Nucleation condensation and transition within the complex network

Nucleation, condensation and λ-transition within the complex network: An application to description of a real-life stock market evolution M. Wiliński, B. Szewczak, T. Gubiec, R. Kutner Faculty of Physics, University of Warsaw & Z. R. Struzik The University of Tokyo and RIKEN Brain Science Institute, Japan VII Sympozjum Fizyki w Ekonomii i Naukach Społecznych UMCS Lublin, 14 -17 May 2014

Schedule ● Motivation and aims ● Why Minimal Spanning Tree (MST) network? ● Movie of MST evolution – empirical evidences ● Dynamic phase diagrams ● Equilibrium vs. detailed balance conditions - „peaceful ocean” phase ● „Macroscopic” dynamic equation for nucleation and condensation ● Crude mechanisms of network evolution – a heuristic approach ● Pictorial summary, conclusions and projects Own bibliography: EPJ ST 205 (2012), 27; APPA 123 (2013), 615; Physica A 392 (2013), 5963; Ar. Xiv. 2013: http: //arxiv. org/abs/1311. 5753

Neuron network: neurons connected by synapsis (source: internet)

Delta Air Lines connections (US) (source: internet)

Interstate highways in US (1970) (źródło: internet)

Sieć połączeń autostradowych i lotniczych (w USA) (źródło: A. & P. Fronczak: Świat sieci złożonych. Od fizyki do internetu, PWN 2009) P(k): prawdopodobieństwo znalezienia w sieci węzła o stopniu k

Rozkłady stopni wierzchołków P(k) A: sieć aktorów, B: sieć bezpośrednich połączeń lotniczych, C: uczelniana sieć WWW (WF PW) (źródło: A. & P. Fronczak: Świat sieci złożonych. Od fizyki do internetu, PWN 2009)

Motto of our study: λ-transition in Helium (źródło: J. Spałek: Emergentność w Prawach Przyrody i Hierarchiczna Struktura Nauki, Postępy Fizyki tom 63 (2012), 8)

Sky. West flight connections (US) (source: internet)

Three-stage strategy ● ● ● First stage: we construct the correlation-based metric space of companies and hence the movie to study the evolution of the complex network. Second stage: we study possible mechanism of structural dynamic phase transitions. Third stage: derivation of „macroscopic” evolution equations

Why Minimal Spanning Tree (MST) network? Nagie drzewo

Why Minimal Spanning Tree (MST) network? Tree is as simple as possible compact network ready to describe a complex system ● Emergence. ● Sensitive of the tree to its reduction or growth – nonlocality. ● Sensivity of the tree to the local perturbation - nonlocality. ● Adaptativity and fitness; attractivity, different preferential rules. ● Full connectivity. ● ● ● Tree is ready to study: flexibility vs. robustness, stability vs. instability, equilibrium vs. non-equilibrium, stationarity vs. nonstationarity, . . . MST is a „hard core” of most other networks. MST: only algorithmic construction and not by the analytical approach (nonintegrability).

Why Minimal Spanning Tree (MST) network? From cross-correlation to metric Cartesian distance d, in K-dimensional space, between companies (vectors, points) i and j: where is a (daily) return of j company at day k. Central formula: where Pearson cross-correlation factor C(i, j) between i and j companies: Conclusion: Distance d is a monotonically decreasing function of the Pearson correlation factor C, e. g. distance is minimal if correlation is maximal.

Why Minimal Spanning Tree (MST) network? Prim's algorithm: connectivity, no loops, minimal length Construction of thread

Why Minimal Spanning Tree (MST) network? Prim's algorithm: connectivity, no loops, minimal length Construction of star: the mechanizm making 'dragon-king' from a poor vertex

Why Minimal Spanning Tree (MST) network? Mean Occupation Layer (MOL) Extreme states of MST

Why Minimal Spanning Tree (MST) network? „The last will be the first” Snap-shot picture of Frankfurt Stock Exchange (FSE) Equilibrium

Why Minimal Spanning Tree (MST) network? Superstar-like (superhub, 'dragon-king') structure for FSE

Why Minimal Spanning Tree (MST) network? Snap-shot picture of Frankfurt Stock Exchange (FSE) Two-part network

Why Minimal Spanning Tree (MST) network? Centralities Mean Occupation Layer (MOL) vs. time for FSE MOL: the average 'handshake' distance from the central vertex

Why Minimal Spanning Tree (MST) network? Centralities Entropy vs. time for FSE P(k): probability of finding (in the network) at time t the vertex having degree k

Equilibrium vs. detailed balance conditions Dynamic power-law - exponent α vs. time a signature of: (i) small world vs. ultra-small world? (ii) multifractality? (iii) complexity?

Why Minimal Spanning Tree (MST) network? Centralities Mean 'handshake' distance: a mean internode distance counted in steps

Why Minimal Spanning Tree (MST) network? Variance

Why Minimal Spanning Tree (MST) network? Centralities Betweeness vs. time for FSE Betweeness of a given vertex: the relative number of paths going through this vertex

How 'dragon-king' takes its edges?

Dynamic phase diagram (daily horizon) 1. Continuous transition to nucleation of edges 2. λ-peak vs. condensation of edges

Dynamic phase diagram (daily horizon) 1. Continuous transition to nucleation of edges 2. λ-peak vs. condensation of edges

Equilibrium vs. detailed balance conditions (DBCs) Power law and binomial distributions GENERAL QUESTION: whether DBCs are valid? ANSWER: yes, DBCs are valid for the market „plankton” PROOF based on empirical data We proof that all systematic currents vanish: 1) DBCs: transition from k to k+l Analogous transition from k - l to k (l = 1, . . . , k-1). 2) From empirical data we have for k < 12 (market „plankton”)

Equilibrium vs. detailed balance conditions (DBCs) Power law and binomial distributions 3) Disconnection of edges statistically independent – binomial distribution 4) Elementary transition rates for disconnection, b(-1|k), from empirical data 5) Connection of edges also statistically independent – binomial distribution 6) Elementary transition rates for connection, b(1|k), from empirical data

Equilibrium vs. detailed balance conditions (DBCs) Power law and binomial distributions GENERAL QUESTION: whether DBCs are valid? ANSWER: yes, DBCs are valid for the market „plankton” PROOF based on empirical data We proof that all systematic currents vanish: 1) DBCs: transition from k to k+l Analogous transition from k - l to k (l = 1, . . . , k-1). 2) From empirical data we have for k < 12 (market „plankton”)

Equilibrium vs. detailed balance conditions (DBCs) Power law and binomial distributions

Equilibrium vs. detailed balance conditions (DBCs) Power law and binomial distributions

„Macroscopic” equation of motion for nucleation and condensation Evolution of the superstar-like structure: a dragon-king (SZG) dynamics For the l. h. s. of the λ-peak For the r. h. s. of the λ-peak Binomial representations for a strong assumption

„Macroscopic” equation of motion for nucleation and condensation Evolution of the superstar-like structure: a dragon-king (SZG) dynamics How depends on ? Complexity reduced to the nonlinear single-variable problem! How Nucleation: Condensation: depends on ?

„Macroscopic” equation of motion for nucleation and condensation Evolution of the superstar-like structure: a dragon-king (SZG) dynamics Nucleation: Allen-Cahn (deterministic) equation Solution: Condensation Solution:

Pictorial summary

General conclusions and projects ● ● Advanced technical analysis, from physical point of view, containing phenomenological approach strongly based on empirical evidences. Nucleation, condensation and λ-transition exist as a dynamical phase transitions. Can we consider them as precursors of crash? ● Modelling was made but more microscopic explanation is needed! ● Analysis of noise is necessary to complete the Langevin equation! ● ● Still relation between MST algorithm and observed real phenomenon and processes is a challenge! Why structural and topological transitions are sometimes separated?