Europhys. Lett., 72 (4), pp. 541–547 (2005) DOI: 10.1209/epl/i2005-10284-x EUROPHYSICS LETTERS 15 November 2005 Fluctuation of incoming ﬂux with multiplicative noise on a scale-free network S. H. Yook( ∗ ) and M. A. de Menezes( ∗∗ ) Department of Physics, University of Notre Dame - Notre Dame, IN 46556, USA received 3 March 2005; accepted in ﬁnal form 22 September 2005 published online 14 October 2005 PACS. 05.70.Ln – Nonequilibrium and irreversible thermodynamics. PACS. 89.75.Hc – Networks and genealogical trees. PACS. 89.75.Da – Systems obeying scaling laws. Abstract. – We study the inﬂuence of topology on the dynamical properties of a diﬀusion pro- cess which can be described by a diﬀusion equation with multiplicative noise on a complex net- work. From numerical simulations we ﬁnd that the ﬂuctuation of the incoming mass on a given node of network scales with the average incoming mass, or ﬂux, in a topology-dependent fashion. By combining numerical results with the Langevin equation of the associated process, we show that inhomogeneity of the underlying structure leads to the appearance of distinct dynamical regions in the system and a crossover behavior in the scaling of ﬂuctuations with average ﬂux. Nonequilibrium statistical mechanics has been an intensively studied subject [1,2], in var- ious physical systems [3–10]. One example of nonequilibrium phenomena which has received great attention in the last decade is self-organized criticality with potential relevance to the scale-invariant features observed in many systems such as the sand pile model [11], river net- works [12–14] and growth dynamics [5]. An early version of self-organized criticality was proposed by Takayasu [15,16]. The model is believed to be relevant to economic systems be- cause its diﬀusion process with aggregation, deposition and evaporation of particles resembles the “eﬃciency” dynamics of competing agents in economic systems [17]. In many cases such as population dynamics or wealth dynamics, it is more natural to assume that the amount of diﬀusion can be described by the multiplicative noise [18]. Moreover, it is well known that the diﬀusive systems with multiplicative noise can be characterized by generalized Levy-Pareto distribution [18–21] and most of the previous works have been focused on the generation of power law by multiplicative noise. In order to make a direct comparison between a model system and empirical observations, however, one should understand how the underlying interaction topology between agents af- fects their dynamical behavior. The eﬀects of the underlying topology of dynamical properties such as relaxation time, the autocorrelation function and the return probability of random ( ∗ ) Permanent address: Department of Physics and Research Institute for Basic Sciences, Kyung Hee University - Seoul 130-701, Korea. ( ∗∗ ) Current address: Instituto de Fisica, Universidade Federal Fluminense, Campus da Praia Vermelha - Av. Litorˆ anea, s/n, Boa Viagem 24210-340, Niter´oi, RJ, Brazil. c  EDP Sciences Article published by EDP Sciences and available at http://www.edpsciences.org/epl or http://dx.doi.org/10.1209/epl/i2005-10284-x