No abstract

Library of Congress Cataloging-in-Publication Data: Nakayama, T. (Tsuneyoshi), 1945-. Fractal concepts in condensed matter physics/ T. Nakayama, K. Yakubo. p. cm. -(Springer series in solid-state sciences, ISSN 0171-1873; 140). Includes bibliographical references and index.(alk.paper) 1. Condensed matter-Mathematics. 2.Fractals. I.

We demonstrate analytically and numerically the possibility that the fractal property of a scale-free network cannot be characterized by a unique fractal dimension and the network takes a multifractal structure. It is found that the mass exponents τ (q) for several deterministic, stochastic, and real-world fractal scale-free networks are nonlinear functions of q, which implies that structural measures of these networks obey the multifractal scaling. In addition, we give a general expression of τ (q) for some class of fractal scale-free networks by a mean-field approximation. The multifractal property of network structures is a consequence of large fluctuations of local node density in scale-free networks.

Using a geographical scale-free network to describe relations between people in a city, we explain both superlinear and sublinear allometric scaling of urban indicators that quantify activities or performances of the city. The urban indicator Y (N ) of a city with the population size N is analytically calculated by summing up all individual activities produced by person-to-person relationships. Our results show that the urban indicator scales superlinearly with the population, namely, Y (N ) ∝ N β with β > 1, if Y (N ) represents a creative productivity and the indicator scales sublinearly (β < 1) if Y (N ) is related to the degree of infrastructure development. These results coincide with allometric scaling observed in real-world urban indicators. We also show how the scaling exponent β depends on the strength of the geographical constraint in the network formation.

The fractal and the small-world properties of complex networks are systematically studied both in the box-covering ͑BC͒ and the cluster-growing ͑CG͒ measurements. We elucidate that complex networks possessing the fractal ͑small-world͒ nature in the BC measurement are always fractal ͑small world͒ even in the CG measurement and vice versa, while the fractal dimensions d B by the BC measurement and d C by the CG measurement are generally different. This implies that two structural properties of networks, fractality and small worldness, cannot coexist in the same length scale. These properties can, however, crossover from one to the other by varying the length scale. We show that the crossover behavior in a network near the percolation transition appears both in the BC and CG measurements and is scaled by a unique characteristic length .