Condensation phenomena in stochastic systems Bath 2016 Quantum Statistics and Condensation Transitions in Networks and Simplicial Complexes Ginestra Bianconi School of Mathematical Sciences, Queen Mary University of London, UK

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

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

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)

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)

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

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

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

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

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

Latecomers with high fitness become hubs

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

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

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.

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

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

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

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

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

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)

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!

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!

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

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

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

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

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

Networks Pairwise interactions define networks

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.

Brain data as simplicial complexes Giusti et al 2016

Protein complexes as simplicies Wan et al. Nature 2015

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

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

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

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

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?

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)

Growing networks describe the emergence of complexity Science 1999

Would growing simplicial complexes describe the emergence of geometry?

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.

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

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

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

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, 10073 (2015)

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

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

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.

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

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.

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

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…

Dimension d=1 Manifold Uniform Preferential attachment attachment s=-1 s=0 s=1 Chain BA model Chain Exponential Scale-free

Dimension d=2 Manifold Uniform Preferential attachment attachment s=-1 s=0 s=1 Chain Exponential Scale-free Scale-free

Dimension d=3 Manifold Uniform Preferential attachment attachment s=-1 s=0 s=1 Chain Scale-free Scale-free Scale-free

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

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

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

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).

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

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

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

Emergent hyperbolic geometry d=3

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

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

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

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

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

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)

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?

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).

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

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

Theory and simulations for NGF in d=3

Emergent geometry at high temperature d=2 b=0.01

Emergent geometry at low temperature d=2 b=5

Emergent geometry at high temperature d=3 b=0.01

Emergent geometry at low temperature d=3 b=5

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.

Collaboration Zhihao Wui Ginestra Bianconii Christoph Rahmedei Giulia Menichettii

References EMERGENT COMPLEX NETWORK GEOMETRIES Z. Wu, G. Menichetti, C. Rahmede, G. Bianconi, Emergent Complex Network Geometry Scientific Reports 5, 10073 (2015) NETWORK GEOMETRY WITH FLAVOR G. Bianconi, C. Rahmede Network geometry with flavor: from complexity to quantum geometry Phys. Rev. E 93, 032315 (2016). MANIFOLDS G. Bianconi, C. Rahmede, Quantum Complex Network Manifolds in d>2 are scale-free Scientific Reports 5, 13979 (2015) PERSPECTIVE G. Bianconi, Interdisciplinary and physics challenges in network theory EPL 111, 56001 (2015). Quantum Complex Network Geometries G. Bianconi, C. Rahmede, Z. Wu, Quantum Complex Network Geometries: Evolution and Phase Transition Phys. Rev. E 92, 022815 (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: 1602.04110 (2016).