equivalently to the admissibility of interrupted communication over paths of length at most $R$. This suggests that DERP is best understood as a family of directed, nonlocal percolation problems rather than a single canonical model.

^^^^4 OR all:\4^^^^. DERP as anomalous directed percolation in driven-dissipative dynamics

In the Rydberg-facilitation realization, atoms are driven from ground to Rydberg state with Rabi frequency $\Omega$ and detuning $\Delta$ with $\Delta \gg \Omega$, interacting through a van der Waals potential $V(r)=C_6/r^6$. Facilitation occurs when PRESERVED_PLACEHOLDER_^^^^4all:\4query4^^^^, which defines the facilitation radius

PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^

The facilitation shell has width

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR all:\4^^^^

volume PRESERVED_PLACEHOLDER_^^^^4all:\4 OR all:\4^^^^, and enhanced facilitation rate

PRESERVED_PLACEHOLDER_^^^^4all:\44^^^^

The epidemic mapping is standard: the order parameter is the active fraction PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^5, with PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^6; the primary control rates are PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^7 and PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^8; and critical scaling obeys PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^9, PRESERVED_PLACEHOLDER_^^^^4 OR all:\4query4^^^^, and PRESERVED_PLACEHOLDER_^^^^4 OR all:\4all:\4^^^^ with PRESERVED_PLACEHOLDER_^^^^4 OR all:\4 OR all:\4^^^^ (&&&^^^^4query4^^^^&&&).

The frozen-gas limit yields a static facilitation graph. Ground-state atoms whose separation falls within PRESERVED_PLACEHOLDER_^^^^4 OR all:\4 OR all:\4^^^^ form an Erdős–Rényi graph with mean degree PRESERVED_PLACEHOLDER_^^^^4 OR all:\44^^^^. At PRESERVED_PLACEHOLDER_^^^^4 OR all:\4^^^^5 there is a percolation threshold; for PRESERVED_PLACEHOLDER_^^^^4 OR all:\4^^^^6 the absorbing-state transition is replaced by a Griffiths phase, whereas for PRESERVED_PLACEHOLDER_^^^^4 OR all:\4^^^^7 the absorbing-state phase transition belongs to conventional DP in PRESERVED_PLACEHOLDER_^^^^4 OR all:\4^^^^8. The simulations in the frozen limit report PRESERVED_PLACEHOLDER_^^^^4 OR all:\4^^^^9, consistent with the PRESERVED_PLACEHOLDER_^^^^4 OR all:\4query4^^^^ DP benchmark PRESERVED_PLACEHOLDER_^^^^4 OR all:\4all:\4^^^^, and PRESERVED_PLACEHOLDER_^^^^4 OR all:\4 OR all:\4^^^^, compared with PRESERVED_PLACEHOLDER_^^^^4 OR all:\4 OR all:\4^^^^ with PRESERVED_PLACEHOLDER_^^^^4 OR all:\44^^^^ (&&&^^^^4query4^^^^&&&).

Atomic motion converts the static facilitation graph into a dynamic one. An excited atom can facilitate fresh neighbors outside its original cluster, generating effective long-range edges. The resulting jump-length distribution has the form

PRESERVED_PLACEHOLDER_^^^^4 OR all:\4^^^^5

with PRESERVED_PLACEHOLDER_^^^^4 OR all:\4^^^^6 the RMS speed. Two velocity scales control the crossover,

PRESERVED_PLACEHOLDER_^^^^4 OR all:\4^^^^7

For PRESERVED_PLACEHOLDER_^^^^4 OR all:\4^^^^8 the network is effectively static and gives PRESERVED_PLACEHOLDER_^^^^4 OR all:\4^^^^9 DP; for $P(r)\sim r^{-\beta}$^^^^4query4^^^^ motion fully mixes contacts during one facilitation time and yields mean-field (MF) behavior; for $P(r)\sim r^{-\beta}$^^^^4all:\4^^^^ the system exhibits extended-range spreading, denoted DERP or anomalous directed percolation (ADP), with continuously varying exponents. The central reported result is a monotone $P(r)\sim r^{-\beta}$^^^^4 OR all:\4^^^^ curve interpolating between a low-velocity DP plateau $P(r)\sim r^{-\beta}$^^^^4 OR all:\4^^^^ and a high-velocity MF plateau $P(r)\sim r^{-\beta}$4 (&&&^^^^4query4^^^^&&&).

The same extended-range logic appears in a $P(r)\sim r^{-\beta}$5-dimensional cellular-automaton study where nearest neighbors are replaced by Lévy-distributed left and right jumps, with $P(r)\sim r^{-\beta}$6 and the identification $P(r)\sim r^{-\beta}$7 for $P(r)\sim r^{-\beta}$8. In that model the critical point $P(r)\sim r^{-\beta}$9 varies continuously from $R$^^^^4query4^^^^ at $R$^^^^4all:\4^^^^ to $R$^^^^4 OR all:\4^^^^ at $R$^^^^4 OR all:\4^^^^, the decay exponent $R$4 decreases from $R$5 to $R$6, and the dynamic exponent $R$7 increases from $R$8 to $R$9, approaching the short-range $\Omega$^^^^4query4^^^^ DP value as long jumps are progressively suppressed (&&&^^^^4 OR all:\4^^^^&&&).

^^^^4 OR all:\4^^^^. Directed-network DERP: bounded directed paths, trusted nodes, and relay structure

In the path-based formulation, DERP is defined on a directed graph $\Omega$^^^^4all:\4^^^^ with trusted nodes present independently with probability $\Omega$^^^^4 OR all:\4^^^^ and untrusted nodes with probability $\Omega$^^^^4 OR all:\4^^^^. A trusted node $\Omega$4 is $\Omega$5-reachable from a node $\Omega$6 in the out-direction if there exists a directed path

$\Omega$7

of length at most $\Omega$8. Relay nodes may be untrusted; they do not interrupt connectivity provided the admissible path-length constraint is respected. In the non-reciprocal setting, immediate backtracking is precluded because traversing an edge and instantly returning through its reciprocal is generically impossible (&&&^^^^4all:\4^^^^&&&).

This yields three natural giant components. The in-giant component (IGC) contains trusted nodes with at least one upstream $\Omega$9-directed path from another trusted node. The out-giant component (OGC) contains trusted nodes with at least one downstream $\Delta$^^^^4query4^^^^-directed path to another trusted node. The strongly connected giant component (SCGC) consists of trusted nodes lying simultaneously in the IGC and OGC. The ^^^^4 OR all:\4query4 OR all:\4^^^^6 formulation also tracks the fraction $\Delta$^^^^4all:\4^^^^ of trusted nodes and the fraction $\Delta$^^^^4 OR all:\4^^^^ of untrusted relay nodes in the SCGC (&&&^^^^4all:\4^^^^&&&).

The ERP background generalizes these ideas through the language of walks with bounded untrusted stretches. In the undirected ERP theory, two trusted nodes are connected if there exists at least one walk between them with at most $\Delta$^^^^4 OR all:\4^^^^ consecutive inactive or untrusted nodes, and bridge-node effects arise because a walk can revisit nodes and break long stretches of untrusted intermediates (Cirigliano et al., 2023, Cirigliano et al., 2024). The directed extension inherits the same range logic but splits connectivity into out-, in-, and strong variants. A plausible implication is that the graph orientation changes not only thresholds but also the combinatorics of admissible relay structures.

The principal application context is communication networks. In the directed DERP paper, the motivating example is quantum communication, where entanglement can be swapped along sequences of repeaters but the total directed path length is limited by decoherence or synchronization constraints. In the ERP literature more generally, the same logic is discussed for noisy data transmission through error-correcting repeaters in classical and quantum networks (Cirigliano et al., 2023, &&&^^^^4all:\4^^^^&&&).

4. Message passing, nonbacktracking structure, and threshold equations

On locally tree-like directed graphs, DERP is exactly solvable by message passing because directionality suppresses immediate backtracking and decouples in- and out-messages. For each directed edge $\Delta$4 and each $\Delta$5, the ^^^^4 OR all:\4query4 OR all:\4^^^^6 formulation introduces $\Delta$6 and $\Delta$7, where $\Delta$8 is the probability that node $\Delta$9 connects its downstream neighbor $\Delta \gg \Omega$^^^^4query4^^^^ to trusted nodes in the IGC at minimum distance $\Delta \gg \Omega$^^^^4all:\4^^^^ from $\Delta \gg \Omega$^^^^4 OR all:\4^^^^, and $\Delta \gg \Omega$^^^^4 OR all:\4^^^^ is the analogous probability for the OGC in the upstream direction. The recursion distinguishes trusted activation at distance one from relay propagation by untrusted nodes. Linearization near criticality gives

$\Delta \gg \Omega$4

which closes exactly because non-reciprocity removes the path-cancellation complications familiar in undirected ERP (&&&^^^^4all:\4^^^^&&&).

The linearized theory is governed by the directed nonbacktracking matrix $\Delta \gg \Omega$5 on directed edges,

$\Delta \gg \Omega$6

with spectral radius $\Delta \gg \Omega$7. Writing $\Delta \gg \Omega$8 and using the block amplitudes $\Delta \gg \Omega$9, one obtains

$V(r)=C_6/r^6$^^^^4query4^^^^

hence the exact threshold condition

$V(r)=C_6/r^6$^^^^4all:\4^^^^

Equivalently,

$V(r)=C_6/r^6$^^^^4 OR all:\4^^^^

The special cases are immediate: for $V(r)=C_6/r^6$^^^^4 OR all:\4^^^^, $V(r)=C_6/r^6$4; for $V(r)=C_6/r^6$5,

$V(r)=C_6/r^6$6

with closed-form solution

$V(r)=C_6/r^6$7

These formulas are exact on locally tree-like directed networks (&&&^^^^4all:\4^^^^&&&).

For the directed configuration model with joint degree distribution $V(r)=C_6/r^6$8, the ensemble branching factor is

$V(r)=C_6/r^6$9

and the threshold becomes

PRESERVED_PLACEHOLDER_^^^^4all:\4query4query4^^^^

In the uncorrelated case, PRESERVED_PLACEHOLDER_^^^^4all:\4query4all:\4^^^^ and PRESERVED_PLACEHOLDER_^^^^4all:\4query4 OR all:\4^^^^; in the maximally correlated case, PRESERVED_PLACEHOLDER_^^^^4all:\4query4 OR all:\4^^^^ almost surely and PRESERVED_PLACEHOLDER_^^^^4all:\4query44^^^^ (&&&^^^^4all:\4^^^^&&&).

A complementary threshold theory appears in the empirical-process analysis of directed percolation on directed Erdős–Rényi graphs with i.i.d. long-range edges. There, exploration of the out-component is encoded by the urn recursion

PRESERVED_PLACEHOLDER_^^^^4all:\4query4^^^^5

and, under PRESERVED_PLACEHOLDER_^^^^4all:\4query4^^^^6, the normalized unvisited fraction follows the ODE PRESERVED_PLACEHOLDER_^^^^4all:\4query4^^^^7 with solution PRESERVED_PLACEHOLDER_^^^^4all:\4query4^^^^8. Component exhaustion is detected by the boundary equation PRESERVED_PLACEHOLDER_^^^^4all:\4query4^^^^9, recovering the threshold PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4query4^^^^, or equivalently PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4all:\4^^^^. The same framework yields the functional approximation

PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4 OR all:\4^^^^

which quantifies finite-PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4 OR all:\4^^^^ fluctuations around the deterministic exploration curve (&&&^^^^4 OR all:\4 OR all:\4^^^^&&&).

5. Universality classes, anomalous exponents, and correlation effects

DERP does not define a single universality class. In the long-range absorbing-state setting, the controlling variable is the spatial-kernel exponent. For long-range directed percolation or contact processes with PRESERVED_PLACEHOLDER_^^^^4all:\4all:\44^^^^ in PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^5, the Rydberg analysis reports three regimes: MF for PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^6, short-range DP for PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^7, and ADP for PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^8, with continuously varying exponents in the ADP window. It further distinguishes ADP-I, where waiting times remain exponential and only space is power-law, from ADP-II, where both space and time acquire heavy tails. In ADP-I, the measured order-parameter exponent agrees with the one-loop field-theoretical prediction

PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^9

up to the onset of ADP-II, beyond which the perturbative approximation ceases to apply because of the temporal heavy tails (&&&^^^^4query4^^^^&&&).

In the PRESERVED_PLACEHOLDER_^^^^4all:\4 OR all:\4query4^^^^ Lévy-flight cellular automaton, the symbol PRESERVED_PLACEHOLDER_^^^^4all:\4 OR all:\4all:\4^^^^ instead labels the jump-length distribution PRESERVED_PLACEHOLDER_^^^^4all:\4 OR all:\4 OR all:\4^^^^ rather than the order-parameter exponent. That study reports a smooth crossover from Lévy-dominated spreading to short-range DP as the jump distribution steepens. It also tests the heuristic relation

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR all:\4 OR all:\4^^^^

finding approximate compliance near PRESERVED_PLACEHOLDER_^^^^4all:\4 OR all:\44^^^^ and systematic deviations for both stronger long-range and more short-range-like regimes (&&&^^^^4 OR all:\4^^^^&&&).

In network DERP, the dominant source of anomalous behavior is degree heterogeneity and in–out correlation rather than a geometric Lévy kernel. On locally tree-like directed networks, the ^^^^4 OR all:\4query4 OR all:\4^^^^6 theory gives exact PRESERVED_PLACEHOLDER_^^^^4all:\4 OR all:\4^^^^5 exponents for trusted and untrusted fractions in one-sided and strongly connected giant components. For uncorrelated well-behaved degree distributions, including Poisson and power laws with PRESERVED_PLACEHOLDER_^^^^4all:\4 OR all:\4^^^^6, the one-sided exponents are

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR all:\4^^^^7

while the SCGC exponents are

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR all:\4^^^^8

For uncorrelated power-law degrees with PRESERVED_PLACEHOLDER_^^^^4all:\4 OR all:\4^^^^9,

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR all:\4query4^^^^

For maximally correlated power-law degrees with PRESERVED_PLACEHOLDER_^^^^4all:\4 OR all:\4all:\4^^^^,

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR all:\4 OR all:\4^^^^

For maximally correlated power laws with PRESERVED_PLACEHOLDER_^^^^4all:\4 OR all:\4 OR all:\4^^^^, the threshold vanishes, PRESERVED_PLACEHOLDER_^^^^4all:\4 OR all:\44^^^^, and the scaling is with PRESERVED_PLACEHOLDER_^^^^4all:\4 OR all:\4^^^^5 itself; defining PRESERVED_PLACEHOLDER_^^^^4all:\4 OR all:\4^^^^6, the exponents become

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR all:\4^^^^7

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR all:\4^^^^8

These formulas show that DERP criticality can be continuously altered either by long-range spatial kernels or by degree correlations, depending on the underlying model class (&&&^^^^4all:\4^^^^&&&).

A common misconception is that DERP simply reproduces ordinary DP or standard site percolation with directional arrows added. The literature instead shows two distinct mechanisms for nonstandard criticality: violation of the short-range, Markovian assumptions of the Janssen–Grassberger conjecture through motion-induced long jumps and heavy-tailed waiting times, and strong sensitivity of threshold and exponents to degree correlations in directed networks (&&&^^^^4query4^^^^&&&, &&&^^^^4all:\4^^^^&&&).

6. Methods, applications, and broader significance

The methodological repertoire around DERP is unusually diverse. In the Rydberg study, universality is established through Monte Carlo simulations, critical decay from a fully excited initial condition, and a machine-learning regressor trained in the MF limit to infer the SOC critical density from subsampled density trajectories. The regressor uses a modified Huber loss with PRESERVED_PLACEHOLDER_^^^^4all:\4 OR all:\4^^^^9 and reports average percentage error PRESERVED_PLACEHOLDER_^^^^4all:\44query4^^^^ (&&&^^^^4query4^^^^&&&). In the PRESERVED_PLACEHOLDER_^^^^4all:\44all:\4^^^^ long-range DP study, a stacked autoencoder with mean squared error loss and Adam optimization predicts PRESERVED_PLACEHOLDER_^^^^4all:\44 OR all:\4^^^^ from cluster images; the predicted critical point is then validated through the quality of power-law density decay, with an example at PRESERVED_PLACEHOLDER_^^^^4all:\44 OR all:\4^^^^ and PRESERVED_PLACEHOLDER_^^^^4all:\444^^^^ yielding PRESERVED_PLACEHOLDER_^^^^4all:\44^^^^5 (&&&^^^^4 OR all:\4^^^^&&&). In network DERP, the main tools are spectral methods for the directed nonbacktracking matrix, fixed-point iteration of message-passing equations, and Monte Carlo simulations on directed configuration-model ensembles (&&&^^^^4all:\4^^^^&&&).

The network side also connects to a broader ERP algorithmics. The undirected ERP framework provides explicit cavity-message iterations for sparse networks, threshold detection through Jacobian spectral radii, and per-iteration cost linear in graph size for sparse instances (Cirigliano et al., 2023, Cirigliano et al., 2024). The empirical-process treatment of directed Erdős–Rényi exploration complements those asymptotic threshold results with finite-size Gaussian corrections to hitting times and exploration trajectories, thereby supplying uncertainty quantification for percolation events at moderate PRESERVED_PLACEHOLDER_^^^^4all:\44^^^^6 (&&&^^^^4 OR all:\4 OR all:\4^^^^&&&).

Experimentally, DERP is already tied to cold-atom observations. The Rydberg work explains why the critical exponent reported by Helmrich et al. in a PRESERVED_PLACEHOLDER_^^^^4all:\44^^^^7 ultracold gas lies between the DP benchmark PRESERVED_PLACEHOLDER_^^^^4all:\44^^^^8 and the MF value PRESERVED_PLACEHOLDER_^^^^4all:\44^^^^9: the relevant thermal regime falls in the anomalous directed-percolation window generated by motion-induced long-range facilitation (&&&^^^^4query4^^^^&&&). On the communication side, ERP and DERP model trusted endpoints linked through untrusted repeaters or hybrid relays, making them directly relevant to noisy classical communication, quantum repeaters, and multiplex classical–quantum infrastructures (Cirigliano et al., 2023, Cirigliano et al., 2024, &&&^^^^4all:\4^^^^&&&).

Directionality also changes the combinatorics in a nontrivial way. Standard percolation intuition often treats directionality as a source of extra complexity. The dedicated DERP analysis shows a more specific statement: when connectivity is defined through directed paths of length PRESERVED_PLACEHOLDER_^^^^4all:\4max_results4query4^^^^, non-reciprocal directionality can significantly reduce the complexity of percolation because it impedes immediate backtracking (&&&^^^^4all:\4^^^^&&&). This is one of the central distinctions between directed extended-range percolation and its undirected ERP counterpart.

Taken together, these results place DERP among the better-characterized classes of directed nonlocal percolation. In one branch of the literature it provides a quantitatively controlled bridge between short-range DP and MF behavior through motion-generated or explicitly imposed long-range kernels. In another it yields exact threshold equations and anomalous critical indices for relay-enabled connectivity on directed networks. The shared lesson is that once locality is relaxed in a directed system, both threshold structure and critical scaling become sensitive to mechanisms that are irrelevant in ordinary short-range percolation: Lévy-flight activation, heavy-tailed waiting times, bounded relay depth, and degree-correlation structure.