Correlation in complex networks follows a linear relation between the degree of a node and the total degrees of its neighbors for six different classes of real networks. This general linear relation is an extension of the Aboav-Weaire law in two-dimensional cellular structures and provides a simple and different perspective on the correlation in complex networks, which is complementary to an existing description using Pearson correlation coefficients and a power law fit. Analytical expression for this linear relation for three standard models of complex networks: the Erdos-Renyi, Watts-Strogatz, and Barabasi-Albert networks is provided. The slope and intercept of this linear relation are described by a single parameter a together with the first and second moment of the degree distribution of the network. The assortivity of the network can be related to the sign of the intercept. Complex networks are a convenient model for studying the topological and structural property of complex systems ͓1,2͔. It consists of nodes which interact among themselves via their connections. Mathematical quantities such as the degree distribution, cluster coefficient, and the average shortest path length are the standard properties in a preliminary characterization of networks. In this paper, we focus on the neighbor connectivity that relates to the degree correlation among the nodes. By extending the Aboav-Weaire Law ͓3,4͔, which was well studied for two-dimensional cellular patterns such as soap froth ͓5,6͔, we find that it is also a good measure to describe the correlation in a wide variety of real and artificial networks in higher dimensions. The complex networks that we have checked are the neural networks ͓7͔, food webs ͓8͔, word co-occurrence ͓9,10͔, scientist collaboration ͓11͔, internet ͓12͔, and yeast protein interaction ͓13͔.We also report here the Aboav's parameters for several standard models: the Erdos-Renyi ͓14͔, Watts-Strogatz ͓15͔, and Barabasi-Albert networks ͓16͔. We find that the results on the assortivity of the network using the Aboav-Wearie law are consistent with conclusions based on the analysis of the Pearson correlation coefficient.Originally, the Aboav-Weaire law was discovered from the empirical analysis of two-dimensional cellular structures in metal grains and later extended to a variety of cellular structures in two and three dimensions. The findings of Aboav provide a linear relation between the total degrees of nearest neighbor nM͑n͒ with the degree of the cell nM͑n͒ =5n + 8, where M͑n͒ is the mean of number of edges of neighboring cells surrounding a cell with n edges ͓3͔. Weaire generalizes Aboav's observation and restates this observation in terms of the variance 2 of the degree distribution of the cellular network ͓4͔:withThis form for the expression of M͑n͒ is usually tested empirically by a plot of nM͑n͒ vs n, which should be linear with slope A and intercept B. As the Aboav-Weaire law can be understood as a statement on the topological correlation of the cellular network, we attempt to general...

We report an experimental measurement of the temporal dependence of the area A us in a twodimensional soap froth which has not been swept by the passage of soap films up to time t, as the froth coarsens from an initial time t 0 within the scaling regime. We find A us scales with the mean cell area ͗A͘ as A us~͗ A͘ 2u 0 , with a first-passage exponent u 0 1. The study of domain coarsening following a quench from a high-temperature disordered phase to a low temperature ordered one is relevant in many areas of physics ranging from the evolution of crystalline domains in metallurgy, to the initial stages of evolution of the Universe in cosmology [1]. Typically, domain patterns evolve into a scaling regime in which statistical pattern properties are time invariant and the average domain size increases as a power law t a . While these features have been well established experimentally and theoretically in classic statistical mechanical problems such as spinodal decomposition and Lifshitz-Slyozov processes, attention has been shifted more recently to properties related to the zerotemperature relaxational dynamics of certain theoretical models, which being unrelated to the usual response functions, are characterized by new nontrivial universal exponents [2-4]. These properties are related to the survival probability of domains, namely, how long until they are overrun by domain walls. The exponents have been calculated in various ways in the case of one-dimensional q-state Potts models [5][6][7], and estimated numerically from computer simulations of the two-dimensional versions of these models [8]. An approximate method for computing these exponents for arbitrary dimensions (for the Ohta-JasnowKawasaki model [9]) was given in [10,11].To our knowledge, only one experimental work has addressed the measurement of an exponent of this type in a physical system [12]. The system consists of patterns of droplets condensing on a surface called breath figures, whose evolution is driven by droplet condensation and coalescence. Breath figures cannot be adequately modeled by domain coarsening of Potts models, and therefore a comparison between existing theoretical predictions based on these models and the experimental results was precluded.In this paper, we present the results of an experimental measurement of exponents related to the zero-temperature Potts models. Our experiment was performed on a two-dimensional soap froth, a system whose evolution is well mimicked by q-state Potts model for large q

We investigate the statistical properties of two-dimensional random cellular systems ͑froths͒ in terms of their shell structure. The froth is analyzed as a system of concentric layers of cells around a given central cell. We derive exact analytical relations for the topological properties of the sets of cells belonging to these layers. Experimental observations of the shell structure of two-dimensional soap froth are made and compared with the results on two kinds of Voronoi constructions. It is found that there are specific differences between soap froths and purely geometrical constructions. In particular these systems differ in the topological charge of clusters as a function of shell number, in the asymptotic values of defect concentrations, and in the number of cells in a given layer. We derive approximate expressions with no free parameters which correctly explain these different behaviors.