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Several real-world directed networks do not have multiple links. For example, in a paper citation network a paper does not cite two identical references, and in a network of friends there exists only a single link between two individuals. This suggest that the growth and evolution models of complex networks should take into account such feature in order to approximate the topological properties of this class of networks. The aim of this paper is to propose a growth model of directed complex networks that takes into account the prohibition of the existence multiple links. It is shown through numerical experiments that when multiple links are forbidden, the exponent γ of the in-degree connectivity distribution, , takes values ranging from 1 to ∞. In particular, the proposed multi-link free (MLF) model is able to predict exponents occurring in real-world complex networks, which range 1.05 < γ < 3.51. As an example, the MLF reproduces somxe topological properties exhibited by the network of flights between airports of the world (NFAW); i.e. γ ≈ 1.74. With this result, we believe that the multiple links prohibition might be one of the local processes accounting for the existence of exponents γ < 2 found in some real complex networks.

We propose a simple network growth process where the preferential attachment contains two essential parameters: homophily, namely, the tendency of sites to link with similar ones, and the number of attaching neighbors. It jointly generalizes the Barabási–Albert model and the scale-free homophilic model with a control parameter which tunes the importance of the homophily on preferential attachment process. Our results support a detailed discussion about different kinds of correlation, in special a fitness correlation introduced in this paper, and comparisons between BA model, scale-free homophilic model, and our present model considering its topological properties: degree distribution, time dependence of the connectivity and clustering coefficient.

We present a fragmentation model that generates almost any inverse power-law size distribution, including dual-scaled versions, consistent with the underlying dynamics of systems with earthquake-like behavior. We apply the model to explain the dual-scaled power-law statistics observed in an Internet access dataset that covers more than 32 million requests. The non-Poissonian statistics of the requested data sizes $m$ and the amount of time $\tau$ needed for complete processing are consistent with the Gutenberg–Richter–law. Inter-event times $\delta t$ between subsequent requests are also shown to exhibit power-law distributions consistent with the generalized Omori law. Thus, the dataset is similar to the earthquake data except that two power-law regimes are observed. Using the proposed model, we are able to identify underlying dynamics responsible in generating the observed dual power-law distributions. The model is universal enough for its applicability to any physical and human dynamics that is limited by finite resources such as space, energy, time or opportunity.

In this paper we present a study of the influence of heterogeneity on the clustering of preferential attachment networks. The study is performed by the numerical analysis of the threshold preferential attachment model, a generalization of the Barabási—Albert model to heterogeneous complex networks. Heterogeneous networks are characterized by the existence of intrinsic properties of the nodes which induce specific affinities in their interactions. We analyze the influence of the affinity parameters on the distribution of degree-averaged clustering coefficients of the threshold model. We show that the introduction of heterogeneity increases the inverse correlation between clustering and connectivity of the nodes, inducing a power-law scaling in the clustering distribution. We also show that a higher level of heterogeneity increases the overall clustering coefficients irrespective of the node degrees. These results exhibit a better agreement of the extended model with the empirical observations of clustering in real networks.

This work investigates a simple inhomogeneous cosmological model within the Lema$\widehat{\mathrm{\text{i}}}$tre–Tolman–Bondi (LTB) metric. The mass-scale function of the LTB model is taken to be $M(r)\propto r^d$ and would correspond to a fractal distribution for $0<d<3$. The luminosity distance for this model is computed and then compared to supernovae (SNe) data. Unlike LTB models which have, in the most general case, two free functions, our model has only two free parameters as the flat Standard Model of cosmology. The best-fit obtained is a matter distribution with an exponent of $d=3.44$ revealing that SNe data do not favor those fractal models.

The distance distributions of the four types of bases A, C, G and T in the complete sequences of human genome are shown to have long-tail power-law but short-distance exponential behaviors. However, the random sequences with identical numbers of bases show only short-distance exponential behaviors. The DNA sequence of E. coli, which is much shorter than human's, is shown to exhibit essentially exponential behavior as its corresponding random sequence. Also, DNA sequence of human's with smaller C + G content is seen to enjoy longer power-law tail in its distributions of C and G. The coefficient in the exponential decaying function shows a linear dependent with the C + G content but the decay exponent in the power-law shows no such dependence.

We perform a phenomenological study of stock price fluctuations of individual companies. We systematically analyze two different databases covering securities from the three major US stock markets. We consider (i) the trades and quotes (TAQ) database, for which we analyze 40 million records for 1000 US companies for the 2-year period 1994–95, and (ii) the Center for Research and Security Prices (CRSP) database, for which we analyze 35 million daily records for approximately 16,000 companies in the 35-year period 1962–96. We study the probability distribution of returns over varying time scales — from 5 min up to 4 years. For time scales from 5 min up to approximately 16 days, we find that the tails of the distributions can be well described by a power-law decay, characterized by an exponent α ≈ 3 — well outside the stable Lévy regime 0 < α < 2. For time scales greater than 16 days, we observe results consistent with a slow convergence to Gaussian behavior.

In recent years, a considerable number of physicists have started applying physics concepts and methods to understand economic phenomena. The term "Econophysics" is sometimes used to describe this work. Economic fluctuations can have many repercussions, and understanding fluctuations is a topic that many physicists have contributed to in recent years. Further, economic systems are examples of complex interacting systems for which a huge amount of data exist and it is possible that the experience gained by physicists in studying fluctuations in physical systems might yield new results in economics. Much recent work in econophysics is focused on understanding the peculiar statistical properties of price fluctuations in financial time series. In this talk, we discuss three recent results. The first result concerns the probability distribution of stock price fluctuations. This distribution decreases with increasing fluctuations with a power-law tail well outside the Lévy stable regime and describes fluctuations that differ by as much as 8 orders of magnitude. Further, this nonstable distribution preserves its functional form for fluctuations on time scales that differ by 3 orders of magnitude, from 1 min up to approximately 10 days. The second result concerns the accurate quantification of volatility correlations in financial time series. While price fluctuations themselves have rapidly decaying correlations, the volatility estimated by using either the absolute value or the square of the price fluctuations has correlations that decay as a power-law and persist for several months. The third result bears on the application of random matrix theory to understand the correlations among price fluctuations of any two different stocks. We compare the statistics of the cross-correlation matrix constructed from price fluctuations of the leading 1000 stocks and a matrix with independent random elements, i.e., a random matrix. Contrary to first expectations, we find little or no deviation from the universal predictions of random matrix theory for all but a few of the largest eigenvalues of the cross-correlation matrix.

We introduce a model of inflation in which the prices of commodities are inflated by a random process. At each time step a price x is selected with a rate of x^α and is inflated by a factor of 1/β where 0<β<1. For α=0, we obtain a general time-dependent solution, where the initial price distribution can be of any form. When α>0, in the long time limit, only the highest price inflates. For α<0, the model exhibits asymptotic scaling behavior. We also consider the effects of a time dependent β, where 0<β(t)<1, for the case α=0. We find that the price distribution approaches a steady state if β(t)-1~0 faster than 1/t.