knowledge manifold
TRANSCRIPT
Knowledge ManifoldLingfei Wu 2016-06-06
1. Two traditions of network modeling: Newton vs. Einstein
2. Hyperbolic networks
3. Geodesics of hidden geometry: image recognition, Internet routing, and disease diffusion
4. The boundary of knowledge
Trivial geometry + non-trivial dynamics = Nontrivial geometry + trivial dynamics
More elegant !
One principle -> one manifold -> a set of dynamics
Einsteinian Newtonian
Densification: the super-linear scaling between edges and nodes (allometric scaling/accelerating growth); Small-world: the coexistence of a short diameter and a high clustering coefficient; Scale-free: the power-law distribution of degree.
W ⇠ N�1. Densification
3. Scale-free P (k) ⇠ k�
N ⇠ el2. Small-world & High CC
Geometry Dynamics
Erdős–Rényi graph, 1959
Barabási–Albert graph, 1999
Random geometric graph Gilbert,1961
Watts and Strogatz model, 1998
Random geometric growth graph Zhang et al.,2015
Small-world
High CC N ⇠ el
W ⇠ N�Densification Scale-free P (k) ⇠ k�
Hyperbolic graph Papadopoulos et al.,2012
Geometry
Trivial geometry + Trivial dynamics (local links)
High CC
Densification
Scale-free high popularity = short distance
Why hyperbolic ?
short diameter
We need a non-trivial space that satisfy
very compact such that for uniformly distributed nodes
N ⇠ el
Angel-Devil (No. 45) M.C. Escher,1945
Why hyperbolic ?
very compact such that for uniformly distributed nodes
N ⇠ el
high popularity = short distance
TED 2009: Margaret Wertheim: The beautiful math of coral
We do not want phenomenal explanations, Why hyperbolic ?
Hyperbolic Geometry of Complex Networks Dmitri Krioukov et al.,2010
Exponential tree = Hyperbolic space
Dendrogram: node similarity clustering
Similarity space
A Global Geometric Framework for Nonlinear Dimensionality Reduction, Joshua B. Tenenbaum et al., 2000
ISOMAP1. Calculate distance between all pairs of nodes
2. Construct network and “trust” only local (short) links
3. Reconstruct (the shortest) paths using the cumulative local links
Only geodesics matters !
Effective distance in disease diffusion
The Hidden Geometry of Complex, Network-Driven Contagion Phenomena, Dirk Brockmann & Dirk Helbing, 2013
2. Construct airport networks and “trust” only local (large traffic) links
Dmn =X
dmnEffective distance
3. Reconstruct (the shortest) paths using the cumulative local links
Only geodesics matters !
Marián Boguñá et al., 2010
1. Embed Internet (at the AS level) into a Hyperbolic space;
2. Routing along the geodesics;
Only geodesics matters !
Questions:
1. Hyperbolic embedding of citation networks ?
link prediction
field identification
2. Lorentz transformation and invariance of light speed in citation networks ?