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High-throughput techniques are leading to an explosive growth in the size of biological databases and creating the opportunity to revolutionize our understanding of life and disease. Interpretation of these data remains, however, a major scientific challenge. Here, we propose a methodology that enables us to extract and display information contained in complex networks. Specifically, we demonstrate that one can (i) find functional modules in complex networks, and (ii) classify nodes into universal roles according to their pattern of intra- and inter-module connections. The method thus yields a ``cartographic representation'' of complex networks. Metabolic networks are among the most challenging biological networks and, arguably, the ones with more potential for immediate applicability. We use our method to analyze the metabolic networks of twelve organisms from three different super-kingdoms. We find that, typically, 80% of the nodes are only connected to other nodes within their respective modules, and that nodes with different roles are affected by different evolutionary constraints and pressures. Remarkably, we find that low-degree metabolites that connect different modules are more conserved than hubs whose links are mostly within a single module., Comment: 17 pages, 4 figures. Go to http://amaral.northwestern.edu for the PDF file of the reprint

We analyze the growth statistics of Swedish trade unions and find a universal functional form for the probability distribution of growth rates of union size, and a power law dependence of the standard deviation of this distribution on the number of members of the union. We also find that the typical size and the typical number of local chapters scales as a power law of the union size. Intriguingly, our results are similar to results reported for other human organizations of a quite different nature. Our findings are consistent with the possibility that universal mechanisms may exist governing the growth patterns of human organizations.

In a recent paper, James Holland Jones and Mark Handcock re-analyze two of the four datasets comprising a database, first analyzed by us, which records the number of distinct sexual partners for Swedish men and women. We argue that the claims of Jones and Handcock can be interpreted in a misleading fashion., Comment: 6 pages

We investigate the statistics of the most connected nodes in scale-free networks. For a scale-free network model with homogeneous nodes, we show by means of extensive simulations that the exponential truncation--due to the finite size of the network--of the degree distribution governs the scaling of the extreme values. We also find that the distribution of maxima obeys scaling and follows the Gumbel statistics. For a scale-free network model with heterogeneous nodes, we show that scaling no longer holds and that the truncation of the degree distribution no longer controls the maximum distribution. Moreover, we find that neither the Gumbel nor the Frechet statistics describe the data.

We formulate a general model for the growth of scale-free networks under filtering information conditions--that is, when the nodes can process information about only a subset of the existing nodes in the network. We find that the distribution of the number of incoming links to a node follows a universal scaling form, i.e., that it decays as a power law with an exponential truncation controlled not only by the system size but also by a feature not previously considered, the subset of the network ``accessible'' to the node. We test our model with empirical data for the World Wide Web and find agreement., Comment: LaTeX2e and RevTeX4, 4 pages, 4 figures. Accepted for publication in Physical Review Letters

We test whether the complexity of cardiac interbeat interval time series is simply a consequence of the wide range of scales characterizing human behavior, especially physical activity, by analyzing data taken from healthy adult subjects under three conditions with controls: (i) a ``constant routine'' protocol where physical activity and postural changes are kept to a minimum, (ii) sympathetic blockade, and (iii) parasympathetic blockade. We find that when fluctuations in physical activity and other behavioral modifiers are minimized, a remarkable level of complexity of heartbeat dynamics remains, while for neuroautonomic blockade the multifractal complexity decreases., Comment: 4 pages with 6 eps figures. Latex file. For more details or for downloading the PDF file of the published article see http://polymer.bu.edu/~amaral/Heart.html and http://polymer.bu.edu/~amaral/Multifractal.html

Many ``real-world'' networks are clearly defined while most ``social'' networks are to some extent subjective. Indeed, the accuracy of empirically-determined social networks is a question of some concern because individuals may have distinct perceptions of what constitutes a social link. One unambiguous type of connection is sexual contact. Here we analyze data on the sexual behavior of a random sample of individuals, and find that the cumulative distributions of the number of sexual partners during the twelve months prior to the survey decays as a power law with similar exponents $\alpha \approx 2.4$ for females and males. The scale-free nature of the web of human sexual contacts suggests that strategic interventions aimed at preventing the spread of sexually-transmitted diseases may be the most efficient approach., Comment: 7 pages with 2 eps figures. Latex file. For more details or for downloading the PDF file of the published article see http://polymer.bu.edu/~amaral/WebofContacts.html . For more results on teh structure of complex networks see http://polymer.bu.edu/~amaral/Networks.html

Stock price changes occur through transactions, just as diffusion in physical systems occurs through molecular collisions. We systematically explore this analogy and quantify the relation between trading activity - measured by the number of transactions $N_{\Delta t}$ - and the price change $G_{\Delta t}$, for a given stock, over a time interval $[t, t+\Delta t]$. To this end, we analyze a database documenting every transaction for 1000 US stocks over the two-year period 1994-1995. We find that price movements are equivalent to a complex variant of diffusion, where the diffusion coefficient fluctuates drastically in time. We relate the analog of the diffusion coefficient to two microscopic quantities: (i) the number of transactions $N_{\Delta t}$ in $\Delta t$, which is the analog of the number of collisions and (ii) the local variance $w^2_{\Delta t}$ of the price changes for all transactions in $\Delta t$, which is the analog of the local mean square displacement between collisions. We study the distributions of both $N_{\Delta t}$ and $w_{\Delta t}$, and find that they display power-law tails. Further, we find that $N_{\Delta t}$ displays long-range power-law correlations in time, whereas $w_{\Delta t}$ does not. Our results are consistent with the interpretation that the pronounced tails of the distribution of $G_{\Delta t} are due to $w_{\Delta t}$, and that the long-range correlations previously found for $| G_{\Delta t} |$ are due to $N_{\Delta t}$., Comment: RevTex 2 column format. 6 pages, 36 references, 15 eps figures

We study the distribution of fluctuations over a time scale $\Delta t$ (i.e., the returns) of the S&P 500 index by analyzing three distinct databases. Database (i) contains approximately 1 million records sampled at 1 min intervals for the 13-year period 1984-1996, database (ii) contains 8686 daily records for the 35-year period 1962-1996, and database (iii) contains 852 monthly records for the 71-year period 1926-1996. We compute the probability distributions of returns over a time scale $\Delta t$, where $\Delta t$ varies approximately over a factor of 10^4 - from 1 min up to more than 1 month. We find that the distributions for $\Delta t \leq$ 4 days (1560 mins) are consistent with a power-law asymptotic behavior, characterized by an exponent $\alpha \approx 3$, well outside the stable L\'evy regime $0 < \alpha < 2$. To test the robustness of the S&P result, we perform a parallel analysis on two other financial market indices. Database (iv) contains 3560 daily records of the NIKKEI index for the 14-year period 1984-97, and database (v) contains 4649 daily records of the Hang-Seng index for the 18-year period 1980-97. We find estimates of $\alpha$ consistent with those describing the distribution of S&P 500 daily-returns. One possible reason for the scaling of these distributions is the long persistence of the autocorrelation function of the volatility. For time scales longer than $(\Delta t)_{\times} \approx 4$ days, our results are consistent with slow convergence to Gaussian behavior., Comment: 12 pages in multicol LaTeX format with 27 postscript figures (Submitted to PRE May 20, 1999). See http://polymer.bu.edu/~amaral/Professional.html for more of our work on this area

We use methods of random matrix theory to analyze the cross-correlation matrix C of price changes of the largest 1000 US stocks for the 2-year period 1994-95. We find that the statistics of most of the eigenvalues in the spectrum of C agree with the predictions of random matrix theory, but there are deviations for a few of the largest eigenvalues. We find that C has the universal properties of the Gaussian orthogonal ensemble of random matrices. Furthermore, we analyze the eigenvectors of C through their inverse participation ratio and find eigenvectors with large inverse participation ratios at both edges of the eigenvalue spectrum--a situation reminiscent of results in localization theory., Comment: 14 pages, 3 figures, Revtex

We comment on a recent Letter by Maunuksela et al. [Phys. Rev. Lett. 79, 1515 (1997)]., Comment: 1 page, 1 figure, http://polymer.bu.edu/~hmakse/Home.html