Network Decomposition and Complexity Measures: An Information Geometrical Approach

Abstract

Abstract:We consider the graph representation of the stochastic model with n binary variables, and develop an information theoretical framework to measure the degree of statistical association existing between subsystems as well as the ones represented by each edge of the graph representation. Besides, we consider the novel measures of complexity with respect to the system decompositionability, by introducing the geometric product of Kullback-Leibler (KL-) divergence. The novel complexity measures satisfy the boundary condition of vanishing at the limit of completely random and ordered state, and also with the existence of independent subsystem of any size. Such complexity measures based on the geometric means are relevant to the heterogeneity of dependencies between subsystems, and the amount of information propagation shared entirely in the system.

Citizen Science and Topology of Mind: Complexity, Computation and Criticality in Data-Driven Exploration of Open Complex Systems

“…For simplicity, we consider the integration of two databases X N and X M , with size N and M ∈ N, X N := {µ i (x)|x ∈ X, i = 1, · · · , N}, X M := {µ i (x)|x ∈ X, i = 1, · · · , M}, respectively. A joint distribution between subsets of X N and X M needs to be determined with respect to common parameters in order to obtain an integrated database including the calculation of up to (N + M)-th order of commonality, such as order-wise correlations [32]. Exhaustive computing follows the argument in Section 4.2, giving the extension of Theorem 6: Theorem 7.…”

“…TDC represents statistical dependencies between two complexity measures in response to a given inter-subjective objective measurement. While significant matching between two commonality orders assures the reproducibility based on the coincidence of observation with these measures, non-significance can also be used to quantify complementarity of different evaluations [32].…”

“…The concept and definition of complexity vary according to the fields, such as algorithmic complexity, statistical complexity, biological complexity, etc. In this paper, we take a generalized definition of complexity measure as the projection from a system's variables to one-dimensional quantity, which is composed to express a distinctive characteristic of the system [32]. This includes classical indices mentioned with the context of complexity, as well as various forms of information expressed as numbers in ICT, such as feature dimensions of machine learning.…”

“…A universally robust model with respect to an arbitrary variable cost function is canonically given by uniform distribution, which is commonly adopted as a prior of Bayesian estimation and random search algorithm [14]. It is also widely prevalent in biological phenomena, as the survival rate depends on the geometric mean of evolutionary fitness, which is maximized with uniformity in space, time, and statistical configuration [32,37].…”

“…Complexity measures are widely studied in information theory, with the underlying principle to abstract a low-dimensional representative index of useful features for functional characterization of complex systems [32]. Usually, complexity measures defined on n real variables are the epimorphism to the one-dimensional real number line, R n → R. The general complexity measure for citizen science is therefore the projection of the database to real value index, X → R, with the condition that this transformation will provide some utility for the management.…”