Quantum!Network!Manifold!

SM&FT 2015 Bari, 10 December 2015 NETWORK GEOMETRY Ginestra Bianconi School of Mathema7cal Sciences, Queen Mary University of London, London, UK Geometry of Complex Networks will have impact In the characterizaon of structure and dynamics of Biological, Social and Technological systems. Quantum Space1me Is a network My own view is that ul7mately physical laws should find their most natural expression in terms of essen7ally combinatorial principles… Thus, in accordance with such a view, should emerge some form of discrete or combinatorial spaceBme. Roger Penrose in On the Nature of Quantum Geometry Quantum Space1me Quq Z. Merali Nature (2013). A Complex Universe I think the next century will be the century of complexity (Stephen Hawking Jan 2000) A theory of Quantum cosmology cannot be logically consistent if it does not describe a complex universe. […] A theory of cosmology must, if it is to be consistent, be a theory of self-organiza7on. (The LiFe OF the Cosmos, Lee Smolin) A primer on Network Theory Complex networks: Between randomness and order LATTICES COMPLEX NETWORKS RANDOM GRAPHS Scale free networks Small world Regular networks With communiDes Totally random Symmetric ENCODING INFORMATION IN Poisson degree THEIR STRUCTURE distribuDon Complex networks have at the same 7me a small characteris7c distance L between the nodes like Cayley trees, Bethe laces, and random graphs i.e. L scales like the logarithm of the number of nodes N or slower and significant density of small loops like laces. The density of triangles is measured by the clustering coefficient C. Was and Strogatz (1998) Scale-free networks Actor networks WWW Internet k k Barabasi-Albert 1999 Faloutsos, Faloutsos and Faloutsos 1999 k finite −γ P(k) ∝ k γ ∈(2,3] 2 k → ∞ € The Ising model on scale-free networks P(k) ∝ k −γ CriDcal Temperature T k(k −1) c ∝ J k € Bianconi 2002,Leone et al. 2002 Dorogovtsev et al. 2002 l € For γ the crical ≤ 3 temperature diverges. The system is in the € Ferromagne1c phase at every temperature Network Geometry Boguna, Krioukov, Claffy Nature Physics (2008) It is believed that most complex networks have an hidden metric such that the nodes close in the hidden metric are more likely to be linked to each other. Hyperbolic geometry of complex networks could contribute to improve roung algorithms Emergent geometry In the framework of emergent geometry networks with hidden geometry are generated by equilibrium or non-equilibrium dynamics that makes no use of the hidden geometry Quantum graphity approach 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 Unsaturated link Saturated link ρij=1 ρij=0 m=2 • ρij=1 if the link is unsaturated, i.e. less than m triangles are incident on it • ρij=0 if the link is saturated, i.e. the number of incident triangles is given by m Process (a) a ρ We choose a link (i,j) with probability [1] ij ij Πi, j = ∑arsρrs and glue a new triangle the link r,s € Growing Growing Simplicial Geometrical Complex Network Process (b) We choose a 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 Growing Simplicial Geometrical Complex Network The model Star1ng from an inial triangle, At each 1me • process (a) takes place and • process (b) takes place with probability p. 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. • Therefore the emergent network geometry for our model with m=2 is a discrete 2d manifold. Scale-free networks In the case m = ∞ a scale-free network with high clustering, significant community structure, finite spectral dimension is € generated. Planar for p=0. Degree distribuon • For m=2 and p=0 we can calculate the degree distribuDon given by 1⎛ 2⎞ k−1 P(k) = ⎜ ⎟ 2⎝ 3⎠ • For and m = ∞ p=0 we can calculate the degree distribuDon given by € 12 P(k) = € (k + 2)(k +1)k € Combinatorial Curvature The combinatorial curvature for a node i of a planar triangulaon is k T R =1− i + i i 2 3 • ki is the degree of the node i, • Ti is the number of triangles to which node i belongs 6 − ki For a node in the bulk Ri = € 6 4 − k For a node at the surface R = i i 6 € € Emergent network geometry and curvature distribuon Exponen1al network Broad degree distribu1on Scale-free network 1 1 R = R = c R = N N R2 < ∞ R2 = ∞ R2 = ∞ € Finite spectral dimension Lij = ki δij − aij ρ(λ) ≈ λ−(d / 2−1) −d / 2 Pc (λ) ≈ λ € Proper1es of emergent network geometries • Small world • Finite clustering • High modularity • Non trivial k-core • Finite spectral dimension Which are proper7es of many real network datasets. Complex Quantum Network Manifolds G.Bianconi, C. Rahmede Arxiv:1506.02648 Higher dimensional discrete manifolds In dimension d the manifold is built by gluing simplices of dimension d In d=2 the simplices are triangles In d=3 they are tetrahedra. Not all the nodes are the same! ε6 ε5 Let assign to each node i ε1 ε an energy ε from a 4 ε g(ε) distribution 3 εε25 Energy of the δ-faces Every δ-face α is associated to an energy which is the sum of the energy of the nodes εα = ∑εi belonging to α i⊂α For example, in d=3 the energy of a link ε is ε1 € 2 ε1+ε2 ε3 the energy of a face is ε1+ε2+ε3 ε 1 ε2 Notaon • Every (d-1)-dimensional face present in the manifolds is assigned a value aα=1 otherwise aα=0. • Every (d-1)-face is saturated (ρα=0) if 2 d-dimensional simplicies are incident to it, otherwise it is unsaturated (ρα=1). • Every (d-1)-face associated to a value nα indicating the number of simplicies exceeding one incident to the face, i.e. nα=1 for saturated faces (2 incident d-dimensional simplicies) nα=0 for unsaturated faces (1 incident d-dimensional simplex) !!!!!!! Complex!Quantum!Network!Manifold! The$evolu)on$for$d=2$$ t=1! ! ! t=2! ! ! ! t=3! ! ! ! ! t=4! −βε α We choose a unsaturated [1] e aα ρα (d-1)-dimensional Πα = −βε α ' face α, with probability ∑e aα'ρα' α'∈Qd,d−1 and glue a new triangle the link € Stascal mechanics of the manifold evoluon Each network history up to Dme t=N has probability e−β E e−β (E − µ N ) P {α(t')} {ε(t')} = = ( t'≤t t'≤t ) Z˜ (t) N! where the total energy E associated to the manifold is given by E n € = ∑εα α α∈Qd,d−1 The entropy variaon of the network sasfy ΔS = β(ΔE −€µ ΔN) + logN ΔN = [β( e − µ) + logN]ΔN where N is proporDonal to t and to the boundary A of the manifold € Quantum Network States To a d-dimensional manifold of N nodes we can associate the Hilbert space N P P ˆ Htot = ⊗Hnode ⊗Hd −1 ⊗H d −1 (similar approach exing in quantum gravity, ex. In “Quantum Graphity” papers) H € The Hilbert space of node node i is associated to a fermionic oscillator with energy εi (node i belonging or not to the manifold) The Hilbert space H d −1 of a (d-1)-dimensional face is associated to a fermionic€ oscillator (face belonging or not to the manifold). ˆ The Hilbert space of a (d-1)-dimensional face is associated to H d −1 fermionic€ oscillator (one or two simplices incident to the face). € Quantum Network States ψ ∈ Htot c o ,a ,n o , a n ψ = ∑ ({ i α α })∏ i ε i ∏ α α {oi ,aα ,nα } i=1,N α oi=0,1 mapped to node i with energy εi belonging or not € to the manifold aα=0,1 mapped to face α belonging or not to the manifold nα=0,1 mapped to the face α incident to 1 or 2 d-dimensional simplices Markovian dynamics of the quantum network states Given an inial condion, ψ (t =1) we consider a Markovian dynamics for the Quantum Network state ψ (t) = U ψ(t −1) € Where enforces the quantum network state normaliza1on at Zˆ (t) me t. € € Complex quantum networks describe the evoluon of the quantum network states Every classical network evoluDon correspond to a given history of the quantum network state up to Dme t. Zˆ (t) = Z˜ (t) = ∑e−β E ∏g(ε(t')) {ε (t'),α (t')}t '≤t t' configura1on space Each contribu1on is propor1onal to the probability of the classical evoluon of the network manifold me € t Generalized degrees The generalized degree kd,δ(α) of a δ-face α in a d-dimensional complex network manifold is given by the number of d-dimensional simplices incident to the δ-face α. If we define the generalized adjacency matrix as ⎧ 1 if the d − dimensional α' simplex belongs to the manifold A ' = ⎨ α ⎩ 0 otherwise The generalized degree is given by € k d,δ (α) = ∑ Aα ' α '⊃α € Generalized degree distribuon case β=0 • If δ=d-1 the generalized degrees follow a binomial distribu1on ⎧ d −1 ⎪ for k =1, P (k) = ⎨ d d,d −1 1 ⎪ for k = 2. ⎩ d • If δ=d-2 the generalized degree follow a exponen1al distribu1on k ⎛ 2 ⎞ d −1 P (k) = ⎜ ⎟ for k ≥1 d,d −2 ⎝ d +1⎠ 2 • If δ<d-2 the generalized degree follow a power-law distribu1on d −1 Γ[1+ (d +1)/(d −δ − 2)] Γ[k + 2/(d −δ − 2)] P (k) = for k ≥1 d,δ d −δ − 2 Γ[1+ 2/(d −δ − 2)] Γ[k +1+ (d +1)/(d −δ − 2)] € Rela1on between degree and generalized degree of nodes The degree of a node i Kd(i) is simply related to its generalized degree kd,0(i) by the relaon Kd (i) = kd,0 (i) + d −1 It follows that: • If the generalized degree kd,0 is scale-free also the degree distribuDon is scale-free.

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