Section 8.2: Shortest path and small world effect By: Ralucca Gera, NPS Most pictures are from Newman’s textbook

The small world effect The typical network average distances between vertices are surprisingly small in real life networks: coined as small world effect Recall Stanley Milgram’s letter-passing experiment had an average of 6 hops (in the inferred network). In math terms the small world effect is a hypothesis that the mean distance 𝑙 is “small” Recall from Chapter 7 that 𝑙= 𝑖, 𝑗 𝑑(𝑖,𝑗) 𝑛 2 , setting 𝑑 𝑖,𝑗 =0, if there is no 𝑖𝑗 path (𝑖 and 𝑗 belong to different components)

Stanley Milgram’s experiment

Funneling was observed by Milgram in his experiment: 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

The small world effect In math terms the small world effect is a hypothesis that the mean distance 𝑙 is “small” Typically networks have been found to have mean distance less than 20 – or in many cases less than 10 – even though the networks themselves have millions of nodes This has implications such as rumor spread in a social networks, response time in the Internet disease spreading in social networks What is 𝑙 for your networks? Thus it is no surprise that real networks have a small 𝑙

Statistics for real networks 𝑙 is the average distance

The small world effect Mathematical models for networks try to mimic: small average path length 𝑙 and high clustering Observed: 𝑙 increases slowly with the number 𝑛 of vertices in the network: 𝑙 ~ log 𝑛 log <𝑘> The diameter of a network is relatively small as well: 𝑑𝑖𝑎𝑚 ~ log 𝑛 (in scale free 𝑑𝑖𝑎𝑚 ~ log ( log 𝑛 ) High average clustering coefficient Average degree Reuven Cohen and Shlomo Havlin Phys. Rev. Lett. 90, 058701 – Published 4 February 2003

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 p = probability of rewiring edges of the lattice Source: Watts, DJ; Strogatz, S H. 1998. Collective dynamics of 'small-world' networks, NATURE 393(668).

Example small word Avg path Avg clust From Ernesto Estrada

Example small word Avg path Avg clust From Ernesto Estrada

Example small word Avg path Avg clust From Ernesto Estrada