Small World Prof Ralucca Gera Applied Mathematics Dept

Small World Prof. Ralucca Gera, Applied Mathematics Dept. Naval Postgraduate School Monterey, California rgera@nps. edu Excellence Through Knowledge

Learning Outcomes ü Understand the definition of small world ü Identify methods to compare your network to the small world model 2

The small world effect • Reuven Cohen and Shlomo Havlin Phys. Rev. Lett. 90, 058701 – February 2003 3

Stanley Milgram’s experiment 4

The small world effect Funneling was observed by Milgram in his experiment: – Most of the shortest paths to a sink vertex i go through one of its neighbors, so there is this funneling towards the destination Chen, H. ; Fan, G. ; Xie, L. ; Cui, J. -H. A Hybrid Path-Oriented Code Assignment CDMA-Based MAC Protocol for Underwater Acoustic Sensor Networks. Sensors 2013, 15006 -15025. 5

Statistics for real networks 6

The small average path • 7 Source: http: //barabasi. com/networksciencebook/chapter/5#origins

Average Distance (function of netw. size N) Analytical prediction BA model ER model 8 Source: L. Barab´asi. http: //barabasi. com/networksciencebook/chapter/5#diameter

How to construct Small-words? This is a model introduced by Watts-Strogatz: Networks that share properties of both regular and random graphs (Watts and his advisor Strogatz) Regular/lattice Small world Random graphs clustering coefficient High Low average path length High Low p = probability of rewiring edges of the lattice Source: Watts, DJ; Strogatz, S H. 1998. Collective dynamics of 'small-world' networks, NATURE 393(668). 9

Example small word Avg path Avg clust 10 From Ernesto Estrada’s presentations

Example small word Avg path Avg clust 11 From Ernesto Estrada’s presentations

Example small word Avg path Avg clust 12 From Ernesto Estrada’s presentations

Computing Small worldness in Network. X • Small-worldness is commonly measured with the coefficient sigma or omega. – Both coefficients compare the average clustering coefficient and shortest path length of a given graph against the same quantities for an equivalent random or lattice graph. 13 https: //networkx. github. io/documentation/latest/reference/algorithms/smallworld. html

References • Newman, “The Structure and Function of Complex Networks” http: //epubs. siam. org/doi/pdf/10. 1137/S 003614450342480 • Source: L. Barabasi. Source: http: //barabasi. com/networksciencebook/chapter/5#origins • Source: Watts, DJ; Strogatz, S H. 1998. Collective dynamics of 'small-world' networks, NATURE 393 (668). • Chen, H. ; Fan, G. ; Xie, L. ; Cui, J. -H. A Hybrid Path-Oriented Code Assignment CDMA-Based MAC Protocol for Underwater Acoustic Sensor Networks. Sensors 2013, 15006 -15025. 14 • Reuven Cohen and Shlomo Havlin. Phys. Rev. Lett. 90, 058701 – Published 4 February 2003 14