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.)

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. 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. SEE (space, space, space, space, space…)

4. Marc Barthelemy’s Review Article on
Spatial Networks
• M. Barthelemy, “Spatial Networks”, Physics Reports, Vol.
499, 1–101 (2011)

7. 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

8. 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?

9. 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

10. 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)

11. Inter-Particle Force
• Experimentally measure
force law to be
• This implies that contact area
scales as
5/4

12. 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]

13. 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

14. 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

15. Puck
Rombach

16. 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

17. 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

18. 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

19. • 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).

20. Spatial Embeddesness Induces
Correlations in Network Diagnostics

21. 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.

22. Contact Networks versus Random
Geometric Graphs (very different)

23. 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

24. Geographical Community Structure
• Different resolution-parameter values give communities
at different scales.
• Averaged results over 100 realizations (spectral
optimization of modularity)

25. Community Structure Constrains Sound
Propagation

26. 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
Â
Wij À gPij
Ã
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

27. Force-Chain Structure and Pressure
“optimal” valu
tion of pressur
¼ 0.7 for 2.7 Â
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
Â
Wij À grBij
Ã
; (4) physical distance (see
are close together in s
bind them; the prese

28. 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

29. 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.)

30. 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

31. 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”)

32. 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.?

33. Summer School on Computational
Algebraic Topology
• https://people.maths.ox.ac.uk/tillmann/CAT-SCHOOL.html
• Some funding available for students and postdocs