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Networks in Space: Granular
Force Networks and Beyond
MASON A. PORTER

(@MASONPORTER)

MATHEMATICAL INSTITUTE

UNIVERSITY ...
SEE
(space, space, space, space, space…)
SoftMatter
www.softmatter.org
ISSN 1744-683X
PAPER
Volume 11 Number 14 14 April 2...
SEE (space, space, space, space, space…)
Marc Barthelemy’s Review Article on
Spatial Networks
•  M. Barthelemy, “Spatial Networks”, Physics Reports, Vol.
499, 1–10...
Force Distribution
Small number of particles carry greater than average force
Force distribution in simulated
granular pac...
Sound Propagation
Liu and Nagel J. Phys. Condens. Matter 1993
•  Sound amplitude is very sensitive to small rearrangements...
Granular Force Networks
•  2D, vertical, 1 layer aggregate of photoelastic disks
•  Internal stress pattern in compressed ...
Experimental Setup
•  Excite acoustic waves from the bottom by sending pulses of 750
Hz waves
•  Bidisperse ensemble of di...
Inter-Particle Force
•  Experimentally measure
force law to be
•  This implies that contact area
scales as
5/4	
  
Measuring Sound
750 Hz input signal detected via:
(1) Change in particle brightness (ΔI)
(2) Piezoelectric (V) [voltage in...
Network Communities
•  Communities = Cohesive
groups or modules
–  In statistical physics, one tries to
derive macroscale ...
Detecting Communities
•  Survey article: MAP, J.-P. Onnela, & P. J. Mucha [2009],
Notices of the American Mathematical Soc...
Puck	
  Rombach	
  
Optimizing Modularity
•  Minimize:
–  Potts Hamiltonian
•  σi = community assignment (spin state) of node i
•  Jij > 0 è ...
Sound Propagation in Granular
Force Networks
•  D. S. Bassett, E. T. Owens, K. E.
Daniels, and MAP [2012], Physical
Review...
Some Network Diagnostics
•  We used 21 diagnostics for the contact (unweighted) networks and
8 diagnostics for the force (...
•  Global efficiency:
–  dij
w = length of weighted shortest path between nodes i and j
•  Geodesic node betweenness:
–  N...
Spatial Embeddesness Induces
Correlations in Network Diagnostics
Random Geometric Graphs (RGGs)
•  Useful to compare properties of real networks to a
random graph ensemble as a “null mode...
Contact Networks versus Random
Geometric Graphs (very different)
A Better Model:
RGG With A Sprinkling of Physics
•  J. Setford, MPhys thesis, University of Oxford, 2014
–  Available at h...
Geographical Community Structure
•  Different resolution-parameter values give communities
at different scales.
•  Average...
Community Structure Constrains Sound
Propagation
Ø  D. S. Bassett, E. T. Owens,
MAP, M. L. Manning, &
K. E. Daniels [2015], Soft
Matter,Vol. 11, No. 14,
2731–2744
Ø  Use...
Force-Chain Structure and Pressure
“optimal” valu
tion of pressur
¼ 0.7 for 2.7 Â
much smaller s
both small and
observatio...
aximum of guniform at g ¼ 0.9 (for g ˛ {0.1, 0.3,.,
essure packings (5.9 Â 10À3
E) and at g ¼ 1.5 for
ackings (2.7 Â 10À4
...
Some Other Approaches for Studying
Granular Force Networks (selected)
•  Tools from computational homology (e.g. persisten...
And Beyond…•  We have also used spatial null
models for studies of brain
networks, disease-occurrence
networks, and intern...
Conclusions
•  Studying community structure allows one to examine
mesoscale structures in granular packings.
–  Provides d...
Future Work: Granular
Networks and Beyond
•  Granular networks
–  Use network diagnostics in time-dependent situations to
...
Summer School on Computational
Algebraic Topology
•  https://people.maths.ox.ac.uk/tillmann/CAT-SCHOOL.html
•  Some fundin...
Networks in Space: Granular Force Networks and Beyond
Networks in Space: Granular Force Networks and Beyond
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Networks in Space: Granular Force Networks and Beyond

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This is my talk for the Network Geometry Workshop (http://ginestra-bianconi-6flt.squarespace.com) at QMUL on 16 July 2015.

(A few of the slides are adapted from slides by my coauthors Dani Bassett and Karen Daniels.)

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Networks in Space: Granular Force Networks and Beyond

  1. 1. Networks in Space: Granular Force Networks and Beyond MASON A. PORTER (@MASONPORTER) MATHEMATICAL INSTITUTE UNIVERSITY OF OXFORD D. S. Bassett, E. T. Owens, K. E. Daniels, and MAP [2012], Physical Review E, Vol. 86, 041306 D. S. Bassett, E. T. Owens, MAP, M. L. Manning, & K. E. Daniels [2015], Soft Matter, Vol. 11, No. 14, 2731–2744 And beyond…
  2. 2. SEE (space, space, space, space, space…) SoftMatter www.softmatter.org ISSN 1744-683X PAPER Volume 11 Number 14 14 April 2015 Pages 2709–2896D. S. Bassett et al. [2015], Soft Matter, Vol. 11, No. 14, 2731–2744
  3. 3. SEE (space, space, space, space, space…)
  4. 4. Marc Barthelemy’s Review Article on Spatial Networks •  M. Barthelemy, “Spatial Networks”, Physics Reports, Vol. 499, 1–101 (2011)
  5. 5. Force Distribution Small number of particles carry greater than average force Force distribution in simulated granular packings with different number of grains Radjai et al. PRL 1996
  6. 6. Sound Propagation Liu and Nagel J. Phys. Condens. Matter 1993 •  Sound amplitude is very sensitive to small rearrangements in the granular packing. •  Is this sensitivity due to sound propagation along the force network?
  7. 7. Granular Force Networks •  2D, vertical, 1 layer aggregate of photoelastic disks •  Internal stress pattern in compressed packing manifests as network of force chains (panel B) •  Force network is a weighted graph in which an edge between 2 particles (nodes) exists if the two particles are in contact with each other; the forces give the weights •  Goal: Use a network-science perspective to try to understand the structure of these force chains
  8. 8. Experimental Setup •  Excite acoustic waves from the bottom by sending pulses of 750 Hz waves •  Bidisperse ensemble of disks –  Diameters: 9 mm, 11 mm –  Density ≈ 1.06 g/cm3 –  Viscoelastic, so Young’s modulus is frequency-dependent •  E(0) ≈ 4 MPa, E(750 Hz) ≈ 50–100 MPa •  Particles in contact = distance between particle centers is less than 1.05 times sum of particle radii (overcounts) •  17 experimental runs •  Details: E. T. Owens & K. E. Daniels, EPL, Vol. 94, 54005 (2011)
  9. 9. Inter-Particle Force •  Experimentally measure force law to be •  This implies that contact area scales as 5/4  
  10. 10. Measuring Sound 750 Hz input signal detected via: (1) Change in particle brightness (ΔI) (2) Piezoelectric (V) [voltage increases linearly with change in stress; flat frequency response]
  11. 11. Network Communities •  Communities = Cohesive groups or modules –  In statistical physics, one tries to derive macroscale and mesoscale insights from microscale information •  Community structure is both modular and hierarchical •  Communities have larger density of internal ties relative to some null model for what ties are present at random –  Modularity
  12. 12. Detecting Communities •  Survey article: MAP, J.-P. Onnela, & P. J. Mucha [2009], Notices of the American Mathematical Society, Vol. 56, No. 9, 1082–1097, 1164–1166 •  Review article: S. Fortunato [2010], Physics Reports, Vol. 486, 75–174
  13. 13. Puck  Rombach  
  14. 14. Optimizing Modularity •  Minimize: –  Potts Hamiltonian •  σi = community assignment (spin state) of node i •  Jij > 0 è “ferromagnetic” interaction between i & j –  è nodes i and j try to be in the same state •  Jij < 0 è “antiferromagnetic” interaction between i & j –  è nodes i and j try to be in different states •  Modularity Optimization: –  Aij = adjacency matrix –  W = (1/2)ΣijAij = sum of all edge weights –  Pij = prob(i connected to j) in null model •  Newman–Girvan: pij = kikj/(2W), where ki = ΣjAij = total edge weight of node i •  “Resolution parameter”: use ϒ*pij
  15. 15. Sound Propagation in Granular Force Networks •  D. S. Bassett, E. T. Owens, K. E. Daniels, and MAP [2012], Physical Review E, Vol. 86, 041306 •  2D granular medium of photoelastic disks •  Two networks –  Underlying topology (unweighted) –  Forces (weighted) •  Both types of networks are needed for characterizing sound propagation
  16. 16. Some Network Diagnostics •  We used 21 diagnostics for the contact (unweighted) networks and 8 diagnostics for the force (weighted) networks •  Contact networks: number of nodes, number of edges, global efficiency, geodesic node betweenness, geodesic edge betweenness, random-walk node betweenness, eigenvector centrality, closeness centrality, subgraph centrality, communicability, clustering coefficient, local efficiency, modularity (optimized using 2 different algorithms), hierarchy, synchronizability, degree assortativity, robustness to targeted and random attacks, Rent exponent, mean connection distance •  Force networks: strength, diversity, path length, geodesic node betweenness, geodesic edge betweenness, clustering coefficient, transitivity, optimized modularity
  17. 17. •  Global efficiency: –  dij w = length of weighted shortest path between nodes i and j •  Geodesic node betweenness: –  Number of shortest paths through node i divided by number of shortest paths •  Weighted clustering coefficient: Some Diagnostics [2]. To subsequently calculate a global weighted clustering coe cient, we computed Cw = 1 N P i Cw(i). B. Weighted Transitivity For weighted networks, we also calculated a di↵erent type of global clustering coe cient. We again let ki be the weighted degree of node i. We also computed Ncyc3 (i) as above, and we then calculated the global weighted transitivity using the formula Tw = P i Ncyc3 (i) P i ki(ki 1) (4) for all nodes i that are part of at least one 3-cycle. For any node i that is not part of any 3-cycle, we set Tw = 0. The code (transitivity_wu.m) that we used for this computation comes from the Brain Connectivity Toolbox [2]. II. TYPOGRAPHICAL ERRORS Our formula (A2) for the geodesic node betweenness centrality of node i in an unweighted network should read Bi = X j,m2G j,m(i) j,m , (5) where all three nodes (j, m, and i) must be di↵erent from each other, j,m is the number of geodesic paths between nodes j and m, and j,m(i) is the number of geodesic paths between j and m that traverse node i. Our computations of node betweenness for unweighted networks in Ref. [1] used equation (5), but the equation in the paper makes it look like we were summing over the nodes. That is not the case, as there is a value of geodesic node betweenness for each node i. The same change corrects equation (A21) for geodesic node betweenness centrality in a weighted network, and again all computations in Ref. [1] used the correct equation. Our formula (A3) for the geodesic edge betweenness centrality of edge (j, m) that connects nodes j and m in an unweighted network should read Be(j, m) = X i,k i,k(j, m) j,m , (6) where j,m is again the number of geodesic paths between nodes j and m, and i,k(j, m) is the number of geodesic paths between i and k that traverse the edge that connects nodes j and m. That is, we used the same type of normalization as in our computation of geodesic node betweenness centrality. The same comment about normalization applies to equation (A22) for geodesic edge betweenness centrality in a weighted network. map: is there a specific BCT .m file to cite for edge betweenness? make sure that we don’t induce any further issues and that we only fix things; need to make notation doesn’t conflict with anything; I don’t like the notation part of the ”solution” abo with the choice of letter for the nodes as well We then calculated a local weighted clustering coe ci Cw(i) = Ncyc3 (i) ki(ki 1) for all nodes i that are part of at least one 3-cycle. For any node i that is not part of any 3-cycle, we set C The code (clustering_coef_wu.m) that we used for this computation comes from the Brain Connectivit [2]. To subsequently calculate a global weighted clustering coe cient, we computed Cw = 1 N P i Cw(i).
  18. 18. Spatial Embeddesness Induces Correlations in Network Diagnostics
  19. 19. Random Geometric Graphs (RGGs) •  Useful to compare properties of real networks to a random graph ensemble as a “null model”, but we need one that is embedded in 2D Euclidean space •  N nodes (same as number of particles) distributed in 2D uniformly at random. There is an edge between any pair of nodes within some distance 2r. We choose r so that number of edges is the same as in the real system.
  20. 20. Contact Networks versus Random Geometric Graphs (very different)
  21. 21. A Better Model: RGG With A Sprinkling of Physics •  J. Setford, MPhys thesis, University of Oxford, 2014 –  Available at http://people.maths.ox.ac.uk/porterm/research/setford- final.pdf –  Paper in preparation •  Start with an RGG, but now relax the centers of the particles –  E.g. using a force law f = δβ for some amount of time to move particles apart •  δ = compression between two particles from initial placement •  β depends on e.g. particle geometry; we used β = 5/4 based on experimental measurements –  Previous network diagnostics for modified RGG and experimental packings now (mostly) match very well •  Still some mysteries: e.g. degree assortativity is twice as large for the force- modified RGG than for the experiments
  22. 22. Geographical Community Structure •  Different resolution-parameter values give communities at different scales. •  Averaged results over 100 realizations (spectral optimization of modularity)
  23. 23. Community Structure Constrains Sound Propagation
  24. 24. Ø  D. S. Bassett, E. T. Owens, MAP, M. L. Manning, & K. E. Daniels [2015], Soft Matter,Vol. 11, No. 14, 2731–2744 Ø  Use a null model that includes more information Ø  Fix topology (i.e. connectivity) but scramble geometry (i.e. edge weights) ›  Wij = weighted adjacency- matrix element = force network ›  Aij = binary adjacency- matrix element = contact network Ø  Communities obtained from optimization of modularity match well with empirical granular force networks in both laboratory and computational experiments by maximizing a quality function known as modulari respect to the assignment of particles to sets called “c nities.” Modularity Q is dened as Q ¼ X i;j  Wij À gPij à d À ci; cj Á ; where node i is assigned to community ci, node j is assig community cj, the Kronecker delta d(ci, cj) ¼ 1 if and only cj, the quantity g is a resolution parameter, and Pij expected weight of the edge that connects node i and j under a specied null model. One can use the maximum value of modularity to q the quality of a partition of a force network into sets of p that are more densely interconnected by strong force expected under a given null model. The resolution par g provides a means of probing the organization of inter-p forces across a range of spatial resolutions. To provide intuition, we note that a perfectly hexagonal packing wit uniform forces should still possess a single commun small values of g and should consist of a collection of particle (i.e., singleton) communities for large values o intermediate values of g, we expect maximizing modul yield a roughly homogeneous assignment of particle communities of some size (i.e., number of particles) bet and the total number of particles. (The exact size depe the value of g.) The strongly inhomogeneous com assignments that we observe in the laboratory and num Publishedon23February2015.DownloadedbyCaliforniaInstitu
  25. 25. Force-Chain Structure and Pressure “optimal” valu tion of pressur ¼ 0.7 for 2.7  much smaller s both small and observation is of the commun of the force-ch The resolu structures in a force chains identies bran chains from f pressures, we resolution par value that app to the commu our calculatio terms of their Paper tuteofTechnologyon21/04/201522:40:58. sc ¼ X i;j˛C  Wij À grBij à ; (4) physical distance (see are close together in s bind them; the prese
  26. 26. aximum of guniform at g ¼ 0.9 (for g ˛ {0.1, 0.3,., essure packings (5.9 Â 10À3 E) and at g ¼ 1.5 for ackings (2.7 Â 10À4 E). In the numerical packings, aximum of guniform at g ¼ 1.1 for all pressures. In our observations in the main text from employ- eighted systemic gap factor g, we nd that the of g is larger when we instead employ guniform 0 to Fig. 5 and 7). We also observe that the curves c gap factor versus resolution parameter exhibit n for the uniformly-weighted gap factor than for ed gap factor. of the resolution parameter tion in the maximum of guniform over packings makes it difficult to choose an optimal resolu- value. We choose to take gopt ¼ 1.1 because (1) it o the maximum of guniform in the numerical 2) it corresponds to the mean of the maximum of aboratory packings. To facilitate the comparison ues of g from the two weighting schemes, we g as ^g and we denote gopt for guniform as ^guniform. rm ¼ 1.1 differs from (and is larger than) ^g ¼ 0.9. ucture at the optimal value of the resolution Fig. 11 In both (A) (frictional) laboratory and (B) (frictionless) numerical packings, we identify larger and more branched force chains at the Soft Matter View Article Online Increasing Pressure •  For larger pressures, we obtain larger and more branched force chains in both the (frictional) laboratory packings described earlier and in frictionless numerical packings
  27. 27. Some Other Approaches for Studying Granular Force Networks (selected) •  Tools from computational homology (e.g. persistent homology): See papers by Konstantin Mischaikow, Lou Kondic, Bob Behringer, and collaborators •  Spatial patterns in breaking of edges in granular contact networks: See papers by Wolfgang Losert, Michelle Girvan, and collaborators •  Multiple network-based approaches by Antoinette Tordesillas and collaborators •  Other groups as well •  (A survey article is now needed.)
  28. 28. And Beyond…•  We have also used spatial null models for studies of brain networks, disease-occurrence networks, and international migration networks •  Brain networks: similar type as the one I discussed today (see D. S. Bassett et al., Chaos, Vol. 23, No. 1, 013142, 2013) •  Diseases: comparison of null models based on gravity and radiation null models for human mobility –  M. Sarzynskya et al., arXiv: 1407.6297 –  It’s complicated. •  Migration: similar null models as with the disease case –  V. Danchev & MAP, in preparation 22 of 45 (a) (b) (c) FIG. 8. Visualization of the three different spatial partitions of Peru’s provinces on a map. (a) Broad climate partition into coast (yellow), mountains (brown), and jungle (green); (b) detailed climate partition, in which we start with the broad partition and then further divide the coast and mountains into northern coast, central coast, southern coast, northern mountains, central mountains, and southern mountains; and (c) the administrative partition of Peru. We obtained province
  29. 29. Conclusions •  Studying community structure allows one to examine mesoscale structures in granular packings. –  Provides diagnostics for structure of granular force chains •  General ideas for spatial networks –  Comparison of properties to null models like random geometric graphs (and suitable modifications thereof) •  Also see work that Carl Dettmann is doing with modifications of RGGs –  Effect of physical constraints on the correlations of network diagnostics –  Development of null models that incorporate physical information •  Funding: EPSRC, McDonnell Foundation, European Commission (“PLEXMATH”)
  30. 30. Future Work: Granular Networks and Beyond •  Granular networks –  Use network diagnostics in time-dependent situations to determine “soft spots” (mini-earthquakes) •  Note: I probably shouldn’t do this in Italy. –  Longer term: In addition to the idealized (2D) situations, get more seriously into geophysics. •  Example: Use community detection to find fault curves and compare with ones done manually by the experts •  Spatial networks more generally –  Development of spatial null models –  How do spatial effects influence network structure, induce correlations between different structural measures, etc.?
  31. 31. Summer School on Computational Algebraic Topology •  https://people.maths.ox.ac.uk/tillmann/CAT-SCHOOL.html •  Some funding available for students and postdocs

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