Large scale dynamical systems (e.g. many nonlinear coupled differential equations) can often be summarized in terms of only a few state variables (a few equations), a trait that reduces complexity and facilitates exploration of behavioral aspects of otherwise intractable models. High model dimensionality and complexity makes symbolic, pen-and-paper model reduction tedious and impractical, a difficulty addressed by recently developed frameworks that computerize reduction. Symbolic work has the benefit, however, of identifying both reduced state variables and parameter combinations that matter most (effective parameters, "inputs"); whereas current computational reduction schemes leave the parameter reduction aspect mostly unaddressed. As the interest in mapping out and optimizing complex input-output relations keeps growing, it becomes clear that combating the curse of dimensionality also requires efficient schemes for input space exploration and reduction. Here, we explore systematic, data-driven parameter reduction by means of effective parameter identification, starting from current nonlinear manifoldlearning techniques enabling state space reduction. Our approach aspires to extend the data-driven determination of effective state variables with the data-driven discovery of effective model parameters, and thus to accelerate the exploration of high-dimensional parameter spaces associated with com- * These two authors contributed equally to this work.Given access to input-output information (black-box function evaluation) but no formulas, one might not even suspect that only the single parameter combination p eff = p 1 p 2 matters. Fitting the model to data f * = (1, 0, 1) in the absence of such information, one would find an entire curve in parameter space that fits the observations. A data fitting algorithm based only on function evaluations could be "confused" by such behavior in declaring convergence. As seen in Fig. 1(a), different initial conditions fed to an optimizer with a practical fitting tolerance δ ≈ 10 −3 (see figure caption for details) converge to many, widely different results tracing a level curve of p eff . The subset of good fits is effectively 1−D; more importantly, and moving beyond the fit to this particular data, the entire parameter space is foliated by such 1−D curves (neutral sets), each composed of points indistinguishable from the model output perspective. Parameter non-identifiability is therefore a structural feature of the model, not an artifact of optimization. The appropriate, intrinsic way to describe parameter space for this problem is through the effective parameter p eff and its level sets. Consider now the inset of Fig. 1(a), corresponding to the perturbed model f ε (p 1 , p 2 ) = f 0 (p 1 , p 2 ) + 2ε(p 1 − p 2 , 0, 0) and fit to the same data. Here, the parameters are identifiable and the minimizer (p 1 , p 2 ) unique: a perfect fit exists. However, the foliation observed for ε = 0 is loosely remembered in the shape of the residual level curves, and the optimizer would be co...
The exploration of epidemic dynamics on dynamically evolving ("adaptive") networks poses nontrivial challenges to the modeler, such as the determination of a small number of informative statistics of the detailed network state (that is, a few "good observables") that usefully summarize the overall (macroscopic, systems-level) behavior. Obtaining reduced, small size accurate models in terms of these few statistical observables -that is, trying to coarse-grain the full network epidemic model to a small but useful macroscopic one -is even more daunting. Here we describe a databased approach to solving the first challenge: the detection of a few informative collective observables of the detailed epidemic dynamics. This is accomplished through Diffusion Maps (DMAPS), a recently developed data-mining technique. We illustrate the approach through simulations of a simple mathematical model of epidemics on a network: a model known to exhibit complex temporal dynamics. We discuss potential extensions of the approach, as well as possible shortcomings.
We discuss the problem of extending data mining approaches to cases in which data points arise in the form of individual graphs. Being able to find the intrinsic low-dimensionality in ensembles of graphs can be useful in a variety of modeling contexts, especially when coarse-graining the detailed graph information is of interest. One of the main challenges in mining graph data is the definition of a suitable pairwise similarity metric in the space of graphs. We explore two practical solutions to solving this problem: one based on finding subgraph densities, and
In order to illustrate the adaptation of traditional continuum numerical techniques to the study of complex network systems, we use the equation-free framework to analyze a dynamically evolving multigraph. This approach is based on coupling short intervals of direct dynamic network simulation with appropriately-defined lifting and restriction operators, mapping the detailed network description to suitable macroscopic (coarse-grained) variables and back. This enables the acceleration of direct simulations through Coarse Projective Integration (CPI), as well as the identification of coarse stationary states via a Newton-GMRES method. We also demonstrate the use of datamining, both linear (principal component analysis, PCA) and nonlinear (diffusion maps, DMAPS) to determine good macroscopic variables (observables) through which one can coarse-grain the model. These results suggest methods for decreasing simulation times of dynamic realworld systems such as epidemiological network models. Additionally, the data-mining techniques could be applied to a diverse class of problems to search for a succint, low-dimensional description of the system in a small number of variables. arXiv:1607.02818v1 [physics.data-an]
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