Influence of network topology on sound propagation in granular materials

Granular media, whose features range from the particle scale to the force-chain scale and the bulk scale, are usually modeled as either particulate or continuum materials. In contrast with each of these approaches, network representations are natural for the simultaneous examination of microscopic, mesoscopic, and macroscopic features. In this paper, we treat granular materials as spatially embedded networks in which the nodes (particles) are connected by weighted edges obtained from contact forces. We test a variety of network measures to determine their utility in helping to describe sound propagation in granular networks and find that network diagnostics can be used to probe particle-, curve-, domain-, and system-scale structures in granular media. In particular, diagnostics of mesoscale network structure are reproducible across experiments, are correlated with sound propagation in this medium, and can be used to identify potentially interesting size scales. We also demonstrate that the sensitivity of network diagnostics depends on the phase of sound propagation. In the injection phase, the signal propagates systemically, as indicated by correlations with the network diagnostic of global efficiency. In the scattering phase, however, the signal is better predicted by mesoscale community structure, suggesting that the acoustic signal scatters over local geographic neighborhoods. Collectively, our results demonstrate how the force network of a granular system is imprinted on transmitted waves.

Figure 1

(a) Image of a 2D vertical aggregate of photoelastic disks confined in a single layer. The driver position is marked with an arrow. Several particles are embedded with a piezoelectric sensor, for which wires are visible. (b) The internal stress pattern within the photoelastic particles manifests as a network of force chains. (c) The dark (blue) lines show a weighted graph, which is determined from image processing and overlaid on the image from panel (b). An edge between two particles (nodes) exists if the two particles are in physical contact with each other; the forces between particles give the weights of the edges.

Figure 2

Example distributions of several network diagnostics for a sample granular packing. The network characteristics that we examine include (a) global efficiency $E_w$, (b) community structure, which we visualize using the quantity $X^2(z+5)$, where $X$ is the community label, (c) geodesic node betweenness $B_w$, and (d) clustering coefficient $C_w$. This figure illustrates their respective sensitivities to system-scale (2D), domain-scale (2D), curve-scale (1D), and particle-scale (0D) structure. The quantity $X^2(z+5)$ allows us to visualize both the community label ($X$) and the intracommunity strength $z$-score $z$ [see Eq. (3)] simultaneously; we chose the constant 5 purely for visual clarity.

Figure 3

(a), (b) Modularity index $Q_w$, (c), (d) number of communities, (e), (f) mean size of communities in terms of number of particles, and (g), (h) variance of community size as a function of (a), (c), (e), (g) the resolution parameter $\gamma$ and (b), (d), (f), (h) the effective fraction of antiferromagnetic edges $\xi \left(\gamma \right)$. Error bars indicate the standard deviation over the 17 experimental runs.

Figure 4

The geographic sizes of communities tend to decrease as the resolution parameter $\gamma$ is increased from (a) $\gamma =0.1$ to (c) $\gamma =2$. We color particles according to their community label. One can examine community structure of the (d) force-chain network using geographic location, (e) intracommunity strength $z$-score $z_i$, (f) participation coefficient $P_i$, and (g) mean force per particle $S_i^{\prime }$. (Recall that $i$ labels the particle.) Example scatter plots for a single experimental run demonstrating that the mean force $S_i^{\prime }$ on particle $i$ is (h) positively correlated with the intracommunity strength $z$-score (the Spearman correlation coefficient for this run is $\rho \approx 0.96$ and the $p$-value is $p\approx 1.9\times {10}^{-283}$) and (i) negatively correlated with the participation coefficient ($\rho \approx -0.18$ and $p\approx 1.8\times {10}^{-5}$). Note that the correlations reported in panels (h) and (i) are for the resolution-parameter value of $\gamma =0.1$. In panel (h), we display the nodes that are assigned to communities 1 through 4 and nodes whose participation coefficients are equal to 0 using different colored markers. We display nodes in any of the four communities whose participation coefficients are greater than 0 (so-called boundary nodes) using dark (purple) markers. The mean correlations for intracommunity strength $z$-score and participation coefficient over experimental runs and values of the resolution parameter are $\rho \approx 0.72\pm 0.02$ and $\rho \approx -0.05\pm 0.04$, respectively. We show the results for one experimental run (No. 2) in this figure, and the results for the other runs are similar.

Figure 5

(a) An example scatter plot between logarithms of the intracommunity strength $z$-score [${\log }_{10}(z+5)$] and the amplitude of the acoustic signal [${\log }_{10}\left(\Delta I\right)$] at $t=1$ for a single experimental run ($j=2$) and resolution-parameter value ($\gamma =0.1$). We added the constant 5 to $z$ to ensure that all values were positive prior to taking the logarithm. The Spearman rank correlation coefficient is $\rho \approx 0.57$ and the $p$-value is $p\approx 2.1\times {10}^{-45}$. (b) Correlation between ${\log }_{10}(z+5)$ and ${\log }_{10}\left(\Delta I\right)$ for all 17 runs as a function of time: $t<40$ is the acoustic signal injection phase, and $t>40$ is the acoustic signal scattering phase. In the bottom part of panel (b), we show a trace of the voltage $V$ of a piezo particle; the injected signal has an amplitude of 0.5V. (c) Correlation between ${\log }_{10}(z+5)$ and ${\log }_{10}\left(\Delta I\right)$ as a function of $\gamma$, where $\gamma \in [0,2]$ is increased in increments of 0.1. We averaged the correlation over all 80 time points at which $\Delta I$ was measured. The mean $\rho$ over runs, $\gamma$ values, and time was $0.34\pm 0.06$. Box plots show the variability over experimental runs. (d) Correlation between ${\log }_{10}\left(\Delta I\right)$ and a variety of network diagnostics: intracommunity strength$z$-score $z=z\left(\gamma \right)$ for $\gamma =0.001$, $\gamma =10$, and $\gamma =100$ (white background, left part of the figure); and weighted (light gray background, middle part of the figure) and unweighted (dark gray background, right part of the figure) versions of global efficiency [$E_w\left(i\right)$ and $E\left(i\right)$], geodesic node betweenness [$B_w\left(i\right)$ and $B\left(i\right)$], and clustering coefficient [$C_w\left(i\right)$ and $C\left(i\right)$]. For completeness, we also show results for the mean force $S^{\prime }$ (left part of the figure), which was the variable previously reported to be correlated with $\Delta I$ [22].

Figure 6

Spearman correlations between signal amplitude $\Delta I$ and (a) the mean force per particle $S^{\prime }$, (b) the intracommunity strength $z$-score $z$ for $\gamma =2$, and (c) the global efficiency $E_w$. Our data encompass all experimental runs and all times, including both injection (left) and scattering (right) phases. We show the Spearman correlations between signal amplitude $\Delta I$ and the three diagnostics shown in panels (a)–(c) in box plots: (d) mean force per particle (${\rho }_{S^{\prime }}$), (e) intracommunity strength $z$-score $z$ (${\rho }_z$), and (f) global efficiency (${\rho }_{E_w}$). We have averaged the correlations that we show in the box plots separately over the injection (left) and scattering (right) phases. For panel (e), note that we also average the correlations over the resolution parameter $\gamma$. Using matlab notation, the precise values of $\gamma$ that we considered are $[\mathtt{0}.\mathtt{001}:\mathtt{0}.\mathtt{001}:\mathtt{0}.\mathtt{009},\mathtt{0}.\mathtt{01}:\mathtt{0}.\mathtt{1}:\mathtt{0}.\mathtt{91},\mathtt{2},\mathtt{3},\mathtt{4}:\mathtt{0}.\mathtt{1}:\mathtt{20},\mathtt{30}:\mathtt{10}:\mathtt{100},\mathtt{200}:\mathtt{100}:\mathtt{1000}]$. The reported $p$-values indicate the results of two-sample $t$-tests.

Figure 7

(a) Spearman correlations between signal amplitude $\Delta I$ and the intracommunity strength $z$-score $z$ averaged over the injection (lower; black) and scattering (upper; red) phases as a function of the resolution parameter $\gamma$. (b) Difference between the correlation in the scattering (${\rho }_z^s$) and injection (${\rho }_z^i$) phases as a function of the number of communities in the partition. We find the greatest sensitivity to phase for partitions with approximately $50–100$ communities (i.e., for communities containing roughly five to eight particles), which corresponds to the upper end of the size scale (two to eight particles) that we identified as potentially interesting using the transformed resolution parameter.

Figure 8

(a) Relationships between 21 binary network diagnostics: degree assortativity ($a$), geodesic node betweenness ($B$), closeness centrality ($C_c$), clustering coefficient ($C$), communicability (${\mathrm{C}}_{\mathrm{o}}$), geodesic edge betweenness ($B_e$), eigenvector centrality ($C_e$), global efficiency ($E_g$), hierarchy ($h$), local efficiency ($E_l$), modularity optimized using the Louvain ($Q_L$) and spectral ($Q_s$) heuristics, number of nodes ($N$), number of edges ($D$), mean connection distance (mcd), random-walk node betweenness ($B_{rw}$), Rent exponent ($p$), robustness to random attack ($R_r$), robustness to targeted attack ($R_t$), subgraph centrality ($C_s$), and synchronizability ($s$). We order the diagnostics to maximize the correlation along the diagonal for better visualization of highly correlated groups of diagnostics. Color indicates the correlation between global network diagnostics over the 17 experimental runs. (b) Reliability, as measured by the coefficient of variation (CV), over the 17 runs for the 21 binary network diagnostics reported in panel (a).

Figure 9

(a) Relationships between eight diagnostics applied to the weighted networks: geodesic node betweenness ($B_w$), clustering coefficient ($C_w$), diversity ($V$), geodesic edge betweenness ($B_{ew}$), global efficiency ($E_w$), modularity ($Q_w$) optimized using the Louvain method, and normalized strength ($S^{\prime }$). We ordered the diagnostics to maximize the correlation along the diagonal for better visualization of highly correlated groups. Color indicates the correlation between global network diagnostics over the 17 experimental runs. (b) Reliability, as measured by the coefficient of variation (CV), over the 17 runs for the eight weighted graph diagnostics reported in panel (a).