Abstract
General models of network navigation must contain a deterministic or drift component, encouraging the agent to follow routes of least cost, as well as a random or diffusive component, enabling free wandering. This paper proposes a thermodynamic formalism involving two path functionals, namely an energy functional governing the drift and an entropy functional governing the diffusion. A freely adjustable parameter, the temperature, arbitrates between the conflicting objectives of minimising travel costs and maximising spatial exploration. The theory is illustrated on various graphs and various temperatures. The resulting optimal paths, together with presumably new associated edges and nodes centrality indices, are analytically and numerically investigated.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows. Theory, algorithms and applications. Prentice Hall (1993)
Alamgir, M., von Luxburg, U.: Phase transition in the family of p-resistances. In: Neural Information Processing Systems (NIPS 2011), pp. 379–387 (2011)
Bollobás, B.: Modern Graph Theory. Springer (1998)
Borgatti, S.P.: Centrality and network flow. Social Networks 27, 55–71 (2005)
Brandes, U., Fleischer, D.: Centrality Measures Based on Current Flow. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 533–544. Springer, Heidelberg (2005)
Chebotarev, P.: A class of graph-geodetic distances generalizing the shortest-path and the resistance distances. Discrete Applied Mathematics 159, 295–302 (2010)
Dubois-Ferrière, H., Grossglauser, M., Vetterli, M.: Valuable Detours: Least-Cost Anypath Routing. IEEE/ACM Transactions on Networking 19, 333–346 (2011)
Iryo, T., Shintaku, H., Senoo, S.: Experimental Study of Spatial Searching Behaviour of Travellers in Pedestrian Networks. In: 1st European Symposium on Quantitative Methods in Transportation Systems, EPFL Lausanne (2012) (contributed talk)
Kemeny, J.G., Snell, J.L.: Finite Markov Chains. Springer (1976)
Farnsworth, K.D., Beecham, J.A.: How Do Grazers Achieve Their Distribution? A Continuum of Models from Random Diffusion to the Ideal Free Distribution Using Biased Random Walks. The American Naturalist 153, 509–526 (1999)
Fouss, F., Pirotte, A., Renders, J.-M., Saerens, M.: Random-Walk Computation of Similarities between Nodes of a Graph with Application to Collaborative Recommendation. IEEE Transactions on Knowledge and Data Engineering 19, 355–369 (2007)
Freeman, L.C.: Centrality in networks: I. Conceptual clarification. Social Networks 1, 215–239 (1979)
Freeman, L.C., Borgatti, S.P., White, D.R.: Centrality in valued graphs: A measure of betweenness based on network flow. Social Networks 13, 141–154 (1991)
Kirchhoff, G.: On a deduction of Ohm’s laws, in connexion with the theory of electro-statics. Philosophical Magazine 37, 463 (1850)
Newman, M.E.J.: A measure of betweenness centrality based on random walks. Social Networks 27, 39–54 (2005)
Noh, J.-D., Rieger, H.: Random walks on complex networks. Phys. Rev. Lett. 92, 118701 (2004)
Saerens, M., Achbany, Y., Fouss, F., Yen, L.: Randomized Shortest-Path Problems: Two Related Models. Neural Computation 21, 2363–2404 (2009)
Travers, J., Milgram, S.: An experimental study of the small world problem. Sociometry 32, 425–443 (1969)
Yen, L., Saerens, M., Mantrach, A., Shimbo, M.: A Family of Dissimilarity Measures between Nodes Generalizing both the Shortest-Path and the Commute-time Distances. In: Proceedings of the 14th SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 785–793 (2008)
Zhou, T.: Mixing navigation on networks. Physica A 387, 3025–3032 (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bavaud, F., Guex, G. (2012). Interpolating between Random Walks and Shortest Paths: A Path Functional Approach. In: Aberer, K., Flache, A., Jager, W., Liu, L., Tang, J., Guéret, C. (eds) Social Informatics. SocInfo 2012. Lecture Notes in Computer Science, vol 7710. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35386-4_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-35386-4_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-35385-7
Online ISBN: 978-3-642-35386-4
eBook Packages: Computer ScienceComputer Science (R0)