The exact solution of the mean geodesic distance for Vicsek fractals

Abstract

The Vicsek fractals are one of the most interesting classes of fractals and the study of their structural properties is important. In this paper, the exact formula for the mean geodesic distance of Vicsek fractals is found. The quantity is computed precisely through the recurrence relations derived from the self-similar structure of the fractals considered. The obtained exact solution exhibits that the mean geodesic distance approximately increases as an exponential function of the number of nodes, with the exponent equal to the reciprocal of the fractal dimension. The closed-form solution is confirmed by extensive numerical calculations.PACS numbers: 83.80. Rs, 61.43.Hv The concept of fractals plays an important role in characterizing the features of complex systems in nature, since many objects in the real world can be modeled by fractals [1]. In the last two decades, a great deal of activity has been concentrated on the studies of fractals [2,3]. It has been shown [4,5,6,7,8,9,10,11] [26,27,28,29,30,31].Despite the importance of this structural property, to the best of our knowledge, the rigorous computation for the mean geodesic distance of Vicsek fractals has not been addressed. To fill this gap, in this present paper we investigate this interesting quantity analytically. We derive an exact formula for the mean geodesic distance characterizing the Vicsek fractals. The analytic method is on the basis of an algebraic iterative procedure obtained from the self-similar structure of Vicsek fractals. The obtained precise result shows that the mean geodesic distance exponentially with the number of nodes. Our research opens the way to theoretically study the mean geodesic distance of regular fractals and deterministic networks [32,33,34]. In particularly, our exact solution gives insight different from that afforded by the approximate solution of stochastic fractals.The classical Vicsek fractals are constructed iteratively [12,15]. We denote by V f,t (t ≥ 0, f ≥ 2) the Vicsek fractals after t generations. The construction starts from (t = 0) a star-like cluster consist of f + 1 nodes arranged in a cross-wise pattern, where f peripheral nodes are connected to a central node. This corresponds to V f,0 . For t ≥ 1, V f,t is obtained from V f,t−1 . To obtain V f,1 , we generate f replicas of V f,0 and arrange them around the periphery of the original V f,0 , then we connect the central structure by f additional links to the corner copy structures. These replication and connection steps are repeated t times, with the needed Vicsek fractals obtained in the limit t → ∞, whose fractal dimension is ln(f +1) ln 3 . In Fig. 1, we show schematically the structure of V 3,2 . According to the construction algorithm, at each time step the number of nodes in the systems increase by a factor of f + 1, thus, we can easily know that the total number of nodes (network order) of V f,t is N t = (f + 1) t+1 . After introducing the Vicsek fractals, we now investi-

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