The interplay between network structure and search dynamics has emerged as a busy subfield of statistical network studies (see e.g. Refs. [1; 9; 11; 10; 13; 14]). Consider a simple graph G = (V,E) (where V is a set of n vertices and E is a set of m edges—unordered pairs of vertices). Assume information packets travel from a source vertex s to a destination t. We assume the packages are myopic agents (at a given time step they have access to information about the vertices in their neighborhood, but not more), and have memory (so they can e.g. perform a depth-first search) but no previous knowledge of the network. Let τ(p) be the time for a packet p to travel between its source and destination. One commonly studied quantity of search efficiency is the expectation value of τ, for randomly chosen s and t. In this chapter we attempt to find efficient ways to index V (with numbers from 1 to n) and utilize these indices for packet navigation. In other words, we try to find ways to compress the global information about the network into numbers 1,…,n so that the information can be used by packets to find short paths to their destinations.
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Notes
- 1.
Every iteration, one step is taken in all branches. The different search branches mark the visited vertices with their indices. A search proceeds only to vertices not marked by any search. When there are no unmarked vertices, the search branch is finished.
- 2.
We do this by randomly exchanging search trees between the two classes and accept changes that improve the partition. The search is continued until their vertex sums differ by at most one or until the partition has not improved for 1000 trials.
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Acknowledgements
PH acknowledges financial support from the Wenner-Gren Foundations, The Swedish Foundation for Strategic Research and the National Science Foundation (grant CCR–0331580).
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Holme, P. (2009). Efficiency of Navigation in Indexed Networks. In: Ganguly, N., Deutsch, A., Mukherjee, A. (eds) Dynamics On and Of Complex Networks. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4751-3_11
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