Lecture 18
More than ordinary space
-a jump into hyperspaces-
1
Characterize the properties of complex systems
by mapping the structure into graphs
in multidimensional spaces, both Euclidean and non-Euclidean.
2
A way to characterize a large, complex, structure is by
disassembling it into smaller pieces.
Space can always be divided into pieces: “tiles”
The study of the structure can be mapped into the
construction of tilings.
3
Tiling are associated to graphs
The study of complex structures can be mapped into the study
of complex graphs, and vice versa.
4
Crystalline systems: division into parts
Elementary cell
Order is simple
The description of ordered
tilings require a little information
about the tiles: the rotational
C4
group combined with the
translational symmetry.
A certain amount of defects can be
introduced and classified.
C6
What can we do if we do not have any symmetry and order?
How can we divide space, in general?
Disorder is complex
5
We can use “Froths”
Space-filling cellular partitions
Topologically stable
Wide-spread in nature
minimally-connected edges
dual of triangulations6
Froths are everywhere and they are dual structures of packings
Voronoï
Delaunay
Indeed from any set of points P one can build a Delaunay (or Delone) triangulation
such that no point in P is inside the circumcircle of any triangle. The dual of the
Delaunay triangulation is the Voronoï cellular partition where the interior of each cell
contains points closer to a given point pi ! P than to any other in P.
This procedure can be generalized to any dimension.
The Voronoï cell is the “region of influence” around each point
7
We are interested in Froths or their duals: tiling with triangular
tiles - triangulations
8
In arbitrary dimensions we call them “simplicial decompositions”
Euclidean tilings
The sum over the angles incident
on a vertex is 2" = 360o
When we embed a graph on a surface, a special kind of cycles
emerges: the faces or the “tiles” are irreducible cycles whose
interiors are a part of the surface.
9
Non-Euclidean tilings
The sum over the
angles incident on a
vertex is not 360o
deficit angle
Smaller than 360o:
excess angle
Elliptic
10
Non-Euclidean tilings
The sum over the
angles incident on a
vertex is not 360o
excess angle
deficit angle
Larger than 360o:
Hyperbolic
11
Euler Formula: - elliptic space
Tetrahedron
V=4
E=6
F=4
Cube
V=8
E = 12
F=6
Dodecahedron
V = 20
E = 30
F = 12
Octahedron
V=6
E = 12
F=8
Icosahedron
V = 12
E = 30
F = 20
Fullerene C540
V = 540
E = 810
F = 272
V-E+F=2
12
Euler Formula: - general
V - E + F = # = 2(1-g)
13
The genus of a connected, orientable surface is the maximum
number of cuttings along closed simple curves which can be
operated without disconnecting the surface.
It is equal to the number of handles.
g=0
g=1
g=2
sphere
0 non-contractible loops
1 cut
0 handle
torus
2 non-contractible loops
2 cuts
1 handle
double torus
4 non-contractible loops
3 cuts
2 handles
14
Topological charge - triangulations
The Euler formula: a topological constraint
Vertices
Faces (triangles)
Euler Poincaré characteristic = 2(1-g)
V - E +F = !
V
"k
Edges
i
= k V = 2E
i=1
3F = 2E
V
1
" = $ (6 # k i )
6 i=1!
!
k=6
<q> > 0
<q> < 0
qi = 6 - ki
k=
5
5
8
Topological charge
k=
k=
!
positive curvature, elliptic tilings
negative curvature, hyperbolic tilings
15
The tiles are flat… where does the curvature go?
16
Gauss-Bonnet formula
For any compact, boundaryless two-dimensional Riemannian
manifold, the integral of the Gaussian curvature over the entire
manifold with respect to area is 2" times the Euler characteristic
of the manifold:
V
" KdA =2#$
Triangulations
M
1
" = $ (6 # k i )
6 i=1
Gaussian curvature K = 1/r2
Q!
" KdA = # 3
M
17
!
The curvature accumulates at the vertices
1 V
(6 " k i ) = $ = 2
#
6 i=1
V
# (6 " k ) = 12
i
i=1
!
12 pentagons (5 coordinated vertices, topological charge q = +1) are necessary to close the ball
!
Positive topological charge => positive curvature
"q
i
= 12(1# g)
i
vertices with degree 7
Negative topological charge
=> negative
! curvature
18
Topological 'Gauss' theorem
–
Q = 6 + V – V+
The information about the charge inside a cluster is on the neat
“flux of edges” through its border
T. Aste, D. Boosé and N. Rivier, "From one cell to the whole froth: a dynamical map'', Phys. Rev. E 53 (1996), p.6181-91.
N. Rivier, T. Aste, Phil. Trans. R. Soc. Lond. A 354 (1996) 2055
19
Regge Calculus and Quantum Gravity
Simplicial approximations of spacetimes which are solutions to
the Einstein field equation. Every Lorentzian manifold admits a
triangulation into simplices and the spacetime curvature can
be expressed in terms of deficit angles.
C. W. Misner, K.S. Thorne & J.A. Wheeler “Gravitation” (W. H. Freeman San Francisco 1973)
T. Regge "General relativity without coordinates". Nuovo Cim. 19 (1961) 558-571.
20
Topological evolution: Alexander Moves
+1
-1
-1
T1
-1
-1
q=3
T2
-1
+1
+1
T1
-1
-1
-1
-1
+1
Topological Charge Exchange
T2
+3
-1
cell-division and coalescence
21
J. W. Alexander, “The combinatorial theory of complexes” Ann. Math. 31 (1930) 292.
!
The Random Froth
Number of triangulations with k external and r internal edges
"n,m
3(m + 2)!(m # 1)! m (4n + 3m # j + 1)!(m + j + 2)(m # 3j)
=
(3n + 3m + 3)! $
j =0 j!( j + 1)!(m # j)!(m # j + 2)!(n # j # 1)!
m= k" 3
n=
r"m
2
The number of triangulations increases exponentially with the
N
256
$
&
number of cells
"#
% 27 '
!
Degree distribution a triangulation made by T1 at random
(k " 2)(2k " 2)! # 3 & k
p(k) = 16
% (
k!(k "1)! $ 16 '
k =6
2
k " k
2
=#
k
21
p(k)(k " k ) =
2
2
!
W.T. Tutte, “A census of planar triangulations” Can. J. Math. 14 (1962) 21.
0.25
0.2
0.15
0.1
0.05
0
3
4
5
6
7
8
9
E. Brézin, C. Itzykson, G. Parisi and B. Zuber, “Planar diagrams” Comm. Math. Phys. 59 (1978) 35.
10
22
D.V. Boulatov, V. A. Kazakov, I. Kostov and A. A. Migdal, “Analytical and numerical study of model of dynamically triangulated random surfaces”, Nucl. Phys. B 275 (1986) 641.
!
C. Godrèche, I. Kostov, and I. Yekutieli, “Topological correlations in cellular structures and planar graph theory” , Phys. Rev. Lett. 69, 2674-2677 (1992)
Are hyperbolic spaces necessary?
Do we really need hyperbolic spaces?
23
Are hyperbolic spaces sufficient?
The genus of a graph is the minimal surface genus such that
the graph can be drawn on such surface without edge
crossings.
Planar graph g=0
K3
K4
K5
K3,3
Kuratowski’s theorem
A finite graph is planar if and only if it does not
contain a subgraph that is an expansion of K5 or K3,3
24
The embedding of KV is possible on an orientable surface Sg of genus
%(V $ 3)(V $ 4) '
g " g =&
(
&
(
12
#
!
Four color suffice to decorate the regions in a plane
avoiding that two neighboring regions (separate by an
edge) have the same color. The four color theorem was the
first major theorem proved exhaustively with a computer.
Some mathematicians still question this kind of proof that
cannot be verified by humans.
Interestingly its generalization to surfaces of genus larger
than 1 (the map color theorem) was proved analytically by
Ringel and Youngs several years before the four color
theorem proof.
# 7 + 1+ 48g %
# of colors " #
%
2
$
&
g = 0 : 4 colors are enough for on the sphere
g =1 : 7 colors are needed for maps on the torus
G. Ringel, Map Color Theorem, Springer-Verlag, Berlin, (1974)
! cap. 4
P. J. Gilbin, Graphs, Surfaces and Homology, Chapman and Hall, 2nd edition (1981)
G. Ringel and J. W. T. Youngs, Proc. Nat. Acad. Sci. USA 60 (1968) 438-445.
25
How can we build complex structures?
26
Pant decomposition
Any surface (g>1) can be decomposed into “pants”
Pants can be “unzipped”
into tilings (and associated
graphs) which can be
represented on the
universal cover.
{6,4}
{4,6}
{7,3}
{3,7}
27
{n,k} Schläfli symbols
g >> 1
degree distribution
g=0
degree distribution N(k)
How a network (freely) develops on high genus surfaces?
N(k)~exp(-$ k)
degree
k
g = N = 5000
degree
N(k)~ k-%
Scale free
28
How “small” is this universe?
Average distance
Small World
low disorder
<j>~Log(N)
high disorder
g = N/3
Log(N)
Average distance
Ultra-Small World
low disorder
high disorder
g = N/3
<j>~Log(Log(N))
29
Log(Log((N))
Finding communities
In the neighboring of hubs there
is a great concentration of
negative charge.
Conversely, communities are like
‘gestating baby universes’ which
are associated to regions with
small (positive) charges.
30
!
Application: minimal networks
1) generate a network on a surface with given genus (pants
decomposition);
2) make the network evolving by means of T1 moves;
3) search for configurations which minimize a ‘cost’ function;
cost = " ai, j d = 2" ai, j (1# c i, j )
2
i, j
i, j
perform a Monte
Carlo dynamics to
maximize the total
correlation
expressed by the
network and search
for the network
which contains the
maximum
information at a
given genus
!
i, j
"a
c
NYSE 100
i, j i, j
i, j
Increasing the genus to
increase the information
content
g
31
Lecture 1 : Science of Complex Systems
Lecture 9 : Network structure and properties
Introduction to complex systems.
When structure fits purpose.
What makes a system complex?
Transport through networks and allometric scaling.
Simple systems with unexpected emerging behaviours.
Lecture 10 : Under Attack
What prediction means?
Epidemic spreading.
What is a complex system?
Percolation.
Vaccination strategies and network resilience.
Lecture 2 : Probabilities and Improbabilities
Basic introduction to probability theory and
Lecture 11 : Group project presentation and discussion
central limit theorem.
Lecture 12 : Group project presentation and discussion
Normal distribution and other statistics.
Lecture 13 : Drunk man walks
Lecture 3 : Fat tails
Random walks and Levy flights: steps into the fractal world.
Extension of the central limit theorem.
Beyond fractals: Multifractals.
Levy stable distributions.
Multifractality at work: scaling properties and exponents.
Where fat tails come from?
Lecture 14 : Science of Risk
Lecture 4 : When things do not average to the mean
Why do earthquakes and market crashes follow similar laws?
Strange things happening with power laws distributions.
Cascade events.
Why do I always wait the bus longer than average?
Fat tails: heavy losses and big gains in financial markets.
Stationary, iid processes and beyond.
Correlations and Implications
Lecture 5 : Statistical Mechanics approaches
Searching for clues: information base networks form correlations.
Why things are in the way they are?
The future as we know it.
Different levels of abstraction: statistical mechanics
Lecture 15 : Complex Matter
approaches.
The pursuit of perfect packing.
Lecture 6 : Maximum entropy, Minimal constraints
Apollonius and concrete.
Entropy maximization principles.
Evolution without a law: constrain satisfaction dynamics.
Lecture 16 : Individual projects discussion and planning
Froths and packings.
Lecture 17 : Insights into disorder
Lecture 7 : Networks everywhere
Granular materials: solid, liquid, gas…and other ‘esoteric’ states.
Introduction to network theory.
Why is disorder is different from randomness?
Six degrees of separation: small world effect.
What is happening at 64%?
Network structures: clusters, communities, motifs.
Isostaticity and mechanical stability.
Lecture 8 : Building Networks
What is happening at 55%?
Random graphs
Lecture 18 : More than ordinary space
Real world networks are not random.
A jump into hyperspaces.
Building non-random, non-regular Networks
Hyperbolic tilings.
Preferential attachment and other network models.
32
Hyperbolic packings.
Scale-free networks.
Wormholes and small worlds.
Small world networks.
The End