Universal Augmentation Schemes for Network Navigability: Overcoming the $\sqrtn$-Barrier
Résumé
Augmented graphs were introduced for the purpose of analyzing the ''six degrees of separation between individuals'' observed experimentally by the sociologist Standley Milgram in the 60's. We define an augmented graph as a pair (G,M) where G is an n- node graph with nodes labeled in {1, . . . , n}, and M is an n × n stochastic matrix. Every node u ∈ V (G) is given an extra link, called a long range link, pointing to some node v, called the long range contact of u. The head v of this link is chosen at random by Pr{u → v} = Mu,v. In augmented graphs, greedy routing is the oblivious routing process in which every intermediate node chooses from among all its neighbors (including its long range contact) the one that is closest to the target according to the distance measured in Theoretical Computer Science 410 (2009) 1970-1981  Contents lists available at ScienceDirect Theoretical Computer Science journal homepage: www.elsevier.com/locate/tcs Universal augmentation schemes for network navigability6 Pierre Fraigniauda,∗, Cyril Gavoilleb, Adrian Kosowskic, Emmanuelle Lebhara, Zvi Lotkerd a CNRS and University Paris Diderot, 75205 Paris, France b University of Bordeaux, 33405 Talence, France c Gdansk University of Technology, 80-952 Gdańsk Wrzeszcz, Poland d Ben Gurion University, Beer-Sheva, Israel the underlying graph G, and forwards to it. The best augmentation scheme known so far √ ensures that, for any n-node graph G, greedy routing performs in O( n) expected number of steps. Our main result is the design of an augmentation scheme that overcomes the O( n) barrier. Precisely, we prove that for any n-node graph G whose nodes are arbitrarily labeled in {1, . . . , n}, there exists a stochastic matrix M such that greedy routing in (G, M ) performs in O ̃(n1/3), where the O ̃ notation ignores the polylogarithmic factors. We prove additional results when the stochastic matrix M is universal to all graphs. In √ √ particular, we prove that the O( n) barrier can still be overcame for large graph classes even if the matrix M is universal. This however requires an appropriate labeling of the √ nodes. If the node labeling is arbitrary, then we prove that the O( n) barrier cannot be overcome with universal matrices.