Internal Cascades: Mechanisms and Dynamics

• Internal cascades are self-amplifying processes triggered by localized events that propagate via endogenous feedback mechanisms.
• They manifest across diverse fields—network science, social media, astrophysics, turbulence, and control theory—with domain-specific characteristics.
• System topology, threshold dynamics, and internal coupling rules critically determine cascade evolution, stability, and recurrence patterns.

Searching arXiv for papers on internal cascades and closely related cascade dynamics. Internal cascades are self-amplifying processes that propagate through endogenous couplings after a localized perturbation, but the term is not used identically across fields. In network science, “internal cascade processes” denote the autonomous unfolding of state changes within a network following an initiating event, without ongoing external input; in social systems the term can refer to reshare and recurrence dynamics generated inside the same network; in astrophysics it denotes electromagnetic cascades developing within gamma-ray sources themselves; in turbulence it refers to nonlinear transfers across scales in systems such as internal gravity waves; and in PDE control it appears in the theory of coupled cascade systems with internal coupling (Motter et al., 2017, Cheng et al., 2016, Fiorillo et al., 29 Aug 2025, Lhachemi et al., 26 Jan 2026). This plurality of meanings suggests that “internal cascades” is best understood as a cross-disciplinary family of mechanisms organized around feedback, thresholding, branching, and scale transfer rather than as a single universal model.

1. Terminological scope and common structural features

In the network-cascade literature, cascades are defined as self-amplifying processes on a network, triggered by localized perturbations or events, which can propagate and impact a substantial part of the system. Three intrinsic features are emphasized: non-additivity, non-locality, and disproportional impact. Within that framework, internal cascade processes refer to the autonomous unfolding of such changes within the network, following an initiating event, without ongoing external input (Motter et al., 2017). This definition is broad enough to cover avalanches, threshold-driven contagion, flow redistribution failures, and basin transitions in dynamical systems.

Outside network failure theory, the same adjective marks different boundaries of “inside.” In social media, the vast majority of content reshares and even new copies arise from within the same network, so recurrence can be analyzed as an internal process of repeated bursts separated by quiescent periods (Cheng et al., 2016). In astrophysics, internal cascades are electromagnetic cascades that develop within the gamma-ray sources themselves, as opposed to propagation-induced cascades in extragalactic photon fields (Fiorillo et al., 29 Aug 2025). In controllability theory, a wave-heat cascade is called “internal” because the coupling acts in the spatial domain rather than at the boundary (Lhachemi et al., 26 Jan 2026). In fluid dynamics, internal gravity waves provide a concrete setting in which cascades are nonlinear transfers of energy and pseudomomentum across scales (Shavit et al., 2023).

A recurring misconception is that an internal cascade must be a failure cascade, must be supercritical, or must reside in a network. The literature supplied here contradicts all three simplifications. Internal cascades can be beneficial or functional, as in spontaneous activity used to generate motion or store memory in neural-network interpretations of event cascades (Onaga et al., 2016). They can arise below the classical epidemic threshold, and they can refer to conservative or weakly dissipative scale-transfer processes rather than node failures (Onaga et al., 2016, Alexakis et al., 2018).

2. Cascading failures in interdependent and load-bearing networks

In interdependent networks, cascading failures propagate through two coupled mechanisms: internal connectivity, meaning failure within a network caused by detachment from the GCC, and inter-network dependency, meaning failure transmitted between layers by dependency links. At the critical point $p=\tilde{p}$, the process starts as a critical branching process with mean offspring number $\overline{m}=1$. If the cascade survives that quasi-neutral stage, accumulated damage makes the network increasingly fragile, $\overline{m}$ becomes supercritical, and the system undergoes abrupt collapse; if extinction occurs first, the network survives (Dilmoney et al., 9 Apr 2025). The paper gives the collapse probability for $n_0$ initially failed nodes in a system of size $N$ as

$$\Pi(n_0)=1-\frac{\Gamma(1/3,z/3)}{\Gamma(1/3)}, \qquad z=C\frac{n_0^3}{N^*}, \qquad N^*=N(\tilde{p}-p_c),$$

with $C\approx 2.5$ empirically. The same analysis yields a plateau or quasi-neutral phase whose duration scales as $N^{1/3}$, together with early-time laws

$n(t)\sim t, \qquad M(t)\sim t^2,$

and an effective stochastic description

$$\frac{dn}{dt}=C\frac{M}{N}n+\eta(t)\sqrt{n}, \qquad \frac{dM}{dt}=n.$$

These results quantify a specific internal-cascade mechanism: demographic noise dominates initially, then deterministic growth takes over if sufficient damage accumulates (Dilmoney et al., 9 Apr 2025).

Load-driven models reach related conclusions through different microscopic rules. In the Bak-Tang-Wiesenfeld sandpile model on interdependent systems, moderate interconnectivity suppresses the largest cascades within each system because a neighboring network acts as a reservoir for excess load, but too much interconnectivity becomes detrimental because it opens pathways for inflicted cascades and increases total capacity, which fuels even larger cascades (Brummitt et al., 2011, Brummitt et al., 2010). The trade-off is therefore local mitigation versus joint vulnerability. In the scale-free case, three properties are jointly necessary to significantly reduce the size of large cascades: scale-free degree distribution, internal network assortativity, and cross-network hub-to-hub connections. Assortative scale-free layers with hub-to-hub interconnectivity exhibit a marked reduction in large cascades, whereas random interlayer connectivity breaks the modular protection and helps failures spread (Turalska et al., 2019).

Threshold-load models make the same dependence on internal architecture explicit. In the random internal event regime, a single small fluctuation can trigger systemic failure if redistributed load repeatedly pushes neighbors over threshold. A central analytic criterion is

$\overline{m}=1$0

where $\overline{m}=1$1 is the number of neighbors and $\overline{m}=1$2 is the safety margin. If this condition is violated, even small random fluctuations may lead to full cascades; if it holds, cascades are generally finite (Tessone et al., 2012). Across these models, a common lesson emerges: the magnitude of the initiating event is often less decisive than topology, capacity asymmetry, safety margin, and cross-layer organization. This suggests that “internal” in systemic-failure settings refers less to the size of the trigger than to the fact that amplification is generated by the system’s own redistribution rules.

3. Subthreshold event cascades and recurrent information cascades

Self-exciting point-process models show that internal cascades need not wait for supercritical reproduction. In inhomogeneous networks governed by a multivariate Hawkes process,

$\overline{m}=1$3

nonstationary cascades of event-occurrences can emerge already in the subthreshold regime. The relevant cascade criterion is expressed through

$\overline{m}=1$4

and the cascade state emerges when $\overline{m}=1$5 (Onaga et al., 2016). For standard Hawkes or SIS-like models, the stationary–nonstationary transition occurs at $\overline{m}=1$6, well below the classical epidemic threshold. Clustering and reciprocal connections facilitate cascade formation, whereas directed or hierarchical architectures impede cascades. The same work develops a rewiring procedure that exchanges pairs of connections to maximize or minimize $\overline{m}=1$7, thereby inciting or suppressing bursts without changing the total number of connections (Onaga et al., 2016).

Information cascades on social platforms supply a complementary notion of internality. Recurrence is defined by multiple bursts of popularity for the same content, separated by lulls of inactivity, and measured through peak-detection criteria based on $\overline{m}=1$8, $\overline{m}=1$9, $\overline{m}$0, and $\overline{m}$1 (Cheng et al., 2016). The empirical analysis reports a mean separation of about 32 days between initial and subsequent bursts for image memes, low overlap in sharers between bursts with Jaccard similarity $\overline{m}$2, and higher overlap of potentially exposed populations with Jaccard $\overline{m}$3. Diversity is quantified using Shannon entropy,

$\overline{m}$4

and overlap by

$\overline{m}$5

The main finding is that moderately viral and diverse content is most likely to recur; very high virality can “immunize” the audience and reduce recurrence. Prediction from the initial burst alone reaches AUC $\overline{m}$6 for whether content will recur, AUC $\overline{m}$7 for the relative size of recurrence, AUC $\overline{m}$8 for when it will recur, and AUC $\overline{m}$9 for individual-copy recurrence (Cheng et al., 2016).

Taken together, these results broaden the concept of internal cascades beyond catastrophic failure. Internal feedback can generate intermittent bursting even when the mean reproduction ratio is below one, and recurrence can be driven primarily by network-internal reintroduction and heterogeneous exposure rather than by repeated external shocks. A plausible implication is that internal cascades are often better characterized by endogenous fluctuation structure than by a single epidemic threshold.

4. Engineered physical realizations and cascade systems with internal coupling

A direct laboratory realization of interdependent cascades is provided by multilayer networks of two disordered superconducting films separated by an insulating $n_0$0 layer. Each layer is a 2D network of Josephson junctions, and sufficiently large driving currents generate Joule heating that acts as a dependency link between corresponding positions in the two layers. A hotspot in one layer raises the temperature of its counterpart in the other, which can trigger switching there, redistribute current, and feed back again, producing adaptive back-and-forth electro-thermal feedbacks (Bonamassa et al., 2022). The experimentally observed consequence is a rich phase diagram of mutual resistive transitions with three regimes: weak interdependence, intermediate interdependence, and strong interdependence. In the strong-interdependence regime, only mutual first-order transitions are present, the phases of the two layers are fully locked, and the system exhibits large hysteresis and bistability between superconducting and normal phases (Bonamassa et al., 2022). This is one of the clearest physical instantiations of an internal cascade as a local event driving abrupt, system-wide reorganization through endogenous feedback.

A formally different but conceptually related use of “cascade” occurs in control theory for coupled hyperbolic–parabolic PDEs. The wave-heat cascade couples a wave equation, boundary-controlled through the Neumann trace at $n_0$1, to a heat equation driven by an internal coupling profile $n_0$2. For both wave-heat and heat-wave cascades, exact controllability is possible if and only if

$n_0$3

so the hyperbolic part dictates the sharp minimal time (Lhachemi et al., 26 Jan 2026). In the wave-heat direction, approximate and exact controllability require that all modal coefficients $n_0$4, and the exact controllability space is a weighted Hilbert space whose weights depend explicitly on $n_0$5. In the reversed heat-wave direction, the analogous condition is $n_0$6, and the weights shift to the hyperbolic modes (Lhachemi et al., 26 Jan 2026).

The significance of this PDE literature is not that it studies avalanches, but that it isolates how internal coupling direction determines where irregularity accumulates. In wave $n_0$7 heat cascades, the parabolic component receives highly irregular weights; in heat $n_0$8 wave cascades, the hyperbolic component does. The paper further shows that the exact controllability space is not invariant along Hilbert Uniqueness Method trajectories: even if both endpoints belong to the controllability space, the associated minimal-energy trajectory may leave it at intermediate times (Lhachemi et al., 26 Jan 2026). This suggests a broader structural point: internal cascades can denote not only spontaneous amplification, but also ordered transmission of dynamical influence through coupled subsystems with direction-dependent regularity loss.

5. Cascades across scales in fluids and high-energy astrophysical sources

In turbulence, cascades are scale-by-scale nonlinear transfers of conserved quantities. The modern classification distinguishes direct, inverse, split, dual, bidirectional, and flux-loop cascades, and ties transitions between them to control parameters such as Reynolds, Rossby, Froude, Péclet, and Alfvén numbers (Alexakis et al., 2018). Here the “internal” aspect lies in redistribution within the turbulent medium itself rather than between separate external reservoirs.

Internal gravity waves provide a specific example in which a second quadratic invariant reshapes transfer directions. For 2D internal gravity waves governed by the Boussinesq equations, the key invariants are total energy and pseudomomentum

$n_0$9

The paper derives a closed kinetic equation for modal energy $N$0,

$N$1

and obtains the isotropic steady spectrum

$N$2

while in effectively sign-definite unidirectional cases the radial spectra are

$N$3

Because the energy cascade exponent is smaller than the pseudomomentum cascade exponent, the paper argues for an inverse energy cascade of internal gravity waves from small to large scales in practically relevant scenarios (Shavit et al., 2023). This is a cascade internal to the wave field, organized by resonant triads and constrained by a sign-indefinite invariant.

High-energy astrophysics uses the term differently but with comparable emphasis on endogenous reprocessing. Internal cascades in gamma-ray-opaque sources develop when high-energy gamma rays trigger pair production on ambient photons and the resulting pairs cool by inverse Compton scattering or synchrotron radiation. The generalized theory distinguishes an equal-reproduction regime, a synchrotron-dominated soft-radiation regime, and a cooling-only regime (Fiorillo et al., 29 Aug 2025). In the classic IC-dominated case, the transparent-regime photon spectrum obeys $N$4. In the synchrotron-dominated regime, the electron and photon spectra become $N$5 and $N$6 above a break, with a transition to $N$7 below it (Fiorillo et al., 29 Aug 2025). The emergence of a universal spectrum requires a linear regime, interaction and loss times much shorter than escape, and energy injection well above the relevant thresholds. Strong magnetic fields invalidate the standard Berezinsky assumptions and drive the source into a different universal class.

Across fluids and astrophysical media, internal cascades therefore denote internally generated transfer channels rather than external forcing histories. The conserved quantity, symmetry, and dominant loss mechanism determine whether flux runs upscale, downscale, or in both directions.

6. Long-range criticality, hybrid spinodals, and driven cascade statistics

Long-range cascades near metastability boundaries admit a unifying critical theory. The spinodal framework identifies two universality classes determined by parity invariance: parity-breaking $N$8-cascades and parity-invariant $N$9-cascades (Bonamassa et al., 2024). The associated hybrid exponents combine mean-field spinodal values for strong exponents with $$\Pi(n_0)=1-\frac{\Gamma(1/3,z/3)}{\Gamma(1/3)}, \qquad z=C\frac{n_0^3}{N^*}, \qquad N^*=N(\tilde{p}-p_c),$$0-dependent corrections for weak exponents: $$\Pi(n_0)=1-\frac{\Gamma(1/3,z/3)}{\Gamma(1/3)}, \qquad z=C\frac{n_0^3}{N^*}, \qquad N^*=N(\tilde{p}-p_c),$$1 These obey the hyperscaling relation

$$\Pi(n_0)=1-\frac{\Gamma(1/3,z/3)}{\Gamma(1/3)}, \qquad z=C\frac{n_0^3}{N^*}, \qquad N^*=N(\tilde{p}-p_c),$$2

with $$\Pi(n_0)=1-\frac{\Gamma(1/3,z/3)}{\Gamma(1/3)}, \qquad z=C\frac{n_0^3}{N^*}, \qquad N^*=N(\tilde{p}-p_c),$$3, $$\Pi(n_0)=1-\frac{\Gamma(1/3,z/3)}{\Gamma(1/3)}, \qquad z=C\frac{n_0^3}{N^*}, \qquad N^*=N(\tilde{p}-p_c),$$4 for $$\Pi(n_0)=1-\frac{\Gamma(1/3,z/3)}{\Gamma(1/3)}, \qquad z=C\frac{n_0^3}{N^*}, \qquad N^*=N(\tilde{p}-p_c),$$5, and $$\Pi(n_0)=1-\frac{\Gamma(1/3,z/3)}{\Gamma(1/3)}, \qquad z=C\frac{n_0^3}{N^*}, \qquad N^*=N(\tilde{p}-p_c),$$6 for $$\Pi(n_0)=1-\frac{\Gamma(1/3,z/3)}{\Gamma(1/3)}, \qquad z=C\frac{n_0^3}{N^*}, \qquad N^*=N(\tilde{p}-p_c),$$7 (Bonamassa et al., 2024). The geometry and lifetime of avalanches depend on symmetry: parity-invariant cascades have larger fractal dimension, larger correlation lengths, and longer lifetimes.

A distinct but compatible description of internally driven cascade statistics is furnished by the propagation-power model. There the system’s susceptibility is summarized by a single propagation power $$\Pi(n_0)=1-\frac{\Gamma(1/3,z/3)}{\Gamma(1/3)}, \qquad z=C\frac{n_0^3}{N^*}, \qquad N^*=N(\tilde{p}-p_c),$$8, evolving between cascades according to

$$\Pi(n_0)=1-\frac{\Gamma(1/3,z/3)}{\Gamma(1/3)}, \qquad z=C\frac{n_0^3}{N^*}, \qquad N^*=N(\tilde{p}-p_c),$$9

and dropping by an amount proportional to event size when a cascade occurs (Burridge, 2013). Cascade sizes are generated by a continuous state branching process with a fixed-$C\approx 2.5$0 asymptotic distribution

$C\approx 2.5$1

At the critical point $C\approx 2.5$2, $C\approx 2.5$3 and the mean cascade size diverges. Averaging over the stationary distribution of $C\approx 2.5$4 produces crossover behavior with an upper-tail asymptotic

$C\approx 2.5$5

The model explains how a system can remain mostly subcritical while still exhibiting heavy-tailed extreme events because slow drive and stochastic fluctuations repeatedly bring it close to criticality (Burridge, 2013).

These two lines of work clarify different aspects of internal cascades. The spinodal theory classifies critical geometry and universality, whereas the propagation-power model explains how internal susceptibility can drift, fluctuate, and self-stabilize to generate crossover statistics. A plausible synthesis is that many internal cascades are governed by both a local branching rule and a slower endogenous variable controlling distance to instability.

7. Period-doubling cascades and the route to chaos

In nonlinear dynamics, a cascade is not a spreading failure but a connected sequence of bifurcations. A period-doubling cascade is a continuous path in parameter-state space of regular periodic orbits with infinitely many period-doubling bifurcations and periods $C\approx 2.5$6 (0910.3570). For smooth one- and two-dimensional parameter-dependent systems, the transition from no chaos to chaos generically involves not one such cascade but infinitely many. When a system moves from a parameter value $C\approx 2.5$7 with no chaos to a parameter value $C\approx 2.5$8 with chaos, virtually all regular periodic orbits at $C\approx 2.5$9 are connected to exactly one cascade by a path of regular periodic orbits (Sander et al., 2010).

The internal structure of these cascades is organized by components in the space of regular periodic orbits. Bounded cascades always come in pairs, connected by a continuous interior path that never reaches the parameter boundaries, whereas unbounded or solitary cascades have a stem that reaches a boundary value (0910.3570, Sander et al., 2010). Solitary cascades are robust under large perturbations, and the number of solitary cascades with a given stem period is conserved under broad classes of perturbations; paired cascades are not robust and can be created or annihilated more easily (Sander et al., 2010). In off-on-off chaos, where chaos appears and disappears as a parameter varies, almost all cascades are paired (0910.3570).

This dynamical-systems usage differs sharply from network-failure usage, yet the shared word “cascade” remains justified by the recursive amplification structure. Each period-doubling bifurcation generates a successor with doubled temporal complexity, and connected families of such doublings organize the onset of chaos. The juxtaposition with failure and transport cascades underscores a final general point: internal cascades are best classified not by domain vocabulary alone, but by how local transformations are recursively connected into system-level reorganization.