Source-linked AI summary
Network Analysis of Particles and Grains
Lia Papadopoulos, Mason A. Porter, Karen E. Daniels, Danielle S. Bassett
TL;DR
Granular materials organize particles and forces across local, mesoscale, and system-wide structures, challenging conventional models that can overlook intermediate-scale organization. This review synthesizes network-based representations and analyses, showing their value for describing heterogeneous granular architecture and dynamics while identifying practical and conceptual boundaries.
Problem
Conventional particulate and continuum models can be agnostic to intermediate-scale organization, although granular behavior spans multiple spatial and temporal scales.
Method
The paper reviews network theory, graph measures, granular network representations, and studies of structural evolution under external perturbations.
Results
Network-based approaches provide insightful qualitative and quantitative descriptions of heterogeneous granular architecture and complex behavior across diverse material complexities and conditions.
Takeaways & Limitations
Physically informed network analysis can support comparisons across experiments, simulations, particle types, dimensions, and loading conditions, and may inform material design.
Takeaways & Limitations
Available experimental techniques may not provide complete inter-particle force information, including vector forces with both normal and tangential components.
Abstract
from arXiv · showhide
The arrangements of particles and forces in granular materials have a complex organization on multiple spatial scales that ranges from local structures to mesoscale and system-wide ones. This multiscale organization can affect how a material responds or reconfigures when exposed to external perturbations or loading. The theoretical study of particle-level, force-chain, domain, and bulk properties requires the development and application of appropriate physical, mathematical, statistical, and computational frameworks. Traditionally, granular materials have been investigated using particulate or continuum models, each of which tends to be implicitly agnostic to multiscale organization. Recently, tools from network science have emerged as powerful approaches for probing and characterizing heterogeneous architectures across different scales in complex systems, and a diverse set of methods have yielded fascinating insights into granular materials. In this paper, we review work on network-based approaches to studying granular matter and explore the potential of such frameworks to provide a useful description of these systems and to enhance understanding of their underlying physics. We also outline a few open questions and highlight particularly promising future directions in the analysis and design of granular matter and other kinds of material networks.
Glossary of Terms
Granular and particulate materials are collections of discrete macroscopic elements whose contact interactions produce nonequilibrium behavior, deformation, and multiscale force organization. The glossary defines key structural, mechanical, imaging, and simulation concepts used to analyze these systems.
- Granular and particulate materials: Granular materials are discrete macroscopic particles that interact only through contact forces and dissipate energy through frictional and inelastic interactions.Their large size also prevents rearrangement from thermal fluctuations.
- Granular and particulate materials: Particulate materials include broader multiphase entities such as bubbles, foams, colloids, and suspensions, whereas granular materials are a subset.
- Loading and deformation: Packing fraction is the particle-volume fraction, while stress ratio is the ratio of shear to normal stresses and frictional failure occurs when shear exceeds frictional resistance.
- Force transmission: Force chains are subsets of contacts carrying the largest forces, often forming filamentary networks aligned with principal stress axes.
- Mechanical states: Jamming marks a transition from an underconstrained, liquid-like state to a mechanically stable, solid-like state as parameters such as packing fraction or contact number change.
- Mechanical states: Isostatic packings have exactly the minimum contacts required for force and torque balance; hyperstatic and hypostatic packings have more and fewer contacts, respectively.
- Loading and deformation: Stress measures applied force per area, strain measures fractional deformation, and pure shear elongates one axis while shortening the perpendicular axis without net rotation.
- Measurement and simulation: Photoelasticity quantifies internal stresses through polarized-light transmission, while DEM and MD simulations compute particle motions under Newtonian forces and neighbor interactions.
1. Introduction
Granular materials exhibit heterogeneous organization across particle, force-chain, domain, and bulk scales, while conventional particulate and continuum models can overlook intermediate-scale structure. The review examines network science as a framework for representing, quantifying, and comparing this organization and its evolution under loading.
- Motivation: Granular materials are nonequilibrium collections of discrete macroscopic particles whose contact interactions produce dissipation and prevent thermally driven rearrangement.
- Motivation: Interactions span local particle neighborhoods, mesoscale force chains and domains, and bulk material behavior under shear or compression.
- Motivation: Particulate and continuum models are often implicitly agnostic to intermediate-scale organization, despite its relevance to static packings and granular dynamics.
- Illustration: Photoelastic imaging visualizes force patterns in sheared quasi-2D packings, with bright particles carrying the strongest forces and force chains aligning along principal stress axes.
- Network approach: Network science can represent pairwise interactions and quantitatively characterize heterogeneous architectures at microscale, mesoscale, and macroscale sizes.
- Implications: Network-based analyses support comparisons across simulations, experiments, particle types, dimensions, and loading conditions, helping distinguish general from system-specific properties.
- Paper scope: The review surveys network theory, graph measures, granular network representations, applications to perturbations, open questions, and future directions.
2.1 What is a network?
A network represents entities as nodes and their relationships as edges, with graphs, matrices, weights, directions, and more general structures providing alternative representations. For granular materials, particles commonly become nodes and contacts become weighted or unweighted edges.
- Graph representations: A graph consists of nodes representing entities and edges representing interactions, while broader network representations include multilayer networks and spatially embedded systems.
- Matrices and weights: An unweighted adjacency matrix records whether node pairs are connected, whereas a weighted matrix records edge weights such as normal or tangential contact-force components.
- Conventions: The review primarily considers undirected networks and excludes negative edge weights, while directed networks are discussed occasionally.
- Matrices and weights: Weighted networks also have a binary connectivity matrix that ignores interaction strength, separating topological structure from weight-dependent geometry.
- Granular representations: Granular contact networks commonly represent particles as nodes and physical contacts as weighted or unweighted edges.
2.2 Some tools for characterizing granular networks
Network measures characterize granular structures from local neighborhoods to global paths and intermediate motifs or communities. These tools can quantify contact organization, weighted interactions, connectivity, and structural changes during compression, deformation, and jamming.
- Network measures: Network methods characterize topological and geometrical organization relevant to granular stability, mechanical response, and wave propagation.
- Local measures: Node degree counts attached edges, while strength sums their weights; in granular contact networks, degree is also called contact or coordination number.
- Paths and connectivity: Paths and walks describe network traversal, with shortest paths, network distance, diameter, and efficiency quantifying connectivity across nodes.
- Additional tools: Random walks and related spectral ideas provide additional approaches for studying network structure and short paths.
- Paths and connectivity: Mean shortest-path distance must be handled carefully on disconnected networks because distances between components are conventionally infinite.
- Weighted measures: Weighted path measures replace unweighted distances with shortest weighted distances, and larger efficiency generally corresponds to smaller mean shortest-path length.
- Granular applications: Network paths and related quantities can track structural changes as granular packings are compressed through the jamming transition and can support applications such as heat-transfer analysis.
2.2.3 Cycles.
Cycles and related network diagnostics characterize local and mesoscale organization in granular materials, with particular relevance to stability, rigidity, and force transmission. The section introduces cycle-based measures alongside clustering and centrality diagnostics for probing these structures.
- Cycle representations: Cycle spaces, bases, and minimum cycle bases represent the organization of simple cycles and their edge-disjoint unions in a network.Minimum cycle bases support analysis of cycle-length distributions and node participation in cycles of different lengths.
- Cycle representations: A cycle-participation vector records how many cycles of each length include a given node, with x_3 and x_4 counting participation in 3- and 4-cycles.Nodes outside any cycle have zero cycle participation.
- Physical significance: Cycles are useful for studying mesoscale granular structure because they involve multiple nodes without typically representing the entire network.Their intermediate scale matches the apparent role of mesoscale features in granular behavior.
- Physical significance: 3-cycles tend to stabilize rigidity under applied forces, whereas 4-cycles can bend or deform.This distinction motivates examining cycle structure in granular networks.
- Related diagnostics: Weighted clustering coefficients measure triangle density using edge weights and have been applied to examine stability in granular packings.The network-wide weighted coefficient is obtained by averaging the local weighted coefficients over nodes.
- Related diagnostics: Geodesic betweenness centrality measures the fraction of shortest paths traversing a node and can probe heterogeneity in granular force patterns.Closeness centrality instead uses inverse summed shortest-path lengths, while subgraph centrality quantifies participation in closed walks.
2.2.6 Subgraphs, motifs, and superfamilies.
Subgraphs and motifs provide building blocks for describing network organization, while community structure captures mesoscale groups of strongly interconnected nodes. In granular networks, physically informed null models and multilayer methods help distinguish spatial domains, force-chain-like structures, and their reconfiguration under loading.
- Subgraphs and motifs: A directed, unweighted graph has 13 different connected 3-node subgraphs, which serve as basic patterns for network analysis.Subgraphs are constructed from subsets of a graph’s nodes and edges.
- Subgraphs and motifs: Motifs are small subgraphs that occur more often than expected under an appropriate null model, often one preserving features such as the degree distribution.Their over-representation can suggest a role in network function, but such interpretations require caution.
- Granular applications: Subgraphs, motifs, and superfamilies have revealed insights into deformation and reconfiguration in granular systems under different loading conditions and perturbations.These methods have been applied in several granular-material studies.
- Community structure: Community structure describes sets of nodes that are more densely or strongly interconnected than expected under a specified null model.Modularity maximization identifies such groups by optimizing an objective involving within-community edge weights and null-model expectations.
- Community structure: Spatial constraints make the Newman–Girvan null model problematic for granular materials because it assumes any pair of nodes can connect.A physically motivated geographical null model instead incorporates spatial information.
- Community structure: Using different null models produces different granular communities: the Newman–Girvan model reveals contiguous domains, whereas the geographical model detects chain-like structures reminiscent of force chains.In the geographical model, P_ij = ρA_ij, with ρ equal to the mean edge weight in the network.
- Multilayer communities: Multilayer community detection tracks force-network communities across system states, allowing persistence and reconfiguration in particle content and mean node strength under applied loads.Interlayer couplings connect corresponding particles across layers, while layer-specific resolution parameters and null models define the optimization.
2.2.8 Flow networks.
This section introduces network concepts for connectivity and higher-order topology, including components, percolation, simplicial complexes, cycles, and persistent homology.
- Connected components and percolation: Percolation theory examines how connected components and percolating clusters emerge as network connectivity or physical parameters change.Granular applications include connectivity percolation, rigidity percolation near jamming, and force-network contact clusters.
- Algebraic topology and computational topology: Simplicial complexes generalize graphs by representing relationships through k-simplices rather than only nodes and dyadic edges.A clique complex fills every fully connected subgraph with a corresponding simplex.
- Algebraic topology and computational topology: Cycles are closed arrangements of simplices, and topology distinguishes cycles surrounding filled regions from those enclosing holes.The latter represent voids in the complex.
- Algebraic topology and computational topology: Persistent homology decomposes a weighted graph into thresholded graphs and tracks the birth and death of topological features across the resulting filtration.For force networks, descending force thresholds produces persistence diagrams for connected components and loops.
- Algebraic topology and computational topology: In two-dimensional granular force networks, flag-complex persistent homology counts loops involving four or more particles while excluding triangular loops.These larger loops are associated with defects relative to perfectly packed monosized disks.
2.3 Some considerations when using network-based methods
Network measures probe different spatial, topological, and geometrical scales, but their precise relationships remain an open issue, especially in spatially embedded systems.
- Spatial and physical information: Network quantities differ in whether they use topology alone, geometry and edge weights, or explicit spatial and physical information.These distinctions matter because granular materials are embedded in real space and obey physical constraints.
- Spatial and physical information: Spatial information can be incorporated by defining edge lengths from physical distances or by using geographical null models in community detection.These choices replace purely topological distances or traditional null models with spatially informed alternatives.
- Scales of network computations: Degree, strength, and clustering quantify local neighborhoods, whereas mean shortest path length and global efficiency probe system-wide organization.Mesoscale methods range from small motifs to larger communities.
- Scales of network computations: Persistent homology is designed to reveal robust structural features across multiple scales.The passage places it between local and global network computations as a multiscale tool.
- Open issues: A precise understanding of the relationships among different network computations remains a major open issue.The issue is particularly important when additional spatial constraints are present.
3. Granular materials as networks
The paper reviews network-based representations of granular materials as complements to traditional models, emphasizing their ability to quantify heterogeneous organization across scales and conditions.
- Overview: Network analysis provides a novel view of granular structure and dynamics that complements and extends traditional granular-material perspectives.The review covers network-based models and approaches developed over the past decade.
- Motivation: Network representations can capture intrinsic heterogeneity, including force chains, and quantify structural changes under external loads or perturbations.These advantages address local, intermediate, and system-wide organization.
- Motivation: Traditional particulate and continuum approaches are often implicitly agnostic to intermediate-scale organization.That organization is relevant to both static packings and granular flows.
- Network representations: Network analysis applies to simulations and experiments, dense quasistatic materials, and granular flows.The framework supports studying how systems evolve over time.
- Network representations: Additional interaction information enables more detailed network constructions for studying heterogeneous force transmission through force chains.A small subset of particles can carry a majority of the force along these chains.
- Scope of the review: The review examines network constructions and diagnostics to quantify multiscale organization and improve understanding of granular physics.It treats network theory as a framework for connecting representations with physical interpretation.
3.1 Contact networks
Contact networks represent granular packings as spatially embedded graphs, linking network structure to coordination, rigidity, jamming, cycles, and force-chain behavior. Network analyses show that contact loops—especially 3-cycles—track stability and deformation across loading, jamming, and aging.
- 3.1 Contact networks: A contact network is a spatially embedded graph whose nodes are particles and whose edges represent interparticle contacts.Particle radii and Euclidean positions constrain the network; contacts can also be estimated experimentally using reasonable thresholds.
- 3.1.1 Coordination number and node degree: Mean node degree is equivalent to the granular coordination number Z, which is connected to mechanical stability, rigidity, and jamming.The coordination number can serve as an order parameter for jamming, with an isostatic value marking the minimum contact number needed for stability.
- 3.1.2 Investigating rigidity of a granular system using a contact network: Contact-network cycles provide mesoscale structural information beyond coordination number, with odd cycles—especially 3-cycles—supporting stability by frustrating grain rotation and providing lateral support.Even cycles can permit rolling without slipping under shear, whereas odd cycles frustrate relative rotations; later studies commonly identify contact loops as stabilizing features.
- 3.1.3 Exploring the role of cycles: During biaxial loading, 3-cycles and 4-cycles decrease toward peak shear stress, while stress-drop intervals produce increases in shorter cycles opening into longer cycles.Force chains with greater 3-cycle participation are less likely to fail by buckling, and 3-cycles are more stable than other cycle lengths in structural-mechanics analyses.
- 3.1.4 Other subgraphs in contact networks: Other subgraph analyses classify local contact conformations into almost-invariant sets and identify transition pathways as packings move toward or away from jammed configurations.Force-chain particles are associated mostly with densely connected conformation subgraphs, and the pathways may support constitutive-modeling efforts.
3.2 Force-weighted networks
Force-weighted network analyses represent granular contacts together with force magnitudes, enabling multiscale characterization of structure, deformation, and transitions such as jamming. Reviewed studies use weighted cycles, communities, efficiency, force thresholds, and persistent homology to connect network organization with mechanical behavior.
- Weighted 3-cycle stability decreases with increasing tilting angle, independently of the effect on mean coordination number.Approximately equal forces on the three edges correspond to the most stable configuration; strongly unequal forces are unstable.
- An increase in 3-force cycles at force-chain buckling may indicate failure onset, although these cycles can initially provide lateral support and frustrate relative rotations.The cycles often concentrate in the shear band and may stabilize force chains before increased loading leads to failure.
- Including contact forces provides a more complete characterization of granular systems than analyzing contact structure alone.
- Force-chain communities distinguish frictional laboratory packings from frictionless simulations and quantify effects of friction and pressure on mesoscale organization.
- Force-network efficiency exceeds that of two null models, suggesting that force-chain spatial organization facilitates heat transport.
- Persistent-homology measures identify jamming through changes in force-network components and capture rapid global rearrangements at the transition followed by smoother reconfiguration above jamming.Betti numbers also quantify effects of friction and polydispersity on force-network organization.
- Betti numbers distinguish disk and pentagon packings across a wide range of force thresholds but do not clearly separate states with similar packing fractions produced by different tap intensities.
3.3 Other network representations and approaches
Beyond force-weighted contact networks, researchers construct flow, displacement, broken-link, functional, kinematic, and phase-space networks to study force transmission, rearrangements, and dynamics. These representations connect network changes to deformation, failure, reversibility, and particle-level motion.
- Maximum-flow–minimum-cut networks represent force transmission capacities and dissipation costs, allowing bottlenecks and constraints on material force transfer to be examined.
- Networks combining contacts with particle displacements track how local rearrangements evolve across strain states until material failure.
- 3.3.2 Broken-link networks: Broken-link networks encode changes in proximity networks during shear and quantify particle rearrangement events through their temporal evolution.
- 3.3.2 Broken-link networks: The largest connected component of a broken-link network grows with the fraction of broken links, identifying a deformation region where rearrangements become globally connected.Because the networks are finite, interpretations as percolation or phase transitions require caution.
- Related networks study the progression from reversible to irreversible dynamics in granular suspensions under oscillatory shear.Experiments varied the shear amplitude across 2°, 4°, 10°, 20°, and 40°.
- Functional and kinematic networks derive edges from particle-property time series or measured motion, while phase-space networks associate communities with slider slip events.
3.4 Comparing and contrasting different network representations and approaches
Network representations encode different aspects and scales of granular organization, so their conclusions are related when they probe similar quantities but differ when they target different structures. The review calls for principled comparisons among network-based, traditional, and alternative representation choices.
- Network representations differ in the physical relationships and information they encode, producing distinct views of granular systems.
- Force-weighted networks retain contact connectivity plus force information, enabling analyses of force-chain organization unavailable to unweighted contact networks.
- Clustering coefficients, 3-cycle densities, and some small-subgraph analyses can yield related conclusions because they probe overlapping local structures.
- Node strength illustrates how network quantities can target different scales, because it is a particle-scale property capturing local force information.
- Future work should directly compare the results and assumptions of network-based, traditional, and alternative network-representation approaches.
3.5 Limitations and practicalities of simulations and experiments
Experiments and simulations provide inter-particle contact and force data, but measurement resolution and contact-identification thresholds constrain how networks can be constructed. Force measurements support more confident contact-network construction, while vector resolution varies across techniques and systems.
- Network studies use inter-particle contact and force data from both experiments and simulations, including measurements under external loading.
- Laser-sheet scanning, photoelasticity, and x-ray tomography can measure particle positions and inter-particle forces.
- Force measurements may provide vector forces, but limited resolution can restrict some techniques or systems to scalar or coarse-grained values.
- Constructing contact networks requires studying thresholding effects to distinguish actual contacts from particles that are merely adjacent.
4. Open problems and future directions
The review identifies future directions in network construction, cross-material applications, and material design. Proposed extensions include richer representations, physical constraints, broader material classes, and network-guided control of multiscale properties.
- 4. Open problems and future directions: Future work is organized around constructing new networks, applying network analysis to additional materials, and designing materials.
- 4.1 Network construction: Granular networks can vary in node and edge definitions, including non-spherical grains, tangential or directional forces, signed weights, and contact-centric duals.
- 4.1 Network construction: Pore networks represent pores as nodes and connecting throats as edges, providing a framework for studying flow through porous materials.
- 4.1 Network construction: Multilayer and multiplex networks can combine heterogeneous node types, multiple relationships, and time dependence, including normal and tangential force relations.
- 4.1 Network construction: Hypergraphs and simplicial complexes can encode interactions among three or more particles rather than restricting analysis to pairwise relations.
- 4.1 Network construction: Network methods should incorporate force balance, physical geometry, spatial embedding, and latent spatial structure through constrained null models and diagnostics.
- 4.2 Beyond granular materials: Network approaches may extend from canonical contact-force granular systems to colloids, glasses, polymers, gels, and other soft materials.
- 4.2 Beyond granular materials: Random geometric graphs and their extensions are identified as useful tools for future network studies.
5. Conclusions
The review synthesizes network science as a productive framework for understanding granular structure, dynamics, and responses across multiple scales. It points toward broader applications in particulate matter and the study of mesoscale interaction patterns in mechanical failure.
- 5. Conclusions: Network science has yielded insights into granular architecture, dynamics, and responses to compression, shear, tapping, and tilting.
- 5. Conclusions: The reviewed work addresses heterogeneous material architecture across particle, mesoscale, and system-wide organization.
- 5. Conclusions: Future efforts should improve understanding of particulate-matter physics and clarify the roles of mesoscale interaction patterns in mechanical failure.