Graph Theoretical Approaches to the Small-World Phenomenon - PowerPoint PPT Presentation

Graph Theoretical Approaches to the Small-World Phenomenon Proseminar Analyse sozialer Netzwerke (eng.: Analysis of Social Networks ) Speaker: Felix Stahlberg Institute for Program Structures and Data Organization (IPD) SOFTWARE DESIGN AND QUALITY GROUP Source: pixelio.de KIT – The cooperation of Forschungszentrum Karlsruhe GmbH and Universität Karlsruhe (TH)

Milgram’s Small-World Experiment [Mil67] Mr. G. S T Boston Omaha Name and selected information concerning T • on a first-name basis • Social and geographic Separation between S and T Felix Stahlberg – Graph Theoretical Approaches to the 2 06.12.2010 Small-World Phenomenon

Scientific Interest (Examples) Neuroscience : Neuronal network of Caenorhabditis elegans Technology : Efficiency of power grids Medicine : Social Networks : Spread of infectious diseases Collaboration of actors Felix Stahlberg – Graph Theoretical Approaches to the 3 06.12.2010 Small-World Phenomenon

Modeling the Problem Phenomenon Mathematical Model ? Felix Stahlberg – Graph Theoretical Approaches to the 4 06.12.2010 Small-World Phenomenon

Mathematical Quantification (Path Length) [DW98] A small-world network is a connected Graph 𝐻 = (𝑊, 𝐹) with A low characteristic path length 𝑀 Average length of shortest paths in 𝐻 𝑒 𝑡 (𝑤 1 ,𝑤 2 ) 𝑤1,𝑤2∈𝑊 In undirected graphs: 𝑀 ≔ |𝑊|( 𝑊 −1) Where 𝑒 𝑡 (𝑤 1 , 𝑤 2 ) is the length of the shortest path from 𝑤 1 to 𝑤 2 . B C A G A D B C F E L=72/42 ≈1.714 L=1 Felix Stahlberg – Graph Theoretical Approaches to the 5 06.12.2010 Small-World Phenomenon

Mathematical Quantification (Clustering) [DW98] A small-world network is a connected Graph 𝐻 = (𝑊, 𝐹) with A high clustering coefficient 𝐷 How close neighbourhoods in 𝐻 are to being a clique 𝐷 𝑤 #𝐹𝑒𝑕𝑓𝑡 𝑗𝑜 𝑂 𝑤 𝐷 ≔ 𝑤∈𝑊 𝑥𝑗𝑢ℎ ∀𝑤 ∈ 𝑊: 𝐷 𝑤 ≔ #𝐹𝑒𝑕𝑓𝑡 𝑗𝑜 𝑂 𝑤 𝑞𝑝𝑡𝑡𝑗𝑐𝑚𝑓 |𝑊| Where 𝑂 𝑤 is the subgraph induced by the neighbourhood of 𝑤 . B C A G A D B C F E C=(6+(2*6)/(7*6))/7 ≈0.898 C=1 Felix Stahlberg – Graph Theoretical Approaches to the 6 06.12.2010 Small-World Phenomenon

Mathematical Quantification (Wider Picture) Global and local Efficiency [LM01] Nonconnected, weighted (nonmetric) graphs Equivalents to 𝑀 and 𝐷 . Measures parallel systems (not sequential like 𝑀 ) Clustering coefficient 𝐷 controversial Felix Stahlberg – Graph Theoretical Approaches to the 7 06.12.2010 Small-World Phenomenon

Cycle Plus A Random Matching Let 𝐻 be a graph formed by adding a random matching to an 𝑜 -cycle. Then with probability tending to 1 as 𝑜 goes to infinity, 𝐻 has diameter 𝐸(𝐻) satisfying log 2 𝑜 − 𝑑 ≤ 𝐸 𝐻 ≤ log 2 𝑜 + log 2 log 𝑜 + 𝑑 𝑥𝑗𝑢ℎ 𝑑 ≤ 10 . [BC88] 𝔽 𝐷 = 2 3 , 𝔽(𝑀) ∈ 𝑃(log 𝑜) Felix Stahlberg – Graph Theoretical Approaches to the 8 06.12.2010 Small-World Phenomenon

Randomly Rewired Lattices (1) Starting from a ring lattice with 𝑜 vertices and 𝑙 edges per vertex, we rewire each edge at random with probability 𝑞 . [DW98] 𝑜 = 12, 𝑙 = 4, 𝑞 = 1 8 𝑜 = 12, 𝑙 = 4, 𝑞 = 0 Felix Stahlberg – Graph Theoretical Approaches to the 9 06.12.2010 Small-World Phenomenon

Randomly Rewired Lattices (2) Starting from a ring lattice with 𝑜 vertices and 𝑙 edges per vertex, we rewire each edge at random with probability 𝑞 . [DW98] 1 0,9 0,8 0,7 0,6 Small-World Networks C(p)/C(0) 0,5 L(p)/L(0) 0,4 0,3 0,2 0,1 0 p 0,001 0,01 0,1 1 Sketch: Small-World Networks in Strogatz-Watts model Felix Stahlberg – Graph Theoretical Approaches to the 10 06.12.2010 Small-World Phenomenon

Algorithmic Aspects Network T S Invisible for S Visible for S Felix Stahlberg – Graph Theoretical Approaches to the 11 06.12.2010 Small-World Phenomenon

Kleinberg’s Model [Kle00] (1) Parameter 𝑞 ≥ 1 , 𝑟 ≥ 0 , 𝑠 ≥ 0 Vertices are identified by tuples (i, j) in a 𝑜 × 𝑜 lattice Metric 𝑒( 𝑗, 𝑘 , 𝑙, 𝑚 ) = 𝑙 − 𝑗 + |𝑚 − 𝑘| A vertex 𝑤 has outgoing edges to All vertices 𝑣 with 𝑒 𝑤, 𝑣 ≤ 𝑞 ( local contacts ) 𝑟 long-range contacts chosen by independent random trials Probability for a vertex 𝑣 to be 𝑤 ’s long -range contact ~𝑒(𝑣, 𝑤) −𝑠 Felix Stahlberg – Graph Theoretical Approaches to the 12 06.12.2010 Small-World Phenomenon

Kleinberg’s Model [Kle00] (2) 𝑜 = 5 𝑞 = 1 𝑟 = 0 Felix Stahlberg – Graph Theoretical Approaches to the 13 06.12.2010 Small-World Phenomenon

Kleinberg’s Model [Kle00] (3) 𝑜 = 5 𝑞 = 2 𝑟 = 2 v Outgoing edges of v Felix Stahlberg – Graph Theoretical Approaches to the 14 06.12.2010 Small-World Phenomenon

Local Algorithms in Kleinberg’s Model 𝑠 = 2 is the only value for which there is a decentralized algorithm capable of producing chains whose length is polynomial in log n . [Kle00] • Cycles plus Random Matchings are inappropriate to explain the small-world phenomenon • The Watts-Strogatz model may be inappropriate to explain the small-world phenomenon Felix Stahlberg – Graph Theoretical Approaches to the 15 06.12.2010 Small-World Phenomenon

Constructive Proof Idea for r=2 S 𝑜 = 11 𝑞 = 1 𝑟 = 1 (Most of the unneccessary long-range contacts removed) T Felix Stahlberg – Graph Theoretical Approaches to the 16 06.12.2010 Small-World Phenomenon

Constructive Proof Idea for r=2 (Analysis) Phase 3 S Phase 2 • #Phases in 𝑃(log 𝑜) • Expected delivery time in each phase in 𝑃(log 𝑜) Phase 1 • Culmulative delivery time in 𝑃(log 2 𝑜) T Phase 0 Felix Stahlberg – Graph Theoretical Approaches to the 17 06.12.2010 Small-World Phenomenon

Conclusion Still in focus of research Applicable to many fields of science Controversial modeling approaches Controversial experiment results Felix Stahlberg – Graph Theoretical Approaches to the 18 06.12.2010 Small-World Phenomenon

References [BC88] B. Bollobás and F. R. K. Chung. The diameter of a cycle plus a random matching. SIAM Journal on Discrete Mathematics , 1(3):328 333, 1988. [DW98] Steven Strogatz. Duncan Watts. Collective dynamics of small- world networks. Nature , 393, June 1998. [Kle00] John Kleinberg. The small-world phenomenon: an algorithm perspective. In STOC ‘00: Proceedings of the thirty -second annual ACM symposium on Theory of computing , pages 163-170, New York, NY, USA, 2000. ACM. [LM01] Vito Latora and Massimo Marchiori. Efiicient behavior of small- world networks. Phys. Rev. Lett. , 87(19):198701, Oct 2001. [Mil67] Stanley Milgram. The small world problem. Psychology Today , 1967 Felix Stahlberg – Graph Theoretical Approaches to the 19 06.12.2010 Small-World Phenomenon

Questions Felix Stahlberg – Graph Theoretical Approaches to the 20 06.12.2010 Small-World Phenomenon