After a very short introduction to some of the more basic unanswered questions in the field of complex systems, we consider the problem of "rare events". At one time such rare events were considered "statistical outliers" because they did not conform to known probability distribution functions. Nowadays it is becoming widely appreciated that even extremely rare events may not be "outliers" but rather may conform to newlyuncovered empirical laws, such as the various power laws characterizing scale invariant phenomena. Further, these laws appear to be "universal" in the sense that they hold across a range of widely different phenomena, consistent with the intriguing possibility that these phenomena have some underlying features in common.
We will illustrate this feature by discussing a few examples drawn from the social sciences, economics, and the physical sciences. For example, in economics, we have demonstrated a power law distribution of returns with exponent 3, outside the Levystable regime. This behavior appears to be "universal" in the sense that it holds for diverse financial markets worldwide, and for different market conditions over the past 75 years. In particular, it appears to encompass economic fluctuations measured to date, including data taking place in times of market crashes [14].
Another example concerns social networks, for which the data on sexual networks appear to be the most nonsubjective [56]. We also discuss focus on a number of topics in threat networks (Al Qaeda) and threatened networks (computer networks, and SARSsusceptible networks), and we argue that the "six degrees of separation" rule that the pathlength scales as the logarithm of the number of nodes N is replaced by a "100 degrees of separation" rule, that the path length scales as a power law N**(1/3) for conditions of strong disorder where there is a distribution of costs for each link of the transmission process as occurs in, e.g., real networks [7].
We also discuss how interdisciplinary "social scientist/physical scientist" collaborations are beginning to gain theoretical insight and understanding of these new empirical laws using concepts drawn from both the social sciences and the physical sciences.
The research reported was done primarily in collaboration with Y. Aberg, L. A. N. Amaral, L. Braunstein, S. V. Buldyrev, R. Cohen, C. Edling, X. Gabaix, S. Havlin, P. Gopikrishnan, F. Liljeros, and V. Plerou. and has been supported by ONR and NSF.
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[2] V. Plerou, P. Gopikrishnan, X. Gabaix, and H. E. Stanley, "Quantifying Stock Price Response to Demand Fluctuations," Phys. Rev. E {\bf 66}, 0271041  0271044 (2002) condmat 0106657.
[3] V. Plerou, P. Gopikrishnan, and H. E. Stanley, "TwoPhase Behaviour of Financial Markets" Nature {\bf 421}, 130 (2003). condmat/0111349.
[4] X. Gabaix, P. Gopikrishnan, V. Plerou, and H. E. Stanley, "A Theory of PowerLaw Distributions in Financial Market Fluctuations," Nature {\bf 423}, 267270 (2003).
[5] F. Liljeros, C. R. Edling, L. A. N. Amaral, H. E. Stanley, and Y. Aberg, "The Web of Human Sexual Contacts" Nature 411, 907908 (2001) condmat/0106507.
[6] Fredrik Liljeros, Christofer R. Edling, H. Eugene Stanley, Y. Aberg, Luis A. Nunes Amaral, "Distributions of number of sexual partnerships have power law decaying tails and finite variance," http://arxiv.org/pdf/condmat/0305528
[7] Lidia A. Braunstein, Sergey V. Buldyrev, Reuven Cohen, Shlomo Havlin, and H. Eugene Stanley, "Optimal Paths in Disordered Complex Networks". Phys. Rev. Letters (accepted). condmat/0305051
