1. Condensation phenomena in stochastic systems Bath Quantum Statistics and Condensation Transitions in Networks and Simplicial Complexes Ginestra Bianconi School of Mathematical Sciences, Queen Mary University of London, UK

2. Between randomness and order ENCODING INFORMATION IN THEIR STRUCTURE
Complex networks
Between randomness and order
LATTICES COMPLEX NETWORKS RANDOM GRAPHS
Scale free networks
Small world
With communities
ENCODING INFORMATION IN THEIR STRUCTURE
Totally random
Poisson degree
distribution
Regular networks
Symmetric

3. Outlook
Quantum statistics and condensation transition in complex networks
Bose-Einstein condensation in complex networks
Fermi-Dirac statistics in growing trees
Condensation transitions in weighted networks
Quantum statistics and condensation transition in network geometry (growing simplicial complexes)
Network geometry with flavor
Complex Quantum Network Manifolds

4. BA scale-free model P(k) ~k-3
(1) GROWTH : At every timestep we add a new node with m edges (connected to the nodes already present in the system).
(2) PREFERENTIAL ATTACHMENT : The probability Π(ki) that a new node will be connected to node i depends on the connectivity ki of that node
P(k) ~k-3
Barabási et al. Science (1999)

5. Growth with uniform attachment
(1) GROWTH : At every timestep we add a new node with m links connected to the nodes already present in the system).
(2) UNIFORM ATTACHMENT : The probability Πi that a new node will be connected to node i is uniform
Exponential
Barabási & Albert, Physica A (1999)

6. Intrinsic properties of the nodes
Not all the nodes are the same!
Let assign to each node
an energy  from a
g(e) distribution and
a fitness h=e-be 6 1 3 2 5 4

7. The Bianconi-Barabasi model
Growth:
At each time a new node and m links are added to the network.
To each node i we assign a energy ei from a g(e) distribution
Preferential attachment towards
high degree high fitness (low energy) nodes:
Each node connects to the rest of the network by m links attached preferentially to well connected, low energy nodes.
G. Bianconi, A.-L. Barabási 2001 6 1 3 2 5 4

8. Master-equation for the degree distribution
The master equation for the degree distribution can be solved using a self-consistent argument

9. Self-consistent solution of the Bianconi Barabasi Model
The degree distribution is given by
As long as the self-consistent equation has a solution for the chemical potential m

10. Properties of Bianconi-Barabasi model
Power-law degree distribution
Fit-get-rich mechanism

11. Latecomers with high fitness become hubs

12. Mapping to a Bose gas 6 1 3 2 5 4 
We can map the fitness model to a Bose gas with
density of states g( );
specific volume v=1;
temperature T=1/.
In this mapping,
each node of energy  corresponds to an energy level of the Bose gas
while each link pointing to a node of energy , corresponds to an occupation of that energy level. 1 3 2 5 4
Network 
Energy diagram
G. Bianconi, A.-L. Barabási 2001

13. Scale-Free Bose-Einstein Fit-get-rich Phase condensate Phase
Bose-Einstein condensation
in complex networks
Scale-Free Bose-Einstein Fit-get-rich Phase condensate Phase
G. Bianconi, A.-L. Barabási 2001

14. Properties of the condensate phase
The self-consistent assumption does not work anymore. The chemical potential m is no longer well defined.The finite size estimation of m becomes positive.

15. Properties of the condensate state
Change of scaling behavior of the maximum degree

16. Time-dependent Properties of the condensate phase
The sequence effective chemical potential is not self-averaging (specifically it is above zero)

17. Time-dependent properties of the condensate phase
The sequence of records with lowest energy dominate the dynamics.

18. The Complex Growing Cayley tree model
Growth:
At each time attach a old node with ri=1 to m links are added to the network and then we set ri=0.
To each node i we assign a energy ei from a g(e) distribution
Attachment towards high energy nodes:
The node i growing at time t is chosen with probability
G. Bianconi, PRE 2002 6 4 5 3 2 7 1

19. Quantum statistics in growing networks
Scale-free network Complex Cayley tree
Bianconi-Barabasi model (2001) Bianconi (2002)
Bose Einstein statistics Fermi statistics

20. MF Equations for the growing scale-free network and the complex growing Cayley tree network
Bianconi-Barabasi model
(ki here indicates the average degree of node i)
Complex Growing
Cayley tree
( ri here indicates the average ri of node i)

21. MF Solution of the Bianconi-Barabasi model
The average degree of node increases in time as a power-law with an exponent depending on its energy,
and a self-consistent constant mB
The self consistent constant mB is determined by the same equation fixing the chemical potential in a Bose gas!

22. MF Solution of the complex growing Cayley model
The average r of node (determining the probability that a node is at the interface) decreases in time as a power-law with an exponent depending on its energy,
and a self-consistent constant mF
The self consistent constant mF is determined by the same equation fixing the chemical potential in a Fermi gas!

23. The energy of the nodes in the bulk of the complex growing Cayley tree follows the Fermi distribution 2

24. Fitness model with reinforcement of the links
G. Bianconi, EPL 2005
A fitness xij is assigned to each link
At each time m’ links chosen are reinforced. They are chosen preferentially in within high fitness and high weights links with probability h6 h1
x12 h3 h2 h5 h4

25. Fitness of links and fitness of nodes
The fitness of the links can be dependent on the fitness of the nodes.
In coauthorship networks more productive authors would collaborate more with more productive authors
In Airport networks connections between hub airports would increase faster their traffic
The considered cases

26. Mean field treatment
The ration m’/m fixes the constant C’ that determines the speed at which links are reinforced
The equation which fixes C’ is given by
In particular C’ is a
decreasing function
of m’/m

27. Condensation of the links
When the self-consistent equation of C’ does not admit a solution we observe a condensation of the weights of a finite fraction of the links

28. Networks
Pairwise interactions define networks

29. Simplicial complexes
Interactions between two or more nodes define simplicial complexes Simplicial complexes are not only formed by nodes and links but also by triangles, tetrahedra etc.

30. Brain data as simplicial complexes
Giusti et al 2016

31. Protein complexes as simplicies
Wan et al. Nature 2015

32. Collaboration networks as simplicial complexes
Actor collaboration networks
Nodes: Actors
Simplicies: Co-actors of a movie
Scientific collaboration networks
Nodes: Scientists
Simplicies: Co-authors of a paper

33. Network Geometry aims at unveiling the hidden metric space of networks
Boguna, Krioukov, Claffy Nature Physics (2008)

34. Hyperbolic geometry N ~ eD S< p Number of nodes Sum of the angles
of a triangle S
S< p
Number of nodes
N ~ eD

35. Hyperbolic networks
A large variety of networks display an hyperbolic network geometry strongly affecting their navigability
Boguna et al. Nature Communication (2010)

36. Emergent geometry the hyperbolic geometry is an emergent property
It is possible that
the hyperbolic geometry is an
emergent property
of the network evolution
which follows dynamical rules
that make no use of the hidden geometry?

37. Simplicial complexes and network geometry
Simplicial complexes are ideal to describe
network geometry
As such they have been widely used in quantum gravity
(spin foams, tensor networks, causal dynamical triangulations)

38. Growing networks describe the emergence of complexity
Science 1999

39. Would growing simplicial complexes describe the emergence of geometry?

40. Emergent Network Geometry
The model describes the underlying structure of a simplicial complex constructed by gluing together triangles by a non-equilibrium dynamics.
Every link is incident to
at most m triangles with m>1.

41. Saturated and Unsaturated links
m=2
if the link is unsaturated, i.e. less than m triangles are incident on it
if the link is saturated, i.e. the number of incident triangles is given by m

42. We choose a unsaturated link and we glue a new triangle the link
Process (a)
We choose a unsaturated link and
we glue a new triangle the link
Growing
Simplicial
Complex
Growing Geometrical Network

43. Process (b) Growing Geometrical Network Growing Simplicial Complex
We choose two adjacent unsaturated links and we add the link between the nodes at distance 2 and all triangles that this link closes as long that this is allowed.
Growing Geometrical Network
Growing
Simplicial
Complex

44. The model Starting from an initial triangle, At each time
process (a) takes place and
process (b) takes place with probability p<1
Z. Wu, G. Menichetti, C. Rahmede, G. Bianconi, Scientific Reports 5, (2015)

45. Discrete Manifolds
A discrete manifold of dimension d=2 is a simplicial complex formed by triangles such that every link is incident to at most two triangles. Exponential degree distribution

46. Scale-free networks
For a scale-free, small-world network with high clustering coefficient and significant community structure is generated.

47. The d-dimensional simplicial complexes
In dimension d the growing simplicial complex built by gluing simplices of dimension d along their (d-1)-face In d=1 the simplices are links In d=2 the simplices are triangles In d=3 they are tetrahedra.

48. Generalized degrees The generalized degree kd,d(a) of a d-face a
in a d-dimensional simplicial complex is given by the number of
d-dimensional simplices incident to the d-face a. 1 2 2 5 4 5 3 3 6
Number of triangles incident
to the nodes k2,0
to the links k2,1 6

49. The incidence number
To each (d-1)-face a we associate the incidence number given by the number of d-dimensional simplices incident to the face minus one If na takes only values na=0,1 the simplicial complex is a discrete manifold.

50. Network Geometry with Flavor
We start with a d-dimensional simplex,
At each time we choose a (d-1)-face
with probability
and we glue to it a new d-dimensional simplex
The flavor s=-1,0,1
Growing
Simplicial
Complex
Network
Geometry with Flavor

51. Flavor s=-1,0,1 and attachment probability
s=-1 Manifold na=0,1
s=0 Uniform attachment na=0,1,2,3,4…
s=1 Preferential attachment na=0,1,2,3,4…

52. Dimension d=1 Manifold Uniform Preferential attachment attachment
s= s= s=1
Chain
BA model
Chain Exponential Scale-free

53. Dimension d=2 Manifold Uniform Preferential attachment attachment
s= s= s=1
Chain
Exponential Scale-free Scale-free

54. Dimension d=3 Manifold Uniform Preferential attachment attachment
s= s= s=1
Chain
Scale-free Scale-free Scale-free

55. Effective preferential attachment emerging in d=3 dimensions
Node i has generalized degree Node i has generalized degree 4
Node i is incident to 5 unsaturated faces Node i is incident to 6 unsaturated faces

56. Random Apollonian networks are the planar projections of NGF manifold in d=3
NGF Manifold d=3 s=-1 Apollonian random network Planar projection of the NGF i

57. Degree distribution of NGF
The NGF with s+d=0, i.e. with (s,d)=(-1,1) are chains
For s+d=1, i.e. (s,d)=(-1,2) and (0,1) we have
For s+d>1 we have

58. Generalized degree distribution for Network Geometry with Flavor (NGF)
For s=1 the NGF are always scale-free
For s=0 the NGF are scale-free for d>1
For s=-1 the NGF are scale-free for d>2
G. Bianconi, C. Rahmede, Scientific Reports (2015);PRE (2016).

59. Generalized degree distribution case b=0, and s=-1
If d=d-1 the generalized degrees follow a binomial distribution
If d=d-2 the generalized degree follow a exponential distribution
If d<d-2 the generalized degree follow a power-law distribution

60. Generalized degree distributions of NGF
Theory versus simulation in NGF s=-1,0,1 and d=3

61. Emergent hyperbolic geometry
The emergent hidden geometry is the hyperbolic Hd space
Here all the links have equal length
d=2

62. Emergent hyperbolic geometry
d=3

63. The pseudo-fractal geometry of the surface of the 3d manifold (random Apollonian network)

64. Energies of the nodes Not all the nodes are the same!
Let assign to each node i
an energy  from a
g(e) distribution 6 5 1 4 3 2 5

65. Energy of the d-faces 1+e2 1+e2+e3
Every d-face a is associated to an energy
which is the sum of the energy of the nodes
belonging to a
For example, in d=3
the energy of a link is
the energy of a face is 2 1
1+e2 3
1+e2+e3 1 2

66. Fitness of the d-faces The fitness of a d-face a is given by
where b=1/T is the inverse temperature
If b=0 all the nodes have same fitness
If b>>1 small differences in energy have large impact on the fitness of the faces

67. Network Geometry with Flavor
Starting from a d-dimensional simplex
We choose a (d-1)-dimensional face a,
with probability
and glue a new d-dimensional simplex to it.
The parameter b=1/T is the inverse temperature.
If b=0 we are recast into the previous model
The flavor s=-1,0,1

68. Bianconi-Barabasi model or the fitness model Is the NGF d=1 s=1
The average degree of the nodes with energy e
follows the Bose-Einstein statistics
G. Bianconi A.L. Barabasi PRL (2001)

69. Network Geometry with Flavor What is the combined effect
Quantum statistics
and
Network Geometry with Flavor
What is the combined effect
of flavor and dimensionality
on the emergence of
quantum statistics?

70. The average of the generalized degree of the NGF over d-faces of energy e follows
G. Bianconi, C. Rahmede, Scientific Reports (2015);PRE (2016).

71. Manifolds in d=3 In NGF with s=-1 and d=3 also called
Complex Quantum Network Manifolds
the average of the generalized degree follow
the Fermi-Dirac, Boltzmann and Bose-Einstein distribution
respectively for
triangular faces, links and nodes

72. Emergence of quantum statistics in CQNM
The average of the generalized degrees follows either Fermi-Dirac, Boltzmann or Bose-Einstein distributions depending on the dimensionality of the d-face
In d=3 the average of the generalized degree follow
the Fermi-Dirac, Boltzmann and Bose-Einstein distribution
respectively for triangular faces, links and nodes

73. Theory and simulations for NGF in d=3

74. Emergent geometry at high temperature
d=2
b=0.01

75. Emergent geometry at low temperature
d=2
b=5

76. Emergent geometry at high temperature
d=3
b=0.01

77. Emergent geometry at low temperature
d=3
b=5

78. Growing simplicial complexes called Network Geometry with Flavor (NGF)
Conclusions
Growing simplicial complexes called Network Geometry with Flavor (NGF)
display:
small world behavior, finite clustering coefficient, high modularity
strong dependence on the dimensionality
NGF with s=-1 are manifolds and are scale-free for d>2
NGF with s=0 ( uniform attachment) are scale-free for d>1
emergent hyperbolic geometry
NGF at finite temperature (with fitness and energy)
have generalized degrees following the Fermi-Dirac, Boltzmann or Bose-Einstein distribution depending on the dimensionality of the d-face and the flavor s.
NGF can undergo phase transition strongly affecting their hidden hyperbolic geometry.

79. Collaboration Zhihao Wui Ginestra Bianconii Christoph Rahmedei
Giulia Menichettii

80. References EMERGENT COMPLEX NETWORK GEOMETRIES
Z. Wu, G. Menichetti, C. Rahmede, G. Bianconi, Emergent Complex Network Geometry
Scientific Reports 5, (2015)
NETWORK GEOMETRY WITH FLAVOR
G. Bianconi, C. Rahmede Network geometry with flavor: from complexity to quantum geometry
Phys. Rev. E 93, (2016).
MANIFOLDS
G. Bianconi, C. Rahmede, Quantum Complex Network Manifolds in d>2 are scale-free
Scientific Reports 5, (2015)
PERSPECTIVE
G. Bianconi, Interdisciplinary and physics challenges in network theory EPL 111, (2015).
Quantum Complex Network Geometries
G. Bianconi, C. Rahmede, Z. Wu, Quantum Complex Network Geometries: Evolution and Phase Transition Phys. Rev. E 92, (2015)
Equilibrium and configuration model
O.T. Courtney and G. Bianconi, Generalized network structures: the configuration model and the canonical ensemble of simplicial complexes arXiv: (2016).