Scale-variant topological portraits of complex networks

Our presentation at NetSci-X 2020.

Transcript

1. Scale-variant topological portraits of complex networks Q. H. Tran*, V. T. Vo, and Y. Hasegawa *zoro@biom.t.u-tokyo.ac.jp The University of Tokyo Waseda, Tokyo 20 January 2020 Hasegawa Laboratory
2. https://www.hasegawa-lab.net/ Information thermodynamics and machine learning Information thermodynamics Thermodynamics of precision Quantum thermodynamics and information Applied algebra Stochastic process Topological data analysis
3. Why the “topological” portrait? ◼ Variant-scale structures ◼ Higher-order interactions 3/18
4. 4/18 Networks Topological Data Analysis Study interactions at variant scales Study the “shape” of data Torres, Brenda Y., et al. Tracking resilience to infections by mapping disease space. PLoS biology 14.4 (2016): e1002436.
5. Persistent homology pipeline Algebraic Topology Filtration over distance with simplices Birth Death Persistence Diagram 0-simplex 1-simplex 2-simplex 3-simplex (solid) (Higher-order interactions) Persistent homology: variant-scale description of data Metric space Sampled data 5/18
6. Network to a metric space Diffusion geometry Original network Induced geometry De Domenico, M. Diffusion geometry unravels the emergence of functional clusters in collective phenomena. Physical Review Letters 118, 168301 (2017). 6/18
7. Diffusion on the network A random walker moves randomly between nodes in continuous time = σ = 1 , … , ( ∣ ) ⊤ ➢ : The probability to find a random walker on which starts from ➢ 0 = 0, … , 1 , … , 0 ⊤ -th = − rw = 1 − 1 σ 0 If = and σ ≠ 0 If ≠ and is adjacent to Otherwise 7/18
8. Interactions Topological scales in the network Small Local, dyadic Micro-scale Medium Non-local, dyadic + non-dyadic Meso-scale Large Global, dyadic + non-dyadic Macro-scale Diffusion on the network = 2 = 3 ◼ ( ∣ ) reflects the interactions between node and other nodes at -scale. = 1
9. Diffusion timescale embedding = 1 , … , ∈ ℝ The shape of finite set of points in the embedded space Idea:Varying will reveal the structure and interactions in the network at variant topological scales : → ℝ ⟼ ( ∣ ) ⟼ ( ∣ ) ⟼ ( ∣ ) 9/18
10. Scale-variant topological portrait Portrait Micro Meso Macro 10/18
11. Kernel method ◼ Can define an inner product ◼ Use in (linear) statistical-learning tasks (e.g., SVM) The space of diagrams ◼ Not a vector space ◼ Difficult to use in (linear) statistical-learning tasks (e.g., classification) ◼ Cannot define an inner product Ω Φ Φ , Feature mapping Φ Feature-mapped space Hilbert space ◼ Use in unsupervised learning tasks (e.g., Kernel PCA, Kernel Change Point Detection) 11/18
12. Scenarios for Applications ◼ Understanding variations of the parameters of network models ◼ Identifying network models ◼ Classifying real-world network data ◼ Detecting transition points in the time-evolving network Girvan-Newman Erdős–Rényi 12/18
13. Identify network models ◼ Configuration models ➢ Generating random networks from a given degree sequence ➢ Preserve the degree distribution but destroy the topological correlation Original models (#networks) Configuration models (#networks) Girvan–Newman (GN-org) (1000) GN-conf (1000) Lancichinetti–Fortunato–Radicchi (LFR-org) (1000) LFR-conf (1000) Watts–Strogatz (WS-org) (1000) WS-conf (1000) 13/18
14. Identify network models Kernel PCA for scale-variant topological portraits Common features of all networks Common features of configuration networks 14/18
15. Transition points detection Drosophila melanogaster ◼ 66 microarray measurements across full life cycle (1-66 time points) (Arbeitman+, Science, 2002) ◼ Four stages in the life cycle ➢ Embryo (1-30) ➢ Larva (31-40) ➢ Pupal (41-58) ➢ Adult (59-66) ◼ Analyze subset of 588 genes related to development → Recover time-evolving networks (Lee Song+, Bioinformatics, 2009) Smooth Change Time Kernel Reweighting 15/18
16. Transition points detection 16/18
17. Transition points detection ◼ The time point to achieve the maximum kernel discriminant ratio agree with the experimentally known transitions ➢ Embryo (1-30) ➢ Larva (31-40) ➢ Pupal (41-58) ➢ Adult (59-66) ◼ Sliding windows contain the diagrams along time points +1 , … , + (corresponding with time-evolving networks +1 , … , +) 17/18
18. Scale-variant Topological Portraits ◼ Able to represent variant topological scales and high-order interactions ◼ Effective to compare small networks ◼ Different from common network measures but can better identify network model 18/18 We need to go further to understand complex networks from higher-order interactions and topological structures.
19. Thank you for listening!