Shiner JS, Davison M. Quantifying the connectivity of scale-free and biological networks. Chaos, Solitons & Fractals. 2004 Jul;21(1):1-8.
Davison M, Essex C, Shiner JS. Global predictability of chaotic epidemiological dynamics in coupled populations. Open Systems & Information Dynamics. 2003 Dec;10(4):311-320.
Davison M, Shiner JS. Many entropies, many disorders. Open Systems & Information Dynamics. 2003 Sept;10(3):281-296.
ABSTRACTS:
Global Predictability of Chaotic Epidemiological Dynamics in Coupled Populations Matt Davison and C. Essex. Department of Applied Mathematics, University of Western Ontario, London, Ontario, Canada,N6A 5B7 J. S. Shiner. The Shiner Group, Bergacher 3, CH-3325 Hettiswil, Switzerland; Abteilung Mathematik und Ingenieurwissenschaften, Fernfachhochschule Schweiz, CH-3900 Brig, Switzerland; Institut für mathematische Statistik und Versicherungslehre der Universität Bern, CH-3012 Bern, Switzerland Abstract. When the dynamics of an epidemic are chaotic, detailed prediction is effectively impossible, except perhaps in the short term. However, a probability distribution underlying the motion does allow for the long term prediction of statistical measures such as the mean or the standard deviation. Even this weaker long term predictability might be lost if distinct populations with chaotic dynamics are coupled. We show that such coupling can result in a phenomenon we call "sensitive dependence on neglected dynamics". In light of this phenomenon, it is somewhat surprising that when two logistic maps are coupled, the long term predictability of the mean and standard deviation is maintained. This is true even though the probability distribution describing the time series depends on the coupling strength. The coupling-strength dependence does reveal itself in the loss of predictability of higher order moments such as skewness and kurtosis. In Press in Open Sys. & Information Dyn. http://www.kluweronline.com/issn/1230-1612 http://www.phys.uni.torun.pl/zfmis/osid/ Many Entropies, Many Disorders Matt Davison; Department of Applied Mathematics, University of Western Ontario, London, Ontario, Canada, N6A 5B7 J. S. Shiner. The Shiner Group, Bergacher 3, CH-3325 Hettiswil, Switzerland; Abteilung Mathematik und Ingenieurwissenschaften, Fernfachhochschule Schweiz, CH-3900 Brig, Switzerland; Institut für Mathematische Statistik und Versicherungslehre der Universität Bern, CH-3012 Bern, Switzerland Abstract. To overcome the deficits of entropy as a measure for disorder when the number of states available to a system can change, Landsberg defined "disorder" as the entropy normalized to the maximum entropy. In the simplest cases, the maximum entropy is that of the equiprobable distribution, corresponding to a completely random system. However, depending on the question being asked and on system constraints, this absolute maximum entropy may not be the proper maximum entropy. To assess the effects of interactions on the "disorder" of a 1-dimensional spin system, the correct maximum entropy is that of the paramagnet (no interactions) with the same net magnetization; for a non-equilibrium system the proper maximum entropy may be that of the corresponding equilibrium system; and for hierarchical structures, an appropriate maximum entropy for a given level of the hierarchy is that of the system which is maximally random, subject to constraints deriving from the next lower level. Considerations of these examples leads us to introduce the "equivalent random system": that system which is maximally random consistent with any constraints and with the question being asked. It is the entropy of the "equivalent random system" which should be taken as the maximum entropy in Landsberg`s "disorder". Open Systems and Information Dynamics 10 (3): 281-296, September 2003 http://ipsapp008.kluweronline.com/content/getfile/5062/27/1/abstract.htm J. S. Shiner. The Shiner Group, Bergacher 3, CH-3325 Hettiswil, Switzerland; Abteilung Mathematik und Ingenieurwissenschaften, Fernfachhochschule Schweiz, CH-3900 Brig, Switzerland; Institut für Mathematische Statistik und Versicherungslehre der Universität Bern, CH-3012 Bern, Switzerland Matt Davison. Department of Applied Mathematics, University of Western Ontario, London, Ontario, Canada, N6A 5B7 Abstract In Press: Chaos, Solitons and Fractals
Quantifying the connectivity of scale-free and biological networks
Scale-free and biological networks follow a power law distribution pk k-a for the probability that a node is connected to k other nodes; the corresponding ranges for a (biological: 1 < a < 2; scale-free: 2 < a 3) yield a diverging variance for the connectivity k and lack of predictability for the average connectivity. Predictability can be achieved with the Rényi, Tsallis and LandsbergVedral extended entropies and corresponding ‘‘disorders’’ for correctly chosen values of the entropy index q. Escort distributions pk k-aq with q > 3/a also yield a nondiverging variance and predictability. It is argued that the Tsallis entropies may be the appropriate quantities for the study of scale-free and biological networks.
http://authors.elsevier.com/sd/article/S096007790300571X
