Introduction

Particle shape plays a crucial role in the collective behavior of granular materials. Unlike the well-studied spherical particles1,2, the behavior of nonspherical particles remains poorly understood. Slender nonconvex particles, in particular, exhibit distinctive mechanical properties due to their entanglement (interlocking). Entanglement manifests macroscopically as deformation-resistant geometric cohesion3,4,5. For example, entangled U-shaped4, Z-shaped6, star-shaped7,8,9, or cross-shaped10 particles can resist uniaxial compression and maintain freestanding columns. Moreover, S- or U-shaped particles can withstand tensile stresses11,12 and be collectively lifted against gravity8,13. These mechanical effects arising from entanglement have recently sparked interest across diverse fields, such as architecture8, living systems14,15, soft robotics16, and metamaterials17,18.

Despite this interest, the structural properties of entangled granular materials, which impact their mechanical behavior, have rarely been quantified. To address this gap, we propose a network-based approach that models entangled granular materials as networks (graphs), where nodes represent particles and edges represent topological links. While network-based approaches have been applied to force chains of convex grains19, they have not yet been extended to the entangled structures of nonconvex grains. The network representation of entangled particles has been briefly introduced in simulation studies for microscopic particles, such as C-shaped colloids20 and kinetoplast DNA21,22. In these studies, two particles are considered entangled when they form a Hopf link—a topological configuration in which two loops are interlinked and cannot be separated without breaking. We adopt this definition to construct entanglement networks, enabling quantitative network analysis.

Here, we investigate the entanglement networks of C-shaped granular particles (C-particles) evolving under vibration through experiments and simulations. The C shape is chosen for its adjustable entanglement capability by varying its opening angle while maintaining the same particle diameter, which allows for proper comparison between different particle batches. By lifting the entangled clusters, we demonstrate that topological links largely reflect the mechanical bonds under tension. Through measurements of key network properties, we reveal that the C-particle networks exhibit a percolation transition, which is effectively captured by our proposed continuum percolation (CP) model of rings. Additionally, we observe that the mean degree grows logarithmically with vibration time, mirroring slow relaxation in disordered systems. These results establish network-based analysis as a promising framework for elucidating the structure and dynamics of entangled granular matter.

Results

We experimentally measure the cluster formation of steel C-particles under controlled vibration. Each particle has a toroidal shape with opening angle θ, centerline diameter D = 9 mm, and thickness d = 1 mm (Fig. 1a). We test nine sets of C-particles with θ ranging from 25° to 135°, each with N = 4000 particles. Initially, disentangled particles are poured in a conical container. The conical shape ensures a consistent geometry, regardless of the amount of particles. To induce particle entanglement, we shake the container with vertical sinusoidal vibration at 20 Hz and 1.55 mm amplitude for a duration of time t. Since particles’ positions and links are difficult to measure, we measure the sizes of mechanically bonded clusters. After stopping the vibration, we slowly lift the clusters one by one from the container (Fig. 1b, c) and weigh them. Because the measurement is destructive, each trial begins with a freshly disentangled sample. This procedure is repeated in 578 experimental trials at different t and θ for sufficient statistics.

Fig. 1: Cluster formation of C-particles under vibration.
figure 1

a 4000 steel C-particles entangle in a conical container under vibration. The C-particle geometry is shown in the callout. C-particles with θ = 25° form a long cluster at t = 8.0 s (b) and a compact, rigid cluster at t = 120 s that retains its in-container shape (c). More results are shown in Supplementary Movie 2. d DEM simulation of sequentially lifted C-particles (θ = 20°, N = 4000) after t = 53 s of vibration under gravity. In this trial, the number of particles in the 442 clusters are s1 = 3097, s2 = 167, s3 = 54, …, s442 = 1. Relative size of the largest lifted cluster S1 increases with vibration time t in experiments (e) and simulations (f, g). Raw data are in Supplementary Fig. 1b–e. Susceptibility χ(t) in experiments (h) and simulations (i). In e–i, markers indicate ensemble averages, and shaded areas indicate standard error bands. j Heat map of ensemble-averaged S1 for lifted clusters at various t and θ. The blue region at small θ indicates that particles are not yet entangled; the blue region at large θ indicates that the clusters are easily broken when lifted.

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To measure the links between particles, we simulate the corresponding system using the discrete element method (DEM) (Fig. 1d). We test 10 sets of frictional C-particles, each comprising a chain of spheres, with θ ranging from 20° to 151°. The particle thickness, vibration amplitude, peak acceleration, and time unit are set to match the experiment. From the positions and orientations of the particles, we determine whether they form Hopf links (Eq. 5, Supplementary Fig. 10)20. Although C-particles are not closed loops, this criterion well approximates their mechanical interlocking, especially when θ is small. The entanglement networks are measured before (in-container) and after lifting the clusters in gravity. The results are explained by Monte Carlo (MC) simulations of the ring CP model. Experiment and simulation details are in “Methods” and Supplementary Information.

Evolution of the largest cluster

Under vibration, initially unentangled C-particles gradually entangle and form clusters, and eventually, the largest cluster dominates the system. In percolation theory, the relative size of this largest cluster, S1 ≡ s1/N, serves as the order parameter, where si represents the number of particles in the ith largest cluster, and N is the total number of particles23. The experiments (Fig. 1e) and simulations (Fig. 1f, g) show that for small θ, lifted S1(t) steadily grows and approaches 1, indicating percolation of entanglement. Additionally, the susceptibility, defined as the mean cluster size \(\chi \equiv {\sum }_{i\ne 1}{s}_{i}^{2}/N\), peaks near the rapid growth of S1(t), i.e., the percolation threshold (critical point), as predicted by percolation theory (Fig. 1h, i)24.

The growth rate of S1(t) depends on θ. For θ ≲ 70°, C-particles with larger θ entangle more easily during vibration, resulting in faster growth of S1 at t > 0 (Fig. 1e, f). However, for θ ≳ 70°, this trend reverses (Fig. 1g, j) because larger θ makes the links break more easily. When only a few tiny clusters detach from the giant cluster, S1 remains close to 1, but when the giant cluster disintegrates during lifting, S1 decreases dramatically. These two cases result in a bimodal S1 distribution at large θ (Supplementary Fig. 1c, e).

The clusters become extremely fragile beyond a certain θ. In experiments at t = 60 s, giant clusters can be lifted stably for θ ≤ 110°, but often disintegrate for θ ≥ 115°. As a result, the ensemble-averaged S1 at t = 60 s plummets around θ = 115° (Supplementary Fig. 1f). This mechanical instability, emerging beyond a certain θ, is also confirmed by simulations. When θ ≳ 150°, the clusters are too fragile to be lifted as giant clusters (Fig. 1g, j, Supplementary Fig. 1g).

This maximum θ for stability depends on particle thickness and surface friction. Ref. 25 has shown that even straight steel rods can form cohesive clusters through friction when the particle aspect ratio exceeds 100. In contrast, our C-particles are thicker, and thus their cohesion relies primarily on normal contact forces within topological links. Reducing particle thickness or increasing the friction coefficient would enhance cohesion, thereby raising the maximum stable θ and shrinking the fragile region in Fig. 1j.

Evolution of the degree distribution

To investigate the entangled structure, we model the granular system as an N-node network comprising multiple disconnected clusters, where nodes represent particles and edges denote topological links. Figure 2a shows a snapshot of the entanglement network from a DEM simulation; the network exhibits relatively homogeneous node degrees (number of links, k) with no prominent hubs.

Fig. 2: Ensemble-averaged degree distribution of the C-particle networks in DEM simulations.
figure 2

a Network representation of the lifted clusters for θ = 20° at t = 82 s. The size and color of a node indicate its connectivity, i.e., degree k. Node positions are set by a force-directed layout for visual clarity and do not reflect actual spatial coordinates. Isolated dots represent disconnected single C-particles. Degree distribution P(k) of the C-particle network (θ = 46°, N = 4000) before (b) and after (c) lifting at various t. Each P(k) closely follows the Poisson distribution in Eq. (1) with the same mean degree 〈k〉 (curves). d Mean degree 〈k〉 (t) is higher for in-container networks (filled markers) than for lifted networks (empty markers). Both can be well fitted by Eq. (2) (curves). Standard errors are smaller than the marker size. More data are in Supplementary Fig. 2, 3.

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This degree homogeneity is evident in the degree distribution P(k), a fundamental metric of network connectivity26. The observed P(k) of the C-particle networks at various t mostly agrees with the Poisson distribution (Fig. 2b, c):

$$P(k)={e}^{-\langle k\rangle }\frac{{\langle k\rangle }^{k}}{k!},$$
(1)

where 〈k〉 is the mean degree of each network. This agreement holds for all in-container networks (before lifting) (Fig. 2b) and stably lifted networks with small θ (Fig. 2c), except for lifted networks with large θ that undergo frequent cluster breakages (Supplementary Fig. 2).

The Poisson P(k) is a feature of both Erdös-Rényi (ER) random networks26 and continuum percolation (CP) models, also known as random geometric graphs27. A similar P(k) is observed in kinetoplast DNA networks, which are quasi-2D assemblies of flexible loops21,22. These results indicate that an underlying spatial randomness is shared by these systems, despite differences in geometry and dynamics. In the following sections, we show that other properties of C-particle networks differ from ER random networks but are similar to a CP model of rings.

Given that the Poisson distribution is solely determined by 〈k〉, knowing how 〈k〉 changes with θ and t is crucial. Figure 2d shows that 〈k〉(t) of the in-container networks can be fitted by

$$\langle k\rangle (t)={k}_{0}+\alpha \, {{{\rm{ln}}}}(1+t/{t}_{0}),$$
(2)

where the fitting parameters k0 and t0 vary with θ, while α is relatively independent of θ (Supplementary Fig. 3a–c). This logarithmic increase persists over a wide range of t, even long after S1(t) (Fig. 1f) has nearly saturated. The lifted networks also display logarithmic 〈k〉(t), but with the curves shifting downward (Fig. 2d) because some links break during the lifting.

To our knowledge, this logarithmic evolution of entangled granular network has not been previously reported. The most relevant prior observation is the logarithmic density increase in vibrated spherical grains28. However, in our system, the mean degree exhibits a clearer logarithmic trend, whereas the density shows significantly larger fluctuations (Supplementary Fig. 8).

Equation (2) is purely empirical, and its theoretical basis remains unclear. We speculate that insights may be gained by considering other glassy disordered systems, in which slow logarithmic relaxation (aging) is common. Examples include stress relaxation in polymers29 and granular media30, structural relaxation in DNA31 and crumpled paper32, as well as magnetization decay in spin glasses33. In these systems, slow relaxation typically arises from sequential transitions between metastable states separated by broadly distributed waiting times, often due to rugged energy landscapes34,35,36. These systems also exhibit history dependence memory effects37. Our observation that lifted C-particle clusters retain their boundary shape (Fig. 1c) and degree distribution (Fig. 2b–d) may suggest a similar form of memory. Whether existing theories of glassy dynamics apply to entangled granular systems remains a puzzle.

Comparison with the ring percolation model

To describe the percolation transition in C-particle networks, we propose a CP model of infinitely thin rings. CP models have been utilized to study various disordered systems, including porous media, semiconductors, and wireless networks38,39. In conventional CP models, objects such as spheres40, disks41, or rods42 are randomly distributed in space and are considered connected when they overlap. In our model, randomly distributed rings in 3D are considered connected when entangled (“Methods”). Our MC simulations show that as ring density increases, the ring CP undergoes a percolation transition at a unique threshold and belongs to the same universality class as conventional CP models (Figs. 3, 4).

Fig. 3: Comparison of C-particle networks in simulations with the ring CP model.
figure 3

Filled markers: in-container networks; empty markers: lifted networks; black curves: ring CP model. N = 4000 for all systems. Data are ensemble-averaged except for (c). a Relative size of the largest in-container cluster, S1(〈k〉), matches the ring CP model (black curve) for various θ; S1(〈k〉) of the lifted networks shifts to larger 〈k〉 as θ increases. Inset: k1/2 of the lifted networks increases with θ. Black horizontal line represents k1/2 of the ring CP model. b Susceptibility χ(〈k〉) of the in-container networks closely follows the ring CP model; χ(〈k〉) of the lifted networks shifts to larger 〈k〉 as θ increases. Inset: Peak height of the lifted networks increases with θ at θ ≤ 72°. Raw data for (a and b) are in Supplementary Fig. 4a–d. c Vertical length, Lz /D, of the largest lifted cluster approximately equals its average eccentricity in each trial. d Lz /D peaks near k1/2 for all θ and closely resembles the average eccentricity of the ring CP model (black curve). Shaded areas in (a, b, and d) indicate standard error bands. e Mean clustering coefficients C(〈k〉) of in-container and lifted networks collapse onto a curve close to the ring CP model (black curve), but much lower than the sphere CP model (dotted curve). f Adjacency spectra for networks with the same 〈k〉: lifted C-particles (θ = 20°, blue area), ring CP model (black outline), and ER random network (red outline). More spectra are compared in Supplementary Fig. 9, with discussions in Supplementary Information. g Cluster size distribution follows ns ~ s−τ near percolation at 〈k〉 ≈ 2.1 for lifted C-particles in simulation (θ = 20°, maroon circles) and experiment (θ = 25°, blue circles), and for the ring CP model (black line). Blue line indicates the slope of the standard 3D percolation (τ = 2.19). More ns data are in Supplementary Fig. 6.

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Fig. 4: Finite-size scaling in the ring CP model and C-particle networks.
figure 4

a–d MC simulation results of the ring CP model with different particle numbers N. a Mean degree 〈k〉 approaches the reduced density \(\eta \equiv n{v}_{{{{\rm{ex}}}}}^{{{{\rm{ring}}}}}\) as N increases39,40. Lines represent empirical fits of the form \(\eta={p}_{1}\langle k\rangle+{p}_{2}{\langle k\rangle }^{{p}_{3}}\) (p1, p2, and p3 are in Supplementary Table 2), used to calculate ηeff(〈k〉; N) in (e, f). b The intersection of S2/S1 gives the percolation threshold ηc = 2.11. Inset: the zoom-in at the intersection. Data collapses of S1 (empty markers), S2 (filled markers) (c) and χ (d) using ηc from (b) and \((\beta,\gamma,\bar{\nu })=(0.41,1.80,2.64)\) of the standard 3D percolation23. In DEM simulations of lifted C-particles (θ = 20°), FSS with the same critical exponents and ηc yields reasonable data collapses of S1 (empty markers), S2 (filled markers) (e), and χ (f), especially for ηeff ≲ ηc. Markers indicate ensemble averages, and shaded areas indicate standard error bands. Data without FSS are shown in Supplementary Fig. 7.

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Although C-particles are neither infinitely thin nor completely randomly located, their entanglement networks are well captured by the ring CP model (Figs. 3, 4). The in-container S1(〈k〉) curves for various θ collapse nicely onto S1 of the ring CP (Fig. 3a). By comparison, the lifted networks are more densely connected, i.e., have higher 〈k〉, than the ring CP networks with the same S1. This deviation is stronger at larger θ, because large openings facilitate the disintegration of the giant cluster, while the decrease in 〈k〉 remains limited owing to the tightly connected core. The percolation threshold can be roughly estimated at S1(k1/2) = 1/2, and the measured k1/2 increases linearly with θ (Fig. 3a inset).

The susceptibility curves χ(〈k〉) of the in-container networks for various θ also collapse onto that of the ring CP model (Fig. 3b). For the lifted networks, χ(〈k〉) shifts to larger 〈k〉 as θ increases, but it always peaks near k1/2(θ) (Supplementary Fig. 4f). This aligns with percolation theory, which predicts a susceptibility peak near the percolation threshold24. The peak height of χ for the lifted networks increases with θ at θ ≤ 72° (Fig. 3b inset), indicating larger non-giant clusters. This occurs because the emergence of the giant cluster, capable of absorbing the non-giant clusters, is delayed to higher 〈k〉 as θ increases.

The vertical length Lz of a lifted cluster reflects the eccentricity of the top node, i.e., the maximum distance from this node to any other node43. Since the top particle is effectively a randomly chosen node, Lz /D approximately measures the node-averaged eccentricity, as confirmed in Fig. 3c. Unlike the monotonically increasing S1, Lz of the largest cluster exhibits a peak near the percolation threshold in both simulations (Fig. 3d) and experiments (Fig. 1b, Supplementary Fig. 5a). When the 〈k〉 axis is normalized by k1/2, the Lz /D curves with different θ collapse onto the average eccentricity of the ring CP’s largest cluster (Fig. 3d), indicating that, given k1/2, Lz(〈k〉) can be predicted by the ring CP model. Meanwhile, the width W of the cluster monotonically increases and plateaus as 〈k〉 increases (Supplementary Fig. 5b). After prolonged shaking, the lifted cluster almost retains its in-container shape (Fig. 1c), which determines the saturated Lz and W.

The clustering coefficient of a node i, Ci, quantifies the tendency of its neighbors to connect with each other, i.e., forming triangles26. The mean clustering coefficient C is the average of Ci over all nodes in the network. Interestingly, the C(〈k〉) curves for all in-container and lifted networks with different θ collapse onto a master curve that closely follows C(〈k〉) of the ring CP (Fig. 3e). This master curve is lower than C(〈k〉) of the sphere CP model but much higher than the vanishing C(〈k〉) = 〈k〉/N of ER random networks when N ≫ 〈k〉40. The similarity in local connectivity patterns between the C-particle networks and the ring CP model is also evident in the eigenvalue spectra of their adjacency matrices (Fig. 3f, Supplementary Fig. 9).

The cluster size distribution ns ≡ Ns/N, where Ns is the number of clusters of size s, is commonly used in percolation studies. According to percolation theory, at the threshold, ns ~ s−τ, where τ is determined by the universality class23. Figure 3g shows that when 〈k〉 is near the threshold observed in Figs. 3a, b or 4b, both the C-particle networks and the ring CP model closely follow ns ~ s−2.19, suggesting that they fall into the standard 3D percolation universality class where τ = 2.1944. Furthermore, ns of in-container clusters agrees with the ring CP model over various 〈k〉, not only at the threshold (Supplementary Fig. 6a–d). ns of lifted clusters also matches the ring CP model before a giant cluster forms, but deviates afterward as the giant cluster partially breaks during lifting (Supplementary Fig. 6e–h).

Finite-size scaling

Phase transitions are affected by the system size, i.e., the number of particles N. By scaling data from various N, finite-size scaling (FSS) can determine the critical point and critical exponents45,46. In CP, the order parameter and susceptibility exhibit the following FSS forms near the percolation threshold ηc:

$${S}_{i} \sim {N}^{-\beta /\bar{\nu }}\,\tilde{{S}_{i}}\left((\eta -{\eta }_{{{{\rm{c}}}}}){N}^{1/\bar{\nu }}\right),$$
(3)
$$\chi \sim {N}^{\gamma /\bar{\nu }}\,\tilde{\chi }\left((\eta -{\eta }_{{{{\rm{c}}}}}){N}^{1/\bar{\nu }}\right).$$
(4)

Here, \(\bar{\nu }\equiv d\nu\) where the space dimension d = 3. β, γ, and ν are the critical exponents. The scaling functions \(\tilde{{S}_{i}}\) and \(\tilde{\chi }\) are independent of N. The reduced density η ≡ nvex, where n is the number density, and vex is the excluded volume39,40. vex is the average accessible volume of the center of a particle when it is connected to another fixed particle. We analytically derive \({v}_{{{{\rm{ex}}}}}^{{{{\rm{ring}}}}}=\pi {D}^{3}/3\) for rings with diameter D (“Methods”, Supplementary Fig. 10). η approaches 〈k〉 as N increases (Fig. 4a) because they both reflect the connection probability.

We first determine ηc of the ring CP model from MC simulations. According to Eq. (3), S2 /S1 is independent of N at ηc46. This is confirmed by the unique intersection of S2(η)/S1(η) curves for different N in Fig. 4b, giving ηc = 2.11. This suggests that the mean degree at the percolation threshold is 〈k〉c = 2.11 when N → ∞, substantially larger than 〈k〉c = 1 of ER random networks26 and comparable to other CP models: 〈k〉c = 2.74 for spheres40 and 〈k〉c = 2.27 for disks in 3D space41. This large 〈k〉c can be attributed to the threshold-increasing effect of spatial constraints47. The threshold can also be estimated from k1/2 (Fig. 3a), the peak of χ (Fig. 3b), or the peak of S2 (Supplementary Fig. 7b, e), but the intersection in Fig. 4b provides the most precise value.

The ring CP model demonstrates the expected FSS behavior, conforming precisely to Eqs. (3) and (4). By using ηc = 2.11 and the critical exponents for standard 3D percolation (β = 0.41, γ = 1.80, and \(\bar{\nu }=2.64\)), we obtain clear data collapses of S1, S2, and χ across all N (Fig. 4c, d, Supplementary Fig. 7). This shows that the ring CP model belongs to the same universality class as lattice percolation and other CP models24.

To check whether the C-particle networks also satisfy FSS, we attempt data collapses using DEM simulations for different N. Unlike CP models, 〈k〉 and cluster sizes in C-particle networks are not determined solely by η (Supplementary Fig. 8). This is because granular particles have correlated positions and orientations due to mechanical interactions under gravity and container constraints. Therefore, instead of η, we employ an effective reduced density, ηeff (〈k〉; N), defined as the value of η in the ring CP model that yields the same 〈k〉 as the C-particle network. We compute ηeff by applying the empirical fit for η(〈k〉; N) from the CP model (Fig. 4a), using the measured 〈k〉 and N for the C-particles. Using ηeff with the same ηc and critical exponents from Fig. 4c,d, FSS yields reasonably good data collapses for S1, S2, and χ (Fig. 4e, f). The robust collapse in the dilute regime (ηeff < ηc) indicates that the C-particle networks exhibit critical behavior similar to the standard percolation universality class. The imperfect collapse of χ in the dense regime (ηeff > ηc) is possibly due to strong interparticle correlations.

Discussion

Our experiments and simulations reveal that the entanglement networks of C-shaped granular particles under vibration exhibit a percolation transition. Striking similarities between the C-particle networks (both in-container and lifted) and our proposed ring CP model are observed across various properties, such as the order parameter and susceptibility of percolation, degree distribution, average eccentricity (vertical length), mean clustering coefficient, and adjacency spectrum. Their cluster size distribution and finite-size scaling follow the standard percolation universality class. These similarities demonstrate that continuum percolation theory is applicable to entangled granular structures.

We have also found that particle shape impacts mechanical bonding when pulled: as the opening angle increases, geometric links act less as mechanical bonds, leading to cluster disintegration and a higher percolation threshold in lifted networks. Additionally, the mean degree of the C-particle networks increases logarithmically with vibration time, exemplifying the widely observed phenomenon of logarithmic aging in disordered systems.

These findings highlight the potential of network-based approaches in studying entangled granular materials. Future studies may incorporate additional complexities, such as the particles’ relative positions, orientations, velocities, and interactions, by assigning features to nodes and edges in the network model. The spatial data can be measured experimentally using X-ray tomography48. Furthermore, graph-based machine learning models can be developed to predict network dynamics and thus the mechanical response of entangled materials49. This network framework can be extended to various other entangled materials, such as mechanically interlocked molecules50,51, polymer chains52, kinetoplast DNA21,22, organisms with branching structures15, and synthetic nonconvex particles from micro53,54,55 to macro scales8. Such advancements will help predict and control their mechanical behavior for diverse applications, including molecular machines50,51, entangled robots16, and granular metamaterials8,17,18.

Methods

Experiment methods

We used C-particles made of type 304 stainless steel, which are strong enough to lift thousands of particles with minimal deformation. To ensure low and uniform surface roughness, we ground freshly made C-particles in a vibrating container for 4 hours until their surfaces became shiny. This duration is sufficient because, after the first hour of shaking, the metal powder generated by grinding was considerably reduced.

To prepare consistent and well-disentangled initial states, the C-particles are trickled through a vibrating funnel with a hole of internal diameter 2.6 cm (Supplementary Fig. 1a and Supplementary Movie 1). The dripping particles are collected by a second funnel, whose bottom hole (diameter: 3.4 cm) is blocked by a tape. The second funnel remains stationary during this process. Once all dripping particles are collected, the second funnel is placed on the electrodynamic shaker (ECON EDS-300 model). Starting at t = 0, a vertical displacement of \(z(t)=A\sin (2\pi ft)\) is applied. We set f = 20Hz and A = 1.55 mm, and the corresponding peak acceleration is A ⋅ (2πf)2 = 2.5g, where g = 9.8 m/s2 (Supplementary Fig. 1a and Fig. 1a).

After vibration, we use a hook made of a paper clip to gently hook and lift the C-particle clusters from the surface one by one (Fig. 1b, c and Supplementary Movie 2). The sizes of the clusters are measured by their weights. We neglect very small clusters ( <3.5 grams) because their sizes are sensitive to the way they are hooked up and do not significantly affect the S1 and χ behaviors. After measurement, all clusters are poured back into the first vibrating funnel (Supplementary Fig. 1a) to be disentangled in preparation for the next trial. To obtain sufficient statistics, we repeat this procedure multiple times for each vibration t, as listed in Supplementary Tables 3–5.

Discrete element method (DEM) simulation

In DEM simulations, the geometries of the funnel container and C-particles are similar to those in our experiment. We use 10 different sets of rigid C-particles with opening angles of 20°, 33°, 46°, 59°, 72°, 98°, 112°, 125°, 138°, and 151° which are composed of 26, 25, 24, 23, 22, 20, 19, 18, 17, and 16 identical small hard spheres, respectively (Supplementary Fig. 12, Table 1). The diameter D of the C-particles is 8.8 times the diameter d of the small spheres. This ratio is similar to the C-particles with D = 9 mm and d = 1 mm in the experiment. The total number of particles N = 4000 is the same as in the experiment. For particles with θ = 20°, N = 500 and 1400 are also simulated for the finite-size scaling. The friction coefficient is set to 0.4 for both particle-particle and particle-container interfaces. Details on the DEM contact model are provided in Supplementary Information.

To prepare a disentangled initial state, C-particles are randomly created with a volume fraction of 0.05 in a vertical cylindrical region with diameter 9D and height 22D right above the container. The particles then fall into the container under gravity (Supplementary Movie 3).

In the entanglement process, the particles are shaken with an amplitude of A = 0.172D (Supplementary Movie 4) and a peak acceleration of 2.5g to replicate the experiment.

After being vibrated for a desirable time, individual clusters near the surface are lifted one by one (Supplementary Movie 5–7). The lifting speed \(0.5\sqrt{gD}\) is equivalent to the speed of a particle that has fallen a relatively short distance, D/8. This speed is chosen to gently lift the cluster while minimizing the overall simulation time. If the bottom of the lifted cluster is more than 1.1d above the top particle remaining in the container, we consider the cluster fully lifted and remove it. This process is repeated until all particles have been removed. The number of trials are listed in Supplementary Table 6, 7.

Monte Carlo (MC) simulation

The continuum percolation (CP) model consisting of closed, infinitely thin rings is simulated using Monte Carlo methods. The ring centers, ri for i = 1, 2, . . . , N, are randomly generated within a cubic volume V with non-periodic boundaries. The normal orientation of the i’th ring is described by the unit vector \({\hat{{{{\boldsymbol{n}}}}}}_{i}=\left[\sin ({\theta }_{i})\cos ({\phi }_{i}),\, \sin ({\theta }_{i})\sin ({\phi }_{i}),\cos ({\theta }_{i})\right]\) where the two angles in the spherical coordinate \({\theta }_{i}={\cos }^{-1}(2{x}_{i}-1)\) and ϕi = 2πxi. Here, xi is a random variable with a uniform distribution in [0, 1]. Snapshots at different reduced densities η are shown in Supplementary Fig. 11. The ensemble averages of the relevant variables are obtained at each η. The numbers of trials are listed in Supplementary Table 8,9.

In contrast to our newly proposed ring CP model, the sphere CP model has already been intensively studied as the simplest CP model in 3D39,40. In each MC simulation of sphere CP model, N = 4000 monodispersed spheres are randomly distributed within a cube with non-periodic boundaries. Two spheres are considered connected if they overlap, i.e., their distance is smaller than their diameter. The number of trials used in Fig. 3e is listed in Supplementary Table 8. The eigenvalue spectra (Supplementary Fig. 9g-i) are averaged over 300 trials.

In addition to ring and sphere CP models, we also simulate ER random networks to compare their spectra with the C-particle network. To achieve a desired mean degree of 〈k〉, N〈k〉/2 connections are randomly attached to N = 4000 nodes with equal probability. The eigenvalue spectra of the networks (Fig. 3f and Supplementary Fig. 9j–l) are averaged over 300 trials.

Entanglement criterion

Entanglement generally refers to geometric constraints between elongated or non-convex bodies that cause mechanical bonding. It can be quantified in various ways. For example, in filamentous materials such as polymers and rods, entanglement can be statistically estimated through tube models52, or measured by the average crossing number over all directions25,56.

For C-particles, we adopt a topological entanglement criterion from Ref. 20: two particles are defined as entangled when the circles passing through their centerlines form a Hopf link (Supplementary Fig. 10). This definition is implemented in our DEM and MC simulations. Mathematically, two circles i and j form a Hopf link if and only if

$$\left({R}^{2}-| {{{\boldsymbol{b}}}}{| }^{2}\right){({{{\boldsymbol{k}}}}\cdot {{{{\boldsymbol{r}}}}}_{ij})}^{2} > | {{{\boldsymbol{k}}}}{| }^{2}{\left(\frac{| {{{{\boldsymbol{r}}}}}_{ij}{| }^{2}}{2}+{{{\boldsymbol{b}}}}\cdot {{{{\boldsymbol{r}}}}}_{ij}\right)}^{2},$$
(5)

where R is their radius, ri and rj are the positions of their centers, and \({\hat{{{{\boldsymbol{n}}}}}}_{i}\) and \({\hat{{{{\boldsymbol{n}}}}}}_{j}\) are the unit vectors normal to their planes. \({{{\boldsymbol{k}}}}\equiv {\hat{{{{\boldsymbol{n}}}}}}_{i}\times {\hat{{{{\boldsymbol{n}}}}}}_{j}\), rij ≡ rj − ri and \({{{\boldsymbol{b}}}}\equiv ({\hat{{{{\boldsymbol{n}}}}}}_{j}\cdot {{{{\boldsymbol{r}}}}}_{ij}/| {{{\boldsymbol{k}}}}{| }^{2})\left[\left({\hat{{{{\boldsymbol{n}}}}}}_{i}\cdot {\hat{{{{\boldsymbol{n}}}}}}_{j}\right){\hat{{{{\boldsymbol{n}}}}}}_{i}-{\hat{{{{\boldsymbol{n}}}}}}_{j}\right]\).

Mean clustering coefficient

In Fig. 3e, the mean clustering coefficient \(C\equiv \mathop{\sum }_{i=1}^{N}{C}_{i}/N\), where N is the total number of nodes in the network, and Ci is the local clustering coefficient of node i26. \({C}_{i}\equiv 2{t}_{i}/({k}_{i}^{2}-{k}_{i})\), where ti is the number of triangles (loops of length 3) attached to node i, and ki is the degree of node i. Ci = 0 when ki is 0 (isolated node) or 1 (leaf node).

Calculation of the excluded volume of rings

In CP models, the connection probability of two objects is determined by the excluded volume vex: when edge effects are negligible, p = vex/V, where V is the total volume of the system. vex is generally defined as the volume around an impenetrable object within which the center of another identical object cannot access because of the presence of the first object. For example, vex = 4πD3/3 for spheres with diameter D40. In the CP model of infinitely thin rings, vex is equivalent to the volume accessible to the center of a ring when it is connected, i.e., forming a Hopf link, to another fixed ring.

Inspired by Onsager’s calculation of the excluded volume of cylinders57,58, we analytically derive the excluded volume of infinitely thin rings, \({v}_{{{{\rm{ex}}}}}^{{{{\rm{ring}}}}}\), as follows: Consider two connected rings of radius R whose centers are C1 and C2, as shown in Supplementary Fig. 10a. To form a Hopf link, the blue ring should enclose either P or Q, but not both. This can be conveniently seen in the X-\(Y{\prime}\) plane as shown in Supplementary Fig. 10b: For given y and γ, C2 can only be located within the shaded area, which is formed by two circles with radius R centered at P and Q. The shaded area can be found by subtracting the overlap area from the total area of two circles: \({A}_{{{{\rm{shaded}}}}}(y)=2\left[\pi {R}^{2}-{A}_{{{{\rm{overlap}}}}}(y)\right]\). The overlap area \({A}_{{{{\rm{overlap}}}}}(y)=2\left[{R}^{2}\theta (y)-x(y)h(y)\right]\), where \(\theta (y)={\cos }^{-1}\left(x(y)/R\right)\), \(x(y)=\sqrt{{R}^{2}-{y}^{2}}\), and h(y) = y. Therefore, the excluded volume for a fixed γ can be calculated as

$${\tilde{v}}_{{{{\rm{ex}}}}}^{{{{\rm{ring}}}}}(\gamma)=2\,\int_{0}^{R}{A}_{{{{\rm{shaded}}}}}(y)\sin \gamma \,dy=\frac{32}{3}{R}^{3}\sin \gamma .$$
(6)

Given that the rings’ orientations are random, we take the average over all orientations to obtain

$${v}_{{{{\rm{ex}}}}}^{{{{\rm{ring}}}}}=\int_{0}^{2\pi }\int_{0}^{\pi }{\tilde{v}}_{{{{\rm{ex}}}}}^{{{{\rm{ring}}}}}(\gamma)\,\frac{\sin \gamma \,d\gamma \,d\phi }{4\pi }=\frac{\pi }{3}{D}^{3},$$
(7)

where D = 2R. We further verified this result using MC simulation.

The excluded volume of disks (circular plates) of diameter D is \({v}_{{{{\rm{ex}}}}}^{{{{\rm{disk}}}}}={\pi }^{2}{D}^{3}/8\)57, slightly larger than \({v}_{{{{\rm{ex}}}}}^{{{{\rm{ring}}}}}\), as expected. Thus the reduced density (equal to the mean degree as N → ∞) of disks is η = π2nD3/8. Using this and the critical number density πncD3/6 = 0.9614 from Ref. 41, we obtain 〈k〉c ≡ ηc = 2.27 for disks.