The average path length of scale free networks |
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| Institution: | 1. Division of Developmental Cognitive Neuroscience, Institute of Development, Aging and Cancer, Tohoku University, Sendai, Japan;2. Division of Medical Neuroimaging Analysis, Department of Community Medical Supports, Tohoku Medical Megabank Organization, Tohoku University, Sendai, Japan;3. Department of Radiology and Nuclear Medicine, Institute of Development, Aging and Cancer, Tohoku University, Sendai, Japan;4. Department of Functional Brain Imaging, Institute of Development, Aging and Cancer, Tohoku University, Sendai, Japan;5. Human and Social Response Research Division, International Research Institute of Disaster Science, Tohoku University, Sendai, Japan;6. Smart Ageing International Research Center, Institute of Development, Aging and Cancer, Tohoku University, Sendai, Japan;7. Japan Society for the Promotion of Science, Tokyo, Japan;8. Faculty of Medicine, Tohoku University, Sendai, Japan;1. Department of Econometrics and Operations Research, Tinbergen Institute, VU University, De Boelelaan 1105, Amsterdam 1081 HV, the Netherlands;2. Paris School of Economics, Centre d’Economie de la Sorbonne, CNRS, Université Paris 1, 106-112 Bd de l’Hôpital, Paris Cedex 13 75647, France |
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| Abstract: | In this paper, the exact solution of average path length in Barabási–Albert model is given. The average path length is an important property of networks and attracts much attention in many areas. The Barabási–Albert model, also called scale free model, is a popular model used in modeling real systems. Hence it is valuable for us to examine the average path length of scale free model. There are two answers, regarding the exact solution for the average path length of scale free networks, already provided by Newman and Bollobas respectively. As Newman proposed, the average path length grows as log(n) with the network size n. However, Bollobas suggested that while it was true when m = 1, the answer changed to log(n)/log(log(n)) when m > 1. In this paper, as we propose, the exact solution of average path length of BA model should approach log(n)/log(log(n)) regardless the value of m. Finally, the simulation is presented to show the validity of our result. |
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