Extraction of Force-Chain Network Architecture in Granular Materials UsingCommunity Detection

Danielle S. Bassett1,2,∗, Eli T. Owens3,4, Mason A. Porter5,6, M. Lisa Manning7, Karen E. Daniels31 Department of Bioengineering, University of Pennsylvania, Philadelphia, PA 19104, USA;

2Department of Electrical Engineering, University of Pennsylvania, Philadelphia, PA 19104, USA;3Department of Physics, North Carolina State University, Raleigh, NC 27695, USA;

4Department of Physics, Presbyterian College, Clinton, SC 29325, USA;5Oxford Centre for Industrial and Applied Mathematics,

Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK;6CABDyN Complexity Centre, University of Oxford, Oxford, OX1 1HP, UK;

7Department of Physics, Syracuse University, Syracuse, NY 13244, USA∗Corresponding author. E-mail address: dsb@seas.upenn.edu

(Dated: April 24, 2015)

Force chains form heterogeneous physical structures that can constrain the mechanical stabilityand acoustic transmission of granular media. However, despite their relevance for predicting bulkproperties of materials, there is no agreement on a quantitative description of force chains. Conse-quently, it is difficult to compare the force-chain structures in different materials or experimentalconditions. To address this challenge, we treat granular materials as spatially-embedded networks inwhich the nodes (particles) are connected by weighted edges that represent contact forces. We usetechniques from community detection, which is a type of clustering, to find sets of closely connectedparticles. By using a geographical null model that is constrained by the particles’ contact network,we extract chain-like structures that are reminiscent of force chains. We propose three diagnostics tomeasure these chain-like structures, and we demonstrate the utility of these diagnostics for identify-ing and characterizing classes of force-chain network architectures in various materials. To illustrateour methods, we describe how force-chain architecture depends on pressure for two very differenttypes of packings: (1) ones derived from laboratory experiments and (2) ones derived from ideal-ized, numerically-generated frictionless packings. By resolving individual force chains, we quantifystatistical properties of force-chain shape and strength, which are potentially crucial diagnostics ofbulk properties (including material stability). These methods facilitate quantitative comparisonsbetween different particulate systems, regardless of whether they are measured experimentally ornumerically.

PACS numbers: 89.75.Fb, 64.60.aq, 81.05.Rm

I. INTRODUCTION

Particulate matter comes in many forms: it rangesfrom frictionless emulsions and frictional granular ma-terials to bonded composites and biological cells. Along-known hallmark of particulate systems is the het-erogeneous distribution of inter-particle forces withinthem [1–6]—a phenomenon that has come to be knownas force chains. The term “force chain” arises fromthe appearance—particularly in two-dimensional (2D)systems—of chains of particles that transfer force fromone particle to another along the chain. In Fig. 1, forexample, the eye readily focuses on the backbone thatarises from the set of bright particles. These particles ex-perience forces that are larger than average, and one cansee such forces by using photoelastic particles and plac-ing them between crossed polarizers. In both two andthree dimensions, it is increasingly common to measureinter-particle forces in a variety of systems – includingfrictional grains [6, 7], frictionless grains [8], and friction-less emulsions [9–11]. This opens the door for achievinga better understanding of force chains in a variety of sys-tems.

It is known that force chains are important for resist-ing shear [5] and directing sound propagation [13–16], butlittle is known about which properties of these chains areuniversal to particulate systems and which are sensitiveto details (such as friction, adhesion, boundary condi-tions, and body forces like gravity). Improved methodsfor automatically identifying force chains and quantifyingtheir specific properties (e.g. size and shape) would yielda deeper understanding of how they impact mechanicalproperties of particulate systems.

Although the presence of force chains is a generic fea-ture of particulate systems, the term “force chain” is of-ten used colloquially, and the field still lacks a quantita-tive definition of the term. Recently, several techniqueshave been proposed that aim to identify which subset ofparticles form a “force-chain network.” Peters et al. [17]calculated force chains in low friction by demanding tworequirements: particles must occur in a “quasi-linear” ar-rangement, and they must contain a concentrated stress.They used an algorithm that requires a choice of a thresh-old value for each of the two conditions, and their notiondoes not allow force chains to branch. Using discrete-element simulations with a flat-punch geometry, Peters

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FIG. 1. (Color Online) We represent granular materialsas spatially-embedded networks [12] whose nodes (particles)are connected by weighted edges that represent contact forces.(A) Image of a 2D vertical aggregate of photoelastic disks thatare confined in a single layer by a pressure of P ≈ 6.7×10−4E.Several particles are embedded with a piezoelectric sensor,for which the wires are visible. (B) The internal stress pat-tern in the photoelastic particles manifests as a network offorce chains. (C) The blue line segments show the edges ofa weighted graph, which we determine from image processingand overlay on the image from panel (B). An edge betweentwo particles (i.e., nodes) exists if the two particles are inphysical contact with each other; the forces between parti-cles give the weights of the edges. Note: the orientation ofthe packings is larger horizontal coordinates to the right, andlarger vertical coordinates upward.

et al. [17] obtained force chains that contain approxi-mately half of the particles in a system. Additionally,they reported that the lengths of these chains satisfy anapproximately exponential distribution. Arevalo et al.[18] examined polygonal structures that underly a force-chain network, which they determined by selecting allinter-particle forces above some threshold. In their nu-merical simulations, they observed that triangular struc-tures dominate the network near a critical packing frac-tion. More recently, topological tools such as computa-tional homology have been used to address the questionof how to determine the structure of force chains [19–21]. For example, using numerical simulations, Kondicet al. [19] were able to distinguish between frictional andfrictionless packings via a topological invariant known asthe 0th Betti number [22]. They demonstrated that the0th Betti number changes with packing density, and theyused a force threshold to isolate strong forces from weakforces.

The above investigations have been informative, butthey also share a common viewpoint that the strongestinter-particle forces form the backbone of a particulatesystem. However, even a linear chain of strong particleswould not be stable against buckling without the par-ticipation of the particles that lie alongside them [23].Therefore, it is necessary to develop techniques that donot include a minimum threshold force to be able to con-sider a particle to be part of a network.

The purpose of the present paper is to use an approachbased on network science [24] to develop an example of anappropriate method. Network science provides a power-ful set of tools to represent complex systems by focusingnot only on the components of such systems, but alsoon the interactions among those components. A networkrepresentation is particularly appropriate for particulatesystems, where particles (network nodes) are connectedto one another by contact forces (network edges).

In Fig. 1, we show an example network, which weconstructed from laboratory experiments. Network ap-proaches in granular materials have already had severalsuccess in describing the dynamics of granular materi-als. For example, prior studies have investigated spatialpatterns in the breaking of edges in a (binary) contactnetwork under shear [25–28] and the influence of net-work topology on acoustic propagation [29]. The latterstudy established the importance of using weighted con-tact networks to take into account the strength of inter-particle forces. Several other papers have also recentlycontributed to this line of inquiry using a variety of dif-ferent network-based approaches [30–33].

In the present paper, we use network representationsto build a set of practical methods to (1) automaticallyextract force chains from force networks (without the useof thresholding) and (2) define a set of three scalar quan-tities that one can use to characterize and classify differ-ent particulate materials. In Section II, we present two

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commonly-used granular systems (which are rather dif-ferent from each other) in which it is possible to measureinter-particle forces. These two case studies—one in a“wet” laboratory and the other based on a standard com-putational model for jammed packings—provide a basisfor describing the utility of our new technique. In SectionIII, we describe our method to automatically extract theforce-chain structure from a force network.

Our method uses community detection, which is a formof clustering for networks [34, 35]. In contrast to pre-vious uses of community detection in particulate sys-tems [29, 36], our method incorporates a geographical nullmodel to account for spatial embeddedness. This geo-graphical null model makes it possible to to identify clus-ters of particles that are more densely interconnected viastrong contact forces than expected given their geograph-ical proximity. We then calculate a gap factor that al-lows one to optimize the resolution at which the detectedcommunities are maximally branched. One can use thegap factor to quantify similarity of a community’s geom-etry to that expected in a force chain. The gap factor islarger when communities are more branch-like and thusmore similar to the expected geometry of a force chain;it is smaller when communities are either more compactor more string-like, and thus less similar to the expectedgeometry of a force chain. In addition to the gap factor,we use two other diagnostics—size and network force—tohelp characterize particulate structures. (We define thethree diagnostics in Section III B.) We then apply ourmethodology to force networks that we measure in twovery different case studies. We demonstrate the utility ofour diagnostics for (1) quantifying changes in force-chainstructure as a function of the confining pressure appliedto a granular material and (2) facilitating the compari-son and classification of force-chain network architecturesacross different media. We conclude and discuss severalimplications of our work in Section V.

II. METHODS

To better understand which features of force chains areuniversal and to demonstrate that our methodology foridentifying force chains is insightful for particulate sys-tems in general, we compare and contrast results fromtwo very different case studies: granular experiments andfrictionless simulations. In both situations, the particlesare confined in two dimensions, and they each consist ofpackings of bidisperse disks that interact with each otherin a Hertzian-like manner. Both cases are jammed underconstant pressure and use approximately the same num-ber of particles (600). The key differences between thetwo systems are (1) the presence of friction and gravita-tional pressure in the experiments and (2) the use of peri-odic boundary conditions in the simulations. Additionaldifferences include a different fabric tensor due to differ-

ent initial conditions and the fact that laboratory exper-iments have a slight fine-scale polydispersity for particlesof the “same” size in addition to the coarse-scale bidis-persity. For each of the experiments and simulations,we measure force-contact networks for (approximately)the same seven different values of confining pressure.

A. Granular Experiments

We perform experiments on a vertical 2D granular sys-tem of bidisperse disks that are confined between twosheets of Plexiglas. The particles are 6.35 mm thick,their diameters are d1 = 9 mm and d2 = 11 mm (whichyields a diameter ratio of approximately 1.22), and theyare cut from Vishay PSM-4 photoelastic material to pro-vide measurements of the internal forces. These particleshave an elastic modulus of E = 4 MPa. We produce newconfigurations by rearranging the particles by hand. Weincrease the pressure by placing additional brass weightson the top surface of the packing. The values of pres-sure, which we report in units of the elastic modulusE (recall that the configuration is two-dimensional), are2.7 × 10−4E, 4.1 × 10−4E, 6.7 × 10−4E, 1.1 × 10−3E,2.2 × 10−3E, 3.8 × 10−3E, and 5.9 × 10−3E. Particlesare constrained by vertical walls to prevent deformationdue to these pressures from occurring in the directionperpendicular to the loading direction. Such constraintson deformation can influence the shape of force chains,which tend to form in the direction of the major principalstress [6, 26, 37]. See Refs. [16, 29, 38] for additionaldetails about the experiments.

For each of 21 particle configurations and the seven val-ues of pressure, we compute particle positions and forcesusing two high-resolution pictures of the system. We useone image, which we take without the polarizers, to de-termine the particle positions and contacts. (See [39] fora description of the technique.) We take particles to bein contact if the force between them is measurable by ourphotoelastic calculations. Using a second image that wetake with the polarizers, we then determine the particlecontact forces by solving the inverse photoelastic problem[39].

B. Frictionless Simulations

We perform numerical simulations of a 50:50 ratio ofbidisperse frictionless disks. We use a diameter ratio of1.22, and particles interact via a Hertzian potential ina box with periodic boundary conditions in both direc-tions and zero gravity [40–42]. This model has been well-studied and it is significantly different from our experi-mental system with respect to friction, gravity, and thefixed boundaries. We generate mechanically-stable pack-ings via a standard conjugate-gradient method [40]. We

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then perform simulations for a fixed packing fraction andvolume, and we analyze 20 mechanically-stable packingsat each packing fraction φ. We choose the seven valuesof the packing fraction so that the mean pressure p atthat packing fraction [43] matches the ones in the exper-iments: [φ, p] = [0.8499, 3× 10−4E], [0.8521, 4× 10−4E],[0.8560, 7 × 10−4E], [0.8621, 11 × 10−4E], [0.8760, 22 ×10−4E], [0.8927, 38 × 10−4E], and [0.9106, 59 × 10−4E],where the modulus E is defined as the energy scale forthe Hertzian interaction ε divided by the mean Voronoıarea of a particle in the packing. The smallest value of φprovides a data point for a jammed packing that is lessdense than what is accessible in our experiments.

III. FORCE-CHAIN EXTRACTION ANDCHARACTERIZATION

A. Force-Chain Extraction

To locate the force chains, we begin by recording whichparticles are in contact with (and exert force on) whichother particles. In network language, we represent eachparticle as a node, and we represent each inter-particlecontact as an edge whose weight is given by the magni-tude of the force at that contact. We thereby constructa force-weighted contact network W from a list of allinter-particle forces. If particle i and j are in contact,then Wij = fij/mean(f), where fij is the normal forcebetween them. If two particles are not in contact, thenWij = 0. In addition, we let Wii = 0. We also constructan unweighted (i.e., binary) matrix B whose elements are

Bij =

{1 , Wij 6= 0 ,

0 , Wij = 0 .

The matrix B is often called an “adjacency matrix” [24],and the matrix W is often called a “weight matrix.”

To obtain force chains from W, we want to determinesets of particles for which strong inter-particle forces oc-cur amidst densely connected sets of particles. We canobtain a solution to this problem via “community de-tection” [34, 35, 44], in which we seek sets of denselyconnected nodes called “modules” or “communities.” Apopular way to identify communities in a network is bymaximizing a quality function known as modularity withrespect to the assignment of particles to sets called “com-munities.” Modularity Q is defined as

Q =∑i,j

[Wij − γPij ]δ(ci, cj) , (1)

where node i is assigned to community ci, node j is as-signed to community cj , the Kronecker delta δ(ci, cj) = 1if and only if ci = cj , the quantity γ is a resolution pa-rameter, and Pij is the expected weight of the edge thatconnects node i and node j under a specified null model.

One can use the maximum value of modularity toquantify the quality of a partition of a force network intosets of particles that are more densely interconnected bystrong forces than expected under a given null model.The resolution parameter γ provides a means of probingthe organization of inter-particle forces across a rangeof spatial resolutions. To provide some intuition, wenote that a perfectly hexagonal packing with non-uniformforces should still possess a single community for smallvalues of γ and should consist of a collection of single-particle (i.e., singleton) communities for large values ofγ. At intermediate values of γ, we expect maximizingmodularity to yield a roughly homogeneous assignmentof particles into communities of some size (i.e., numberof particles) between 1 and the total number of particles.(The exact size depends on the value of γ.) The stronglyinhomogeneous community assignments that we observein the laboratory and numerical packings (see Section IV)are a direct consequence of the disorder in the packings.

An important choice in maximizing modularity opti-mization is the null model Pij [45, 46]. The mostcommon null model for modularity optimization is theNewman-Girvan (NG) null model [34, 35, 47, 48]

PNGij =

kikj2m

, (2)

where ki =∑jWij is the strength (i.e., weighted de-

gree) of node i and m = 12

∑ijWij . The NG null model

is most appropriate for networks in which a connectionbetween any pair of nodes is possible. Importantly, manynetworks include (explicit or implicit) spatial constraintsthat exert a strong influence on which edges are present[12]. For particulate systems, numerous edges are simplyphysically impossible, so it is important to improve uponthe NG null model for such applications. We use the termgeographical constraints to describe the explicit spatialconstraints in such systems. These constraints exert asignificant effect on network structure, so it is importantto take them into account when choosing a null model.For granular materials (and other particulate systems),each particle can only be in contact (i.e., Wij 6= 0) withits immediate neighbors. We therefore use a null model,which we call the geographical null model, to account forthis constraint. The geographical null model is

Pij = ρBij , (3)

where ρ is the mean edge weight in a network and Bis the binary adjacency matrix of the network. (Such anull model was used previously for applications in neu-roscience [46].) Recall that the adjacency matrix encap-sulates the presence or absence of contact between eachpair of particles. For a granular material, ρ = f := 〈fij〉is the mean inter-particle force. Because we have nor-malized the edge weights in the force network (Wij =fij/mean(f)), we note that in our case ρ = 1.

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Maximizing Q yields a so-called “hard partition” of anetwork into communities in which the total edge weightinside of modules is as large as possible relative to thechosen null model. A hard partition assigns each node toexactly one community. (An alternative is a “soft parti-tion” [49], which allows each node in a network to be as-sociated with multiple communities.) For the geographi-cal null model in (3), maximizing Q assigns the particlesinto communities that have inter-module particle forcesthat are larger than the mean force. Such communitiesrepresent the force chains in a granular system.

Because maximizing Q is NP-hard [50], the success ofthe maximization is subject to the limitations of the em-ployed computational methods. In the present paper,we use a Louvain-like locally greedy algorithm [51]. Ad-ditionally, given the numerous near-degeneracies in themodularity landscape that tends to inflict networks thatare constructed from empirical data [52] (i.e., many dif-ferent partitions often yield comparably large values ofQ), we report community-detection results that are en-semble averages over 20 optimizations.

B. Properties of Force Chains

We characterize communities using several diagnostics:size, network force, and a gap factor (a novel notion thatwe introduce in the present paper). The size sc of acommunity c is simply the number of particles in thatcommunity. The systemic size s is given by the mean ofsc over all communities. The modularity Q is composedof sums of magnitudes of bond forces and therefore alsohas units of force. Therefore, we use the term networkforce to indicate the contribution of a community c tomodularity. The network force of a community is givenby the formula

σc =∑i,j∈C

[Wij − γρBij ] , (4)

where C is the set of nodes in community c. The sys-temic network force σ is the mean of σc over all com-munities. Communities that correspond to force chainscomposed of densely packed particles with large forces be-tween them have large values of network force, whereascommunities that correspond to force chains composed ofsparsely packed particles with small forces between themhave small values of network force.

To identify the presence of gaps and the extent ofbranching in the geometry of a force-chain network,we calculate the Pearson correlation between “physicaldistance” (which we measure using the standard Eu-clidean metric) and “hop distance” (which is often calledthe “topological distance” and counts distance measuredonly along network edges), and we examine how the cor-relation depends on the force-chain topology. Communi-ties with compact or linear-chain characteristics (see the

HopDistance

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FIG. 2. (Color Online) Schematic illustration of the gap fac-tor, which we measure via the Pearson correlation between thehop distance Lt and physical distance Lp in granular force net-works. Network communities with gaps, branches, and ringscan have a larger hop distance than physical distance. Theythus reside in the upper triangle.

main diagonal in Fig. 2) occur when there is a perfectcorrelation between hop distance and physical distance.

The perfect correlation arises because particles thatare one hop away from each other (i.e., they are adja-cent to each other in the binary contact network) arealso 1 particle-distance away. Small linear chains occurclose to the origin, where both hop distance and physi-cal distance are small, whereas long linear chains occurin the upper right quadrant of Fig. 2. Note that we usethe term “compact” in the spirit of the mathematicalsense of the term, although our exact meaning is some-what different: “compact” communities of particles areat the opposite extreme as linear chains. Small compactblobs contain particles that are 1 hop-distance away fromone another and 1 particle-distance away from one an-other, and they therefore occur near the origin of Fig. 2.Larger compact blobs contain particles that are severalhop-distances away (and an equal amount of particle-distances away), and they therefore occur in the upperright quadrant of Fig. 2.

In contrast to compact blobs and linear chains, forcechains with gaps, branches, and rings have a larger hopdistance than physical distance (see the upper triangle inFig. 2). Particles that are close together in space do notnecessarily have strong forces to bind them; the pres-ence of more complicated shapes decreases the correla-tion between the physical and hop distance. (Note forparticulate systems that the lower triangle is unphysical,as it would require forces between particles that are notin contact with one another to achieve a larger physicaldistance than hop distance.)

To identify communities that are composed ofbranched structures, we measure the amount of correla-tion between the hop distance and the physical distance

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FIG. 3. (Color Online) Sample community-detection resultsfor a single granular experiment at 4.1 × 10−4E. We colorcommunities according to (A) community size sc, (B) net-work force σc, and (C) community gap factor gc. In thesecalculations, we use γopt = 0.9, which we choose to maximizethe systemic gap factor g across all pressures (rather thanfor a single pressure). This example illustrates that the threenetwork diagnostics can reveal very different spatial distribu-tions in the data and thereby supports examining all threediagnostics.

in the set of all node pairs in a given community. Tocompute the hop distance of a community, we define thecommunity contact network Bc. Its elements Bcij are en-tries of the matrix B for which the corresponding nodeshave both been assigned to the same community c. Wethen calculate the path lengths between possible pairs ofnodes in a community c using the hop distance on thematrix Bc. The resulting matrix of pairwise distances isLt. To compute physical distance, we calculate the Eu-clidean distances between all possible pairs of nodes in acommunity. The resulting matrix of pairwise distances isLp.

In defining the community gap factor, we choose toweight each community by its size. We thus weight largecommunities more heavily than small communities in lin-ear proportion to the number of particles that they con-tain. In this case, the gap factor gc of a community cmeasures the presence of gaps and the extent of branch-ing in a community. We calculate it using with the for-mula

gc = 1− rc scsmax

, (5)

where rc is the value of the Pearson correlation betweenthe upper triangle of Lt and the upper triangle of Lp,and smax is the size of the largest community. (Note thatwe exclude the diagonal elements of the matrices Lp andLt.) To provide further illustration of this quantity, a setof communities colored by their respective gap-factors isshown in Fig. 3(C).

We define the (weighted) systemic gap factor as

g = 1− 1

n

∑c

rc scsmax

, (6)

where the quantity n is the total number of communities(including singletons) and we again note that large com-munities contribute more than small communities to the

averaging in Eq. (6). An alternative choice for the sys-temic gap factor is to weight all communities uniformlyto calculate an alternative systemic gap factor

guniform = 1− 1

n

∑c

rc . (7)

As we discuss in more detail in the appendix, branchedcommunities typically include more nodes than linearcommunities, so the weighted gap factor g tends to belarger for communities with more branching. By con-trast, guniform tends to be larger for more linear commu-nities. For the remainder of the main manuscript, wefocus on the weighted gap factor g because it varies farless across packings. Further studies of guniform wouldalso be interesting.

IV. CHARACTERIZING FORCE CHAINS

We now characterize the force-chain networks of theexperiments and simulations that we described in SectionII using the methodology that we described in SectionIII. Our community-detection procedure consists of twostages. First, we maximize modularity (1) with the geo-graphical null model (3) for different values of the resolu-tion parameter γ: from γ = 0.1 to γ = 2.1 in incrementsof ∆γ = 0.2. We then choose a resolution that approxi-mately maximizes the systemic gap factor g [see Eq. (6)].This ensures that we extract communities of particlesthat have strong and dense force-weighted contacts withone another (i.e., our network partition has a large valueof the modularity Q), and that they are spatially sparse(i.e., they tend to have a small topological-physical dis-tance correlation rc). The communities that we obtaintend to take the form of chain-like structures that arereminiscent of force chains; this provides visual supportthat our technique is successful. Our communities aremuch better than what one obtains using the NG nullmodel. In previous work using the NG null model[29],we observed that these latter communities always tendto be compact in form. We know, however, that commu-nities with other qualitative features are very common.(See the schematic in Fig. 2.)

The resolution parameter γ sets the spatial resolutionof the communities [34, 53, 54] [see Eq. (1)]. By tun-ing γ, one can either examine large communities (usingsmall values of γ) or small communities (using large val-ues of γ). In Fig. 4A, we show an example computa-tion using data from experiments. (Note that we do notshow single-particle communities.) Observe that smallvalues of γ produce communities that are dominated bycompact structures (small g), whereas large values of γproduce communities that are dominated by linear struc-tures (small g as well). This suggests that we can identifyan optimal value γopt that maximizes the systemic gap

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FIG. 4. (Color Online) Sample community-detection resultsfor a single granular experiment at 4.1×10−4E illustrate mul-tiresolution structure in chain-like communities. (A) Com-munity structure as a function of the resolution parameter γ.Color indicates the logarithm (base 10) of the network forceσc of community c. (We don’t show single-particle communi-ties.) (B) The systemic gap factor g [see Eq. (6)] as a functionof the resolution parameter γ exhibits a maximum at γ = 0.7(for γ ∈ {0.1, 0.3, . . . , 2.1}). The error bars indicate the stan-dard deviation of the mean over the laboratory packings. Inthe inset, we show an image of the 2D vertical packing ofphotoelastic disks that we use for panels (A) and (B).

factor g. This also corresponds to the choice for whichthe detected communities are most similar to the force-chain structure that we observe visually (see Fig. 4B).For this particular packing, we identify γopt = 0.7 as thebest choice in our examination of γ ∈ {0.1, 0.3, . . . , 2.1}.As one can see in Fig. 4A, we observe at all values of γthat the detected communities vary in their size, networkforce, and gap factor. We also note that single-particlecommunities are common for all values of γ near 1 (andbecome increasingly common as γ increases after it ex-ceeds 1).

Community detection via modularity maximizationwith a resolution parameter chosen to yield a maximalsystemic gap factor provides an automated approach fordetecting force chains in granular media. Once identified,we then calculate the size s, network force σ, and gap fac-tor g to describe key features of a force-chain network.In the following subsections, we quantitatively examinehow the distributions of s, σ, and g vary as a function ofpressure for both granular experiments and frictionlesssimulations.

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FIG. 5. (Color Online) (A) Images of experimental 2D ver-tical packings of photoelastic disks. These images reveal themanifestation of the internal stress pattern in a set of photoe-lastic particles as a network of force chains. (B) Communitystructure as a function of the resolution parameter γ for thefollowing pressures: (top) 6.7× 10−4E, (middle) 2.2× 10−3E,and (bottom) 5.9×10−3E. Color indicates the logarithm (base10) of the network force σc of community c. (C) Gap factor asa function of both the resolution parameter γ and the pres-sure. The error bars indicate a standard deviation of themean over the laboratory packings. The arrow emphasizesincreasing pressure.

A. Granular Experiments

Our experiments are at seven different pressures, whichrange from 2.7×10−4E to 5.9×10−3E. As we illustrate inFig. 5A, we observe that the communities become morecompact (so they are closer to the diagonal line in Fig. 2)as pressure increases. At a small value of the resolutionparameter (γ = 0.7), the detected communities meta-morphose from branched structures to compact domainsas pressure increases. At a large value of the resolutionparameter (γ = 1.9), the detected communities tend toshrink from many-particle chains to 2-particle chains aspressure increases. This provides a way to quantify ourearlier observations that force-chain networks at higher

8

pressures are more homogeneous (observations at smallγ) and less chain-like (observations at large γ) than theyare at lower pressures.

At all pressures (and despite the aforementioned dif-ferences), we find a maximum in the systemic gap fac-tor g as a function of the resolution parameter γ (seeFig. 5B). The shape of the curve of the gap factor versusthe resolution-parameter has a more pronounced peak athigher pressures than at lower pressures: larger slopesdescend from and (especially) lead up to the values of γnear the maximum. At all pressures, small values of γselect compact communities and large values of γ selecttwo-particle communities. In between, we observe a valueof γ at which the most branch-like structures appear; werefer to this value that approximately maximizes the gapfactor as the “optimal” value. This optimal value changesslightly as a function of pressure; for example, γopt = 0.9at 5.9 × 10−3E and γopt = 0.7 for 2.7 × 10−4E. High-pressure packings also exhibit a much smaller systemicgap factor than low-pressure packings at both small andlarge values of the resolution parameter. This observa-tion is consistent with both the increased compactnessof the communities and the increasingly homogeneousnature of the force-chain structure as one increases thepressure.

The resolution that maximizes the gap factor identifiesstructures in a force network that are most reminiscentof the force chains that are apparent by eye; in otherwords, it identifies branching communities. Therefore,to extract force chains from force networks across dif-ferent packings and pressures, we examine communitystructure for a range of resolution parameters and iden-tify the resolution-parameter value that approximatelymaximizes the gap factor. We refer to the communitiesdetected at γopt as the “force chains” in our calculations,and we characterize their properties in terms of their size,network force, and gap factor. For an illustrative exam-ple of a single packing whose communities are color-codedby either size, network force, or gap factor (see Fig. 3).

As we illustrate in Fig. 6, we observe that the sizeand network force of the force chains have approximatelyexponential distributions for all seven pressures. Thisidentifies that the majority of communities are relativelysmall and weak, but a few communities are relativelylarge and strong. By contrast, the gap-factor distributionis skewed: most communities have a large gap factor, andonly a few communities have a small gap factor. In high-pressure packings, communities tend to be less broadlydistributed with respect to both size and network force;this is consistent with the homogenization of force chainsas pressure increases.

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B. Frictionless Simulations

To illustrate quantitative differences in force-chainstructure between laboratory and numerical packings, weuse our methodology to identify force chains in friction-less simulations. In addition to their lack of friction, thenumerical packings also differ from our experiments inthat they use periodic boundary conditions, have zerogravity, have zero fine-scale polydispersity, and have adifferent fabric tensor due to their different initial condi-tions. (A commonality between the two types of pack-ings is that they are both bidisperse.) As in SectionIV A, we find communities via modularity maximizationat different values of the resolution parameter for differ-ent packing fractions, which we select to match the sevenpressures from the experiments.

As we show in Fig. 7A, the simulated packings changequalitatively as a function of the resolution parameter.For relatively small values of the resolution parameter(γ = 0.7), the granular force network has collapsed intoa single compact community. In contrast, for relativelylarge values of the resolution parameter (γ = 1.3), weidentify many small branch-like communities that arereminiscent of force chains. Unlike in the granular ex-periments, we do not observe a strong qualitative changein community structure as a function of pressure (regard-less of the value of γ). Compare Figs. 5B and 7B.

In order to select the most branched community struc-ture, we again identify a value γopt that is associated withthe maximum value of the systemic gap factor g. As withthe experimental packing, there is a clearly identifiable

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maximum (see Fig. 7B), which occurs at γ = 0.9 forγ ∈ {0.1, 0.3, . . . , 2.1}. We also observe that the shape ofg(γ) is more consistent across pressures for the simula-tions than it is for the experiments.

As in the laboratory packings, we refer to the commu-nities at γopt as our “force chains,” and we character-ize their properties in terms of size, network force, andgap factor (see Fig. 8). Across the set of pressures from2.7 × 10−4E to 5.9 × 10−3E, we observe that chain sizeand network force have approximately exponential distri-butions. This indicates that the majority of communitiesare relatively small and weak, but a few communities arerelatively large and strong. The gap-factor distributionhas a left skew, which indicates that most communities

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for the resolution-parameter value (γ = 0.9) that maximizesthe gap factor in the seven different pressures separately. (C)Histogram of the gap factor gc for the communities at γ = 0.9.The arrows emphasize increasing pressure.

have a large gap factor and only a few communities havea small gap factor.

In contrast to the laboratory experiments, the shape ofthe distributions for size, network force, and gap factor infrictionless simulations do not change dramatically as afunction of pressure. Nevertheless, the frictionless simu-lations to exhibit a systematic difference in the mean sizeand mean network force for communities as a function ofpressure [see Fig. 8B].

C. Comparison Between Laboratory andNumerical Force Chains

Using our methodology of extracting force chains, itis possible to quantitatively compare and contrast twoparticulate systems (such the aforementioned granularexperiments and frictionless simulations). In Fig. 9,we summarize the three main diagnostics (size, networkforce, and gap factor) that we calculate for each of ourtwo case studies. We anticipate that these diagnosticsmight provide a helpful means of characterizing how wella given set of simulations—e.g., with anisotropic cellshapes [55, 56] or different models of friction [57, 58]—captures experimental features. It also provides a sys-tematic way to compare the properties of different par-ticulate systems or to monitor the temporal evolution ofa particulate system.

The first key difference between our results for the ex-perimental and numerical packings is that the granularexperiments have force chains with smaller mean size and

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network force (compare Figs. 9A and B). This result isconsistent with the more homogeneous nature of forcechains in the frictionless packings than in the laboratorypackings. Laboratory packings exhibit a few large andstrong force chains (see Figs. 6A and B), but the ma-jority of a packing tends to be dominated by singletoncommunities (see Fig. 6A). By contrast, the frictionlesspackings have community-size and network-force distri-butions that are far less skewed (see Fig. 8). Additionally,the majority of such a packing is dominated by the forcechains (see Fig. 8A); there are few singletons. These dif-ferences in the homogeneity of the packings and in thedistributions of size and network force leads on averageto larger and stronger force chains in the frictionless nu-merical packings than in the laboratory ones.

We also observe that the gap factor is smaller on av-erage in the laboratory packings than in the frictionlesspackings. This result is also consistent with the homoge-nization of structure in frictionless packings, but it mightalso be influenced by differences between the two types ofpackings from the presence versus absence of gravity. Thefrictionless packings are gravity-free, whereas the labora-tory packings are influenced by gravity. In contrast tothe frictionless packings, the laboratory packings there-fore often exhibit linear vertical chains, which decreasethe systemic gap factor. (See our discussion in Appendix2.)

Finally, we observe that the size and network forceof chains increases with pressure in laboratory packings,whereas the systemic gap factor decreases with pressurefor such packings. These observations indicate the pres-ence of a changing length scale, as is to be expected[59, 60]. As pressure increases, the force structure be-

comes more homogeneous, leading to extended sections ofmaterial with densely packed particles that exert strongforces on one another. However, the force structure be-comes less branch-like, leading to a decrease in the sys-temic gap factor. The frictionless numerical packings alsoexhibit increases in size and network force of chains withpressure; however, unlike in the laboratory packings, thefrictionless packings do not exhibit a decrease in systemicgap factor with pressure. This quantitative difference be-tween the two types of packings yields a technique formeasuring the visual differences between Figs. 5A and7A, in which it is clear that the geometry of the forcechains is affected less by pressure in the simulated set-ting than in the laboratory setting.

Figure 9B also demonstrates that the size of communi-ties that resemble force chains decreases as pressure ap-proaches 0, which provides some elucidation for a long-standing question in the field of disordered matter. Inparticulate systems at 0 temperature, the point at whichpressure P = 0 is the jamming transition, and a lot of re-cent work has been devoted to attempting to understandthe nature of this transition [61]. There is now strong ev-idence that there is a growing length scale [59, 60], whichcorresponds to the size of regions that are mechanicallyunstable, near the transition. As we discuss in SectionV, it is natural to postulate that our force-chain com-munities are negatively correlated with weak regions ina particulate packing. Therefore, we expect force chainsto shrink as mechanically unstable regions grow.

V. CONCLUSIONS AND DISCUSSION

In this paper, we treated granular materials asspatially-embedded networks in which nodes (particles)are connected by edges whose weights we determine fromcontact forces. We developed and applied a network-based clustering method, in which we detect tightly-connected “communities” of particles via modularitymaximization with a geographical null model, to ex-tract chain-like structures that are reminiscent of forcechains in both numerical and laboratory packings. Fromthese chain-like structures, we calculated three key di-agnostics (size, network force, and gap factor), and weillustrated their utility for identifying and characteriz-ing force-chain network architectures across two typesof packings (laboratory and numerical) and across sevendifferent pressures. To characterize force chains, we iden-tified an “optimal” resolution-parameter value that ap-proximately maximizes the gap factor in each of thesescenarios. An alternative approach would be to examinethe structure of force chains that are identified at a set ofresolution parameters that one optimizes separately byconsidering other parameters (e.g., size, network force,etc.). This is an interesting direction for future study.

After we identified an optimal value for the resolu-

11

tion parameter, we systematically evaluated and com-pared properties of force-chain communities. A commonfeature that we observed in both the (frictional) labo-ratory and (frictionless) numerical packings is that thedistribution of force-chain sizes is consistent with an ex-ponential distribution. In the laboratory packings of 2Dgranular materials, we found that high-pressure force net-works exhibit compact rather than branching communi-ties, which is consistent with the notion that increasingpressure causes a breakdown in the long-range heteroge-neous structure in a material. In contrast, the geometryof the force chains in the frictionless numerical packingsappear to be affected less by pressure. The force chainsin the numerical packings also tend to be both larger andstronger than their counterparts in laboratory packings.Together, these results support the conclusion that theforce-chain structure, topology, and geometry is differentin the two systems.

Methodological Insights

An important conclusion of our paper is that choos-ing an appropriate null model is critical for extractinginformation from community detection in networks. Wehave developed and applied a geographical null model forthe detection of network communities, which are stronglyreminiscent of force chains, in particulate materials. Thepower of this null model lies in its ability to fix a net-work’s topology (which is given by a binary contactnetwork) while scrambling its geometry (which is givenby a force-weighted contact network). The geographi-cal null model thereby includes more of the fundamentalphysics of particulate systems than the Newman-Girvannull model that is commonly used in modularity max-imization. An interesting future direction would be toincorporate different physical constraints and principles(e.g., force balance) and to examine the different resultsthat one obtains by including different combinations ofrelevant physical ideas. A key question in community de-tection is what is the minimal set of physical ideas—and,more generally, the minimal amount of problem-specificinformation—that one can include in a null model to getanswers that are more insightful than using a one-size-fits-all null model like the Newman-Girvan model.

Practical Utility

We expect that our methodology will provide a frame-work for understanding which features of force chains areuniversal versus which are governed by detailed particle-particle or particle-environment interactions. For exam-ple, one can use our methodology to provide a quantifica-tion of the similarity of force chains between a simulationand a given experiment. It can also provide similar quan-

tifications between two experiments (or two simulations)with different types of particles or boundary conditions.It is also a viable tool to help predict differences in macro-scropic behavior based on subtle differences in the meanor distributions of force-chain diagnostics.

Our methodology also provides additional informationthat is not available via traditional methods for identify-ing force chains. For example, we know which particlesare strongly connected within communities, and we canalso observe the spatial arrangement of the various com-munities that comprise a force network. This allows oneto ask whether the arrangement of communities helps togovern the linear and nonlinear response of a disorderedpacking. For example, it is possible that large, strongcommunities indicate a region of relatively high mechan-ical stability or that boundaries between these communi-ties indicate areas of weakness. In the future, these typesof investigations should be helpful for obtaining a betterunderstanding of the onset of flow or failure in particulatesystems.

ACKNOWLEDGEMENTS

DSB was supported by the Alfred P. Sloan Foundation,the Army Research Laboratory through contract num-ber W911NF-10-2-0022 from the U.S. Army Research Of-fice, and the National Science Foundation award #BCS-1441502. KED and ETO were supported by NSF-DMR-0644743 and NSF-DMR-1206808, and they thank JamesPuckett for the development of the photoelastic image-processing code. MLM was supported by the AlfredP. Sloan Foundation, NSF-BMMB-1334611, and NSF-DMR-1352184. MAP was supported by the James S. Mc-Donnell Foundation (research award #220020177), theEPSRC (EP/J001759/1), and the FET-Proactive projectPLEXMATH (FP7-ICT-2011-8; grant #317614) fundedby the European Commission. The content is solely theresponsibility of the authors and does not necessarily rep-resent the official views of the funding agencies.

APPENDIX 1: EFFECT OF COMMUNITYWEIGHTING ON FORCE-CHAIN EXTRACTION

In this appendix, we describe an alternative set of mea-surements that we make using the uniformly-weightedsystemic gap factor guniform, which we defined in Eq. (7).

Gap Factor Versus Resolution Parameter

In Fig. 10, we observe that the curve of the maxi-mum of guniform versus the resolution parameter γ variessignificantly with pressure in the (frictional) laboratorypackings but not in the (frictionless) numerical packings.

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FIG. 10. (Color Online) (A) In the laboratory packings, the systemic uniformly-weighted gap factor guniform exhibits a maximumas a function of the resolution parameter γ between roughly γ = 0.9 and γ = 1.3 (for γ ∈ {0.1, 0.3, . . . , 2.1} that is differentfor different pressures. The error bars indicate the standard deviation of the mean over the laboratory packings. (B) In thenumerical simulations, guniform exhibits a maximum (which varies little with pressure) as a function of the resolution parameterγ at γ = 1.1. The error bars indicate the standard deviation of the mean over the numerical packings.

In the laboratory packings, we observe a maximum ofguniform at γ = 0.9 (for γ ∈ {0.1, 0.3, . . . , 2.1}) in high-pressure packings (5.9× 10−3 E) and at γ = 1.5 for low-pressure packings (2.7× 10−4 E). In the numerical pack-ings, we observe a maximum of guniform at γ = 1.1 for allpressures. In comparison to our observations in the maintext from employing the size-weighted systemic gap fac-tor g, we find that the optimal value of γ is larger whenwe instead employ guniform (compare Fig. 10 to Fig. 5 andFig. 7). We also observe that the curves of the systemicgap factor versus resolution parameter exhibit larger vari-ation for the uniformly-weighted gap factor than for thesize-weighted gap factor.

Optimal Value of the Resolution Parameter

The large variation in the maximum of guniform overpackings and pressures makes it difficult to choose anoptimal resolution-parameter value. We choose to takeγopt = 1.1 because (1) it corresponds to the maximum ofguniform in the numerical packings and (2) it correspondsto the mean of the maximum of guniform in the laboratorypackings. To facilitate the comparison of optimal valuesof γ from the two weighting schemes, we denote γopt for gas γ and we denote γopt for guniform as γuniform. Note thatγuniform = 1.1 differs from (and is larger than) γ = 0.9.

Force-Chain Structure at the Optimal Value of theResolution Parameter

The force chains that we identify for the optimal valuefor the uniformly-weighted gap factor (at γuniform = 1.1)differ from those that we identified in the main text forthe optimal value of the size-weighted gap factor (atγ = 0.9). We show our comparison in Fig. 11. For both

laboratory and numerical packings, the force chains thatwe identify at γ = 0.9 are larger and more branchedthan the ones that we identify at γ = 1.1 (which aresmaller and more linear). Indeed, the communities thatwe identify at γ = 1.1 have more singletons than thecommunities that we identify at γ = 0.9. These resultsfollow from the difference in the two weighting schemesfor calculating a systemic gap factor. The size-weightedsystemic gap factor g weights larger communities moreheavily than smaller ones, and the larger communitiestend to be the more branched communities that we iden-tify at smaller values of the resolution parameter (e.g.,at γ = 0.9). In contrast, the uniformly-weighted sys-temic gap factor guniform gives equal weight to small andlarge communities, and it therefore uncovers the linearcommunities that are evident at larger values of the res-olution parameter (e.g., at γ = 1.1). We can thereforeuse the size-weighted gap factor to identify larger, morebranched force chains and the uniformly-weighted gapfactor to identify smaller, more linear force chains.

APPENDIX 2: METHODOLOGICALCONSIDERATIONS

Robustness of Community Structure to Errors inthe Estimation of Contact Forces

In our frictional laboratory experiments, we estimatethat errors in the force measurements could be as largeas ±30% of the contact force fij ; the high variabilityarises from the nonlinear fitting process. (Recall that wetake particles to be in contact if the force between them ismeasurable by our photoelastic calculations. We then de-termine the particle contact forces by solving the inversephotoelastic problem using images taken with polarizers[39].) Somewhat surprisingly, we find that the errors in

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A Bγ=0.9 γ=1.1 γ=0.9 γ=1.1

Laboratory Numerical

Pres

sure

2.7

4.1

6.7

11

22

38

59

1 5Network Force

x10 E-4

FIG. 11. (Color Online) In both (A) (frictional) laboratoryand (B) (frictionless) numerical packings, we identify largerand more branched force chains at the optimal resolution de-termined by (left ; γ = 0.9) the size-weighted gap factor g,and we identify smaller and less branched force chains atthe optimal resolution determined by (right ; γ = 1.1) theuniformly-weighted gap factor guniform. These observationsare consistent across all pressure values, but they are espe-cially evident at high pressures in the laboratory packingsand are least evident at low pressures in the laboratory pack-ings. In the numerical packings, we observe little variation fordifferent values of pressure. In both panels, we highlight thenetwork force for the force chains that we identify in examplepackings.

the force estimates are independent of both the local forceand the global pressure. To ensure that our results arequalitatively robust to these variations, we construct 20simulated force networks for each experimental network(21 packings, seven pressures) by adding Gaussian noisewith width a fij/3 to each contact. For each of these sim-ulated networks, we reevaluate the estimated communitystructure (from which we infer the force chains).

To determine whether the estimates are robust to thisamount of noise, we compare the community structureof the actual force networks with those of the simulatedforce networks using the z-score of the Rand coefficient[62]. For comparing two partitions α and β, we calculatethe Rand z-score in terms of the network’s total numberM of pairs of nodes, the number Mα of pairs that arein the same community in partition α, the number Mβ

of pairs that are in the same community in partition β,and the number wαβ of pairs that are assigned to thesame community both in partition α and in partition β.The z-score of the Rand coefficient comparing these twopartitions is

zαβ =1

σwαβ

(wαβ −

MαMβ

M

), (8)

where σwαβis the standard deviation of wαβ (as in [62]).

Let the mean partition similarity z denote the mean valueof zαβ over all possible partition pairs for α 6= β.

We observe that the assignment of particles to com-munities (and consequently to force chains) in the ex-perimental laboratory force networks is, on average, sta-tistically similar (the z-scores for similarity are largerthan 18) to the assignment of particles to communities inthe simulated networks constructed by adding Gaussiannoise with width fij/3 to each contact (see Fig. 12A).These results indicate that our estimates of network com-munities is relatively robust to the empirical measure-ment error in the inter-particle contact forces.

To further clarify the robustness of our algorithm tovariations in inter-particle forces on the order of ±30% ofthe contact forces, we perform similar calculations for thenumerical packings. For each numerical packing, we con-struct 20 simulated force networks by adding Gaussiannoise with width fij/3 to each contact. For each of thesesimulated networks, we reevaluate the estimated com-munity structure (from which we infer the force chains).We observe that the assignment of particles to communi-ties (and consequently to force chains) in the numericalforce networks is, on average, statistically similar (the z-scores for similarity are larger than 9) to the assignmentof particles to communities in the simulated networksconstructed by adding Gaussian noise with width fij/3to each contact (see Fig. 12B). These results indicate thatour estimates of network communities is relatively robustto measurement error in the inter-particle contact forcesin the form of ±30% of the contact force.

Effects of Gravity on Community Structure inLaboratory Packings

In the laboratory packings, gravity can play an impor-tant role in the heterogeneity of observed force chains.To determine the impact of gravity on network diagnos-tics, we calculate the mean vertical position of particles ineach network community for each packing, pressure, andoptimization of the modularity quality function for theresolution-parameter value (γ = 0.9) that maximizes thegap factor. For each pressure value separately, we collatethese four diagnostics (size, network force, gap factor,and mean vertical position) for every community that weidentify over packings and modularity optimizations. Todecrease the potential for false positives, we then identify

14

Resolution Parameter

Pres

sure

0.1 0.5 0.9 1.3 1.7 2.1

2.7

4.1

6.7

11

22

38

5950

100

150

200

x10 E-4

Resolution Parameter

Pres

sure

0.1 0.5 0.9 1.3 1.7 2.1

2.7

4.1

6.7

11

22

38

59 20

40

60

80

100

120

140

160x10 E-4

A

B

z-score of Rand coe�cient

z-score of Rand coe�cient

FIG. 12. (Color Online) In both (A) laboratory and (B) nu-merical packings, we calculate the similarity between the as-signment of particles to communities for the actual packingsas compared to the packings in which we have added Gaussiannoise with width fij/3 to each contact. We estimate partitionsimilarity using the z-score of the Rand coefficient; we com-pute the z-score for each pair of actual-noisy networks, foreach pressure, for each packing, and for each value of the res-olution parameter. In this figure, we show the mean z-scoreof the Rand coefficient over the 21 packings and over the 20noisy instantiations of the actual force networks.

the unique community sizes, and average network force,gap factor, and mean vertical position over all communi-ties of that size. This process serves as a data-reductionprocedure. It decreases the degrees of freedom in sub-sequent statistical testing, and it thereby decreases thepotential for false positives. We then ask whether thenetwork diagnostics (size, network force, or gap factor)are significantly correlated with the mean vertical posi-tion of the communities. We use a Spearman rank corre-lation to increase our robustness to outliers in the data.For low values of pressure we observe a significant corre-lation between each of the three community-level networkdiagnostics (size, gap factor, and strength) and the meanvertical position of the particles in that community (see

2.7 4.1 6.7 11 22 38 59

-0.4

-0.2

0

0.2

0.4

0.6

Pressure

Corr

elat

ion

Betw

een

Dia

gnos

ticVa

lues

and

Ver

tical

Pos

ition

SizeNetwork ForceGap Factor

x10 E-4

**

*

**

*

*

FIG. 13. (Color Online) Spearman correlation between net-work diagnostics (community size, network force, and gap fac-tor) and the mean vertical position of the community as afunction of pressure. Asterisks indicate significant correla-tions at the level of p < 0.05. (The p-value is uncorrected.)

Fig. 13).

At pressures of 2.7 × 10−4E and 4.1 × 10−4E, largercommunities with larger network force tend to be iden-tified at a lower vertical position of the packing; thisis consistent with the compaction effects of gravity. Atpressures of 4.1×10−4E, communities with large gap fac-tors tend to be observed at high vertical positions in thepackings (where communities tend to be small and lesscompact). For values of pressure between 22×10−4E and59×10−4E, we observe no relationship between mean ver-tical position of particles in a community and communitysize, network force, or gap factor. These results indicatethat gravity can play a role in the shape of force chains,particularly at low pressures.

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