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Local Graph Stability in Exponential Family Random Graph Models
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| Pengarang | : | Yue Yu |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 4) |
| Halaman | : | 1369-1415 |
| Abstrak | : | Exponential family random graph models (ERGMs) are widely used to model networks by parameterizing graph probability in terms of a set of user-selected sufficient statistics. Equivalently, ERGMs can be viewed as expressing a probability distribution on graphs arising from the action of competing social forces that make ties more or less likely, depending on the state of the rest of the graph. Such forces often lead to a complex pattern of dependence among edges, with non-trivial large-scale structures emerging from relatively simple local mechanisms. While this provides a powerful tool for probing macro-micro connections, much remains to be understood about how local forces shape global outcomes. One very simple question of this type is that of the conditions needed for social forces to stabilize a particular structure: that is, given a specific structure and a set of alternatives (e.g., arising from small perturbations), under what conditions will said structure remain more probable than the alternatives? We refer to this property as local stability and seek a general means of identifying the set of parameters under which a target graph is locally stable with respect to a set of alternatives. Here, we provide a complete characterization of the region of the parameter space inducing local stability, showing it to be the interior of a convex cone whose faces can be derived from the change scores of the sufficient statistics vis-a?-vis the alternative structures. As we show, local stability is a necessary but not sufficient condition for more general notions of stability, the latter of which can be explored more efficiently by using the “stable cone” within the parameter space as a starting point. In addition to facilitating the understanding of model behavior, we show how local stability can be used to determine whether a fitted model implies that an observed structure would be expected to arise primarily from the action of social forces, versus by merit of the model permitting a large number of high probability structures, of which the observed structure is one (i.e., entropic effects). We also use our approach to identify the dyads within a given structure that are the least stable, and hence predicted to have the highest probability of changing under the current social forces. The utility of the “stable cone” for ERGM parameter optimization is then demonstrated on a physical model of amyloid fibril formation. This demonstration features a visualization of the stable region, whereby it is shown that the majority of the region of the model's parameter space where ERGM simulations produce the highest fibril yield lies within the “stable cone.” |
Ideal Free Dispersal under General Spatial Heterogeneity and Time Periodicity
| Pengarang | : | Robert Stephen Cantrell |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 3) |
| Halaman | : | 789-813 |
| Abstrak | : | A population is said to have an ideal free distribution in a spatially heterogeneous but temporally constant environment if each of its members has chosen a fixed spatial location in a way that optimizes its individual fitness, allowing for the effects of crowding. In this paper, we extend the idea of individual fitness associated with a specific location in space to account for the full path that an individual organism takes in space and time over a periodic cycle, and we extend the mathematical formulation of an ideal free distribution to general time periodic environments. We find that, as in many other cases, populations using dispersal strategies that can produce a generalized ideal free distribution have a competitive advantage relative to populations using dispersal strategies that cannot do so. A sharp criterion on the environmental functions is found to be necessary and sufficient for such ideal free distribution to be feasible. In the case the criterion is met, we show the existence of dispersal strategies that can be identified as producing a time-periodic version of an ideal free distribution, and such strategies are evolutionarily steady and are neighborhood invaders from the viewpoint of adaptive dynamics. Our results extend previous works in which the environments are either temporally constant, or temporally periodic but the total carrying capacity is temporally constant. |
Asymptotic Reduction of a Lithium-Ion Pouch Cell Model
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 3) |
| Halaman | : | 765-788 |
| Abstrak | : | A three-dimensional model of a single-layer lithium-ion pouch cell is presented which couples conventional porous electrode theory describing cell electrochemical behavior with an energy balance describing cell thermal behavior. Asymptotic analysis of the model is carried out by exploiting the small aspect ratio typical of pouch cell designs. The analysis reveals the scaling that results in a distinguished limit and highlights the role played by the electrical conductivities of the current collectors. The resulting model comprises a collection of one-dimensional models for the through-cell electrochemical behavior which are coupled via two-dimensional problems for the Ohmic and thermal behavior in the planar current collectors. A further limit is identified which reduces the problem to a single volume-averaged through-cell model, greatly reducing the computational complexity. Numerical simulations are presented which illustrate and validate the asymptotic results. |
Multiscale Analysis of Nutrient Uptake by Plant Roots with Sparse Distribution of Root Hairs: Nonstandard Scaling
| Pengarang | : | John R. King |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 4) |
| Halaman | : | 1361-1388 |
| Abstrak | : | In this paper we undertake a multiscale analysis of nutrient uptake by plant roots by considering different scale relations between the radius of root hairs and the distance between them. We combine the method of formal asymptotic expansions and rigorous derivation of macroscopic equations. The former prompt us to study a distinguished limit (which yields a distinct effective equation), allow us to determine higher-order correctors, and provide motivation for the construction of correctors essential for rigorous derivation of macroscopic equations. The results of our asymptotic analysis are validated by direct comparison with full-geometry numerical simulations. |
Optimal Mixing in Transport Networks: Numerical Optimization and Analysis
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 3) |
| Halaman | : | 741-764 |
| Abstrak | : | Many foraging microorganisms rely upon cellular transport networks to deliver nutrients, fluid, and organelles between different parts of the organism. Networked organisms ranging from filamentous fungi to slime molds demonstrate a remarkable ability to mix or disperse molecules and organelles in their transport media. Here we introduce mathematical tools to analyze the structure of energy efficient transport networks that maximize mixing and sending signals originating from and arriving at each node. We define two types of entropy on flows to quantify mixing and develop numerical algorithms to optimize the combination of entropy and energy on networks, given constraints on the amount of available material. We present an in-depth exploration of optimal single source-sink networks on finite triangular grids, a fundamental setting for optimal transport networks in the plane. Using numerical simulations and rigorous proofs, we show that if the constraint on conductances is strict, the optimal networks are paths of every possible length. If the constraint is relaxed, our algorithm produces loopy networks that fan out at the source and pour back into a single path that flows to the sink. Taken together, our results expand the class of optimal transportation networks that can be compared with real biological data and highlight how real network morphologies may be shaped by trade-offs between transport efficiency and the need to mix the transported matter. |
Analysis and Numerical Simulations of a Reaction-Diffusion Model with Fixed Active Bodies
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 4) |
| Halaman | : | 1339-1360 |
| Abstrak | : | We propose an interaction model for two substances (molecules) of concentrations ???? and ????. The substance ???? is assumed to be involved in the growth process of ????, and in contrast, the substance ???? inhibits the production of ????. Our model is based on a reaction-diffusion system for the interaction of two considered substances (molecules). We analyze its well-posedness and perform a numerical scheme to simulate the behavior of the solutions. As an application, we investigate the role of A????-oligomers in the early stage of Alzheimer's disease propagation, more precisely for the impact of A????-oligomers in neuron impairment as studied in [M. Andrade-Restrepo et al., Modeling the spatial propagation of A???? oligomers in Alzheimer's disease, in CEMRACS 2018 - Numerical and Mathematical Modeling for Biological and Medical Applications: Deterministic, Probabilistic and Statistical Descriptions, Marseille, France, 2018, pp. 1--10]. |
Hyperbolic Quadrature Method of Moments for the One-Dimensional Kinetic Equation
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 82 (No. 2) |
| Halaman | : | 750-771 |
| Abstrak | : | A tractable solution is proposed to a classical problem in kinetic theory, namely, given any set of realizable velocity moments up to order 2?????, a closure for the moment of order 2????? +1 is constructed for which the moment system found from the free-transport term in the one-dimensional (1-D) kinetic equation is globally hyperbolic and in conservative form. In prior work, the hyperbolic quadrature method of moments (HyQMOM) was introduced to close this moment system up to fourth order (???? ≤2). Here, HyQMOM is reformulated and extended to arbitrary even-order moments. The HyQMOM closure is defined based on the properties of the monic orthogonal polynomials ???????? that are uniquely defined by the velocity moments up to order 2????? −1. Thus, HyQMOM is strictly a moment closure and does not rely on the reconstruction of a velocity distribution function with the same moments. On the boundary of moment space, ???? double roots of the characteristic polynomial ????2?????+1 of the Jacobian matrix of the system are the roots of ????????, while in the interior, ????2?????+1 and ???????? share ???? roots. The remaining ???? +1 roots of ????2?????+1 bound and separate the roots of ????????. An efficient algorithm, based on the Chebyshev algorithm, for computing the moment of order 2????? +1 from the moments up to order 2????? is developed. The analytical solution to a 1-D Riemann problem is used to demonstrate convergence of the HyQMOM closure with increasing ????. |
Effective Medium Theory for Embedded Obstacles in Elasticity with Applications to Inverse Problems
| Pengarang | : | Qingle Meng |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 82 (No. 2) |
| Halaman | : | 720-749 |
| Abstrak | : | We consider the time-harmonic elastic wave scattering from a general (possibly anisotropic) inhomogeneous medium with an embedded impenetrable obstacle. We show that the impenetrable obstacle can be effectively approximated by an isotropic elastic medium with a particular choice of material parameters. We derive sharp estimates to rigorously verify such an effective approximation. Our study is strongly motivated by the related studies of two challenging inverse elastic problems including the inverse boundary problem with partial data and the inverse scattering problem of recovering mediums with buried obstacles. The proposed effective medium theory readily yields some interesting applications of practical significance to these inverse problems. |
One-Dimensional Ferronematics in a Channel: Order Reconstruction, Bifurcations, and Multistability
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 82 (No. 2) |
| Halaman | : | 694-719 |
| Abstrak | : | We study a model system with nematic and magnetic order, within a channel geometry modeled by an interval, [−????,????]. The system is characterized by a tensor-valued nematic order parameter ???? and a vector-valued magnetization ????, and the observable states are modeled as stable critical points of an appropriately defined free energy which includes a nemato-magnetic coupling term, characterized by a parameter ????. We (i) derive ????∞ bounds for ???? and ????; (ii) prove a uniqueness result in specified parameter regimes; (iii) analyze order reconstruction solutions, possessing domain walls, and their stabilities as a function of ???? and ???? and; (iv) perform numerical studies that elucidate the interplay of ???? and ???? for multistability. |
On A Transient Nonlinear One-Dimensional Reaction-Diffusion Equation with A Point-Source Initial Condition
| Pengarang | : | Michael Vynnycky |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 82 (No. 2) |
| Halaman | : | 677-693 |
| Abstrak | : | We revisit the analysis of a transient nonlinear reaction-diffusion equation describing two-particle coalescence or, alternatively, annihilation in an infinite domain. On nondimensionalizing the governing equations, we find that the problem is, in general, free of dimensionless parameters, indicating that the original work, which was purported to be for the slow reaction limit, is instead better interpreted as being for small times. By means of analysis using similarity-like variables, we also find that the delta-peaked initial condition used in the original analysis leads to an unexpected and easily overlooked restriction on the reaction order, which cannot be greater than 3; the same applies for two-term regular perturbation expansions that are valid for short times, and which we also determine. The full equations are then solved numerically for all times and the solutions are found to agree well with the analytical results that are available for short times. |