scholarly journals Information Geometry of Nonlinear Stochastic Systems

Entropy ◽  
2018 ◽  
Vol 20 (8) ◽  
pp. 550 ◽  
Author(s):  
Rainer Hollerbach ◽  
Donovan Dimanche ◽  
Eun-jin Kim
Keyword(s):  
Geometric Structure ◽  
Stochastic Systems ◽  
Power Laws ◽  
Information Geometry ◽  
Scaling Exponent ◽  
Stochastic Noise ◽  
Initial Stage ◽  
Constant Rate ◽  
Information Change ◽  
Transient Relaxation

We elucidate the effect of different deterministic nonlinear forces on geometric structure of stochastic processes by investigating the transient relaxation of initial PDFs of a stochastic variable x under forces proportional to -xn (n=3,5,7) and different strength D of δ-correlated stochastic noise. We identify the three main stages consisting of nondiffusive evolution, quasi-linear Gaussian evolution and settling into stationary PDFs. The strength of stochastic noise is shown to play a crucial role in determining these timescales as well as the peak amplitude and width of PDFs. From time-evolution of PDFs, we compute the rate of information change for a given initial PDF and uniquely determine the information length L(t) as a function of time that represents the number of different statistical states that a system evolves through in time. We identify a robust geodesic (where the information changes at a constant rate) in the initial stage, and map out geometric structure of an attractor as L(t→∞)∝μm, where μ is the position of an initial Gaussian PDF. The scaling exponent m increases with n, and also varies with D (although to a lesser extent). Our results highlight ubiquitous power-laws and multi-scalings of information geometry due to nonlinear interaction.

scholarly journals RECIPROCITY AND THE EMERGENCE OF POWER LAWS IN SOCIAL NETWORKS

2006 ◽  
Vol 17 (07) ◽  
pp. 1067-1076 ◽  
Author(s):  
MICHAEL SCHNEGG
Keyword(s):  
Social Networks ◽  
Power Laws ◽  
Scaling Exponent ◽  
Policy Makers ◽  
Public Health Programs ◽  
Scale Free ◽  
The Past ◽  
Naturally Occurring ◽  
Growing Networks ◽  
Law Degree

Research in network science has shown that many naturally occurring and technologically constructed networks are scale free, that means a power law degree distribution emerges from a growth model in which each new node attaches to the existing network with a probability proportional to its number of links (= degree). Little is known about whether the same principles of local attachment and global properties apply to societies as well. Empirical evidence from six ethnographic case studies shows that complex social networks have significantly lower scaling exponents γ ~ 1 than have been assumed in the past. Apparently humans do not only look for the most prominent players to play with. Moreover cooperation in humans is characterized through reciprocity, the tendency to give to those from whom one has received in the past. Both variables — reciprocity and the scaling exponent — are negatively correlated (r = -0.767, sig = 0.075). If we include this effect in simulations of growing networks, degree distributions emerge that are much closer to those empirically observed. While the proportion of nodes with small degrees decreases drastically as we introduce reciprocity, the scaling exponent is more robust and changes only when a relatively large proportion of attachment decisions follow this rule. If social networks are less scale free than previously assumed this has far reaching implications for policy makers, public health programs and marketing alike.

scholarly journals Isometric Signal Processing under Information Geometric Framework

Entropy ◽  
2019 ◽  
Vol 21 (4) ◽  
pp. 332 ◽  
Author(s):  
Hao Wu ◽  
Yongqiang Cheng ◽  
Hongqiang Wang
Keyword(s):  
Signal Processing ◽  
Probability Distribution ◽  
Geometric Structure ◽  
Information Geometry ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Statistical Manifold ◽  
Intrinsic Parameter ◽  
Geometric Framework ◽  
Necessary And Sufficient

Information geometry is the study of the intrinsic geometric properties of manifolds consisting of a probability distribution and provides a deeper understanding of statistical inference. Based on this discipline, this letter reports on the influence of the signal processing on the geometric structure of the statistical manifold in terms of estimation issues. This letter defines the intrinsic parameter submanifold, which reflects the essential geometric characteristics of the estimation issues. Moreover, the intrinsic parameter submanifold is proven to be a tighter one after signal processing. In addition, the necessary and sufficient condition of invariant signal processing of the geometric structure, i.e., isometric signal processing, is given. Specifically, considering the processing with the linear form, the construction method of linear isometric signal processing is proposed, and its properties are presented in this letter.

scholarly journals Information Geometry of Spatially Periodic Stochastic Systems

Entropy ◽  
2019 ◽  
Vol 21 (7) ◽  
pp. 681
Author(s):  
Rainer Hollerbach ◽  
Eun-jin Kim
Keyword(s):  
Equilibrium Point ◽  
Stochastic Systems ◽  
Initial Conditions ◽  
Planck Equation ◽  
Information Geometry ◽  
Initial Position ◽  
Equilibrium Solutions ◽  
Gaussian Peak ◽  
Unstable Fixed Point ◽  
Spatially Periodic

We explore the effect of different spatially periodic, deterministic forces on the information geometry of stochastic processes. The three forces considered are f 0 = sin ( π x ) / π and f ± = sin ( π x ) / π ± sin ( 2 π x ) / 2 π , with f - chosen to be particularly flat (locally cubic) at the equilibrium point x = 0 , and f + particularly flat at the unstable fixed point x = 1 . We numerically solve the Fokker–Planck equation with an initial condition consisting of a periodically repeated Gaussian peak centred at x = μ , with μ in the range [ 0 , 1 ] . The strength D of the stochastic noise is in the range 10 - 4 – 10 - 6 . We study the details of how these initial conditions evolve toward the final equilibrium solutions and elucidate the important consequences of the interplay between an initial PDF and a force. For initial positions close to the equilibrium point x = 0 , the peaks largely maintain their shape while moving. In contrast, for initial positions sufficiently close to the unstable point x = 1 , there is a tendency for the peak to slump in place and broaden considerably before reconstituting itself at the equilibrium point. A consequence of this is that the information length L ∞ , the total number of statistically distinguishable states that the system evolves through, is smaller for initial positions closer to the unstable point than for more intermediate values. We find that L ∞ as a function of initial position μ is qualitatively similar to the force, including the differences between f 0 = sin ( π x ) / π and f ± = sin ( π x ) / π ± sin ( 2 π x ) / 2 π , illustrating the value of information length as a useful diagnostic of the underlying force in the system.

scholarly journals A study of the initial stage of oil spreading on calm water through methods of both instantaneous and constant rate of oil release.

1983 ◽  
Vol 16 (6) ◽  
pp. 502-507
Author(s):  
TETSUO AKIYAMA ◽  
SATORU MITSUMORI ◽  
TORU TERADA ◽  
KOZO KOIDE
Keyword(s):  
Initial Stage ◽  
Constant Rate ◽  
Calm Water

scholarly journals Porous Structure Evolution of Cellulose Carbon Fibres during Oxidation in Carbon Dioxide

2003 ◽  
Vol 21 (4) ◽  
pp. 363-371 ◽  
Author(s):  
Krzysztof Babeł
Keyword(s):  
Porous Structure ◽  
Isothermal Oxidation ◽  
Micropore Volume ◽  
Structure Evolution ◽  
Regenerated Cellulose ◽  
Porous Structures ◽  
Carbon Fibres ◽  
Initial Stage ◽  
Constant Rate ◽  
Third Stage

The isothermal oxidation of regenerated cellulose carbon fibres in the presence of CO2 or steam was described, together with the changes in the porous structures of the active fibres that evolve during different stages of the process. Three such stages were defined. In the initial stage of heating and out-gassing, changes in the porous structure were related to the violent pyrolysis that occurred. As a result, a considerable number of micropores (accessible to nitrogen) were generated together with a small number of mesopores. The next stage (principal activation stage) involved burning at a constant rate accompanied by an uniform increase in the micropore volume. In the third stage, an increased rate of oxidation was observed. This was accompanied by the development of transitional pores linked with the simultaneous limitation of micropore evolution. This stage was not efficient.

DEXP: A fast method to determine the depth and the structural index of potential fields sources

Geophysics ◽  
2007 ◽  
Vol 72 (1) ◽  
pp. I1-I11 ◽  
Author(s):  
Maurizio Fedi
Keyword(s):  
Potential Field ◽  
Extreme Points ◽  
Power Laws ◽  
Potential Fields ◽  
Newtonian Potential ◽  
Specific Power ◽  
Scaling Exponent ◽  
Fast Method ◽  
Structural Index ◽  
Derivatives Of

We show that potential fields enjoy valuable properties when they are scaled by specific power laws of the altitude. We describe the theory for the gravity field, the magnetic field, and their derivatives of any order and propose a method, called here Depth from Extreme Points (DEXP), to interpret any potential field. The DEXP method allows estimates of source depths, density, and structural index from the extreme points of a 3D field scaled according to specific power laws of the altitude. Depths to sources are obtained from the position of the extreme points of the scaled field, and the excess mass (or dipole moment) is obtained from the scaled field values. Although the scaling laws are theoretically derived for sources such as poles, dipoles, lines of poles, and lines of dipoles, we give also criteria to estimate the correct scaling law directly from the data. The scaling exponent of such laws is shown to be related to the structural index involved in Euler Deconvolution theory. The method is fast and stable because it takes advantage of the regular behavior of potential field data versus the altitude [Formula: see text]. As a result of stability, the DEXP method may be applied to anomalies with rather low SNRs. Also stable are DEXP applications to vertical and horizontal derivatives of a Newtonian potential of various orders in which we use theoretically determined scaling functions for each order of a derivative. This helps to reduce mutual interference effects and to obtain meaningful representations of the distribution of sources versus depth, with no prefiltering. The DEXP method does not require that magnetic anomalies to be reduced to the pole, and meaningful results are obtained by processing its analytical signal. Application to different cases of either synthetic or real data shows its applicability to any type of potential field investigation, including geological, petroleum, mining, archeological, and environmental studies.

scholarly journals POWER LAWS ARE LOGARITHMIC BOLTZMANN LAWS

1996 ◽  
Vol 07 (04) ◽  
pp. 595-601 ◽  
Author(s):  
MOSHE LEVY ◽  
SORIN SOLOMON
Keyword(s):  
Experimental Data ◽  
Monte Carlo ◽  
Degrees Of Freedom ◽  
Stochastic Systems ◽  
Power Laws ◽  
Random Processes ◽  
Fine Tuning ◽  
Power Law Distribution ◽  
Natural Origin ◽  
Dispersion Law

Multiplicative random processes in (not necessarily equilibrium or steady state) stochastic systems with many degrees of freedom lead to Boltzmann distributions when the dynamics is expressed in terms of the logarithm of the elementary variables. In terms of the original variables this gives a power-law distribution. This mechanism implies certain relations between the constraints of the system, the power of the distribution and the dispersion law of the fluctuations. These predictions are validated by Monte Carlo simulations and experimental data. We speculate that stochastic multiplicative dynamics might be the natural origin for the emergence of criticality and scale hierarchies without fine-tuning.

scholarly journals A Study of Cake Filtration Parameters Using the Constant Rate Process

Processes ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 746 ◽  
Author(s):  
Faiz Mahdi ◽  
Timothy Hunter ◽  
Richard Holdich
Keyword(s):  
Calcium Carbonate ◽  
Filter Cake ◽  
Cake Filtration ◽  
Initial Stage ◽  
Constant Rate ◽  
Medium Resistance ◽  
Material Sample ◽  
The Mean ◽  
Analytical Approaches ◽  
Constant Rate Process

The minerals calcium carbonate and talc were filtered under various conditions of filtrate flow rate and suspension concentration, using constant rate conditions with the aid of a peristaltic pump to draw the filtrate. Cake concentrations of between 0.41 and 0.53 v/v for calcium carbonate and 0.19 and 0.26 v/v for talc were recorded. The mean sizes of the two different minerals were very similar, but the average specific resistances obtained from the experiments were 5.9 × 1010 and 7.4 × 1011 m/kg for calcium carbonate and talc, respectively. These results do not agree with what would be predicted from an analytical equation for permeability, such as Kozeny-Carman. In addition, discontinuities were observed in all cases on the curves of filtrate volume with time for the initial stage of filtration. This behaviour is attributed to retarded packing compressibility (RPC) complicating the analysis of the filter medium resistance. RPC is an important component in determining the filter cake resistance and its functionality with cake forming pressure. It is found that there are additional effects that enhance the resistance to permeation in different cake materials, which is not recognised in the standard analytical approaches. These complexities can be related to shape, polydispersity, or agglomeration within the material sample and not to the experimental equipment or procedure. Furthermore, a complete and straightforward methodology is presented in this work for investigating the significance, or otherwise, of medium resistance on the later stages of the filtration.

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