Control Principles of Complex Networks

A reflection of our ultimate understanding of a complex system is our ability to control its behavior. Typically, control has multiple prerequisites: It requires an accurate map of the network that governs the interactions between the system's components, a quantitative description of the dynamical laws that govern the temporal behavior of each component, and an ability to influence the state and temporal behavior of a selected subset of the components. With deep roots in nonlinear dynamics and control theory, notions of control and controllability have taken a new life recently in the study of complex networks, inspiring several fundamental questions: What are the control principles of complex systems? How do networks organize themselves to balance control with functionality? To address these here we review recent advances on the controllability and the control of complex networks, exploring the intricate interplay between a system's structure, captured by its network topology, and the dynamical laws that govern the interactions between the components. We match the pertinent mathematical results with empirical findings and applications. We show that uncovering the control principles of complex systems can help us explore and ultimately understand the fundamental laws that govern their behavior.Comment: 55 pages, 41 figures, Submitted to Reviews of Modern Physic

Lagrangian Mean Curvature flow for entire Lipschitz graphs II

We prove longtime existence and estimates for solutions to a fully nonlinear Lagrangian parabolic equation with locally $C^{1,1}$ initial data $u_0$ satisfying either (1) $-(1+\eta) I_n\leq D^2u_0 \leq (1+\eta)I_n$ for some positive dimensional constant $\eta$, (2) $u_0$ is weakly convex everywhere or (3) $u_0$ satisfies a large supercritical Lagrangian phase condition.Comment: 17 page

Rigidity of Entire self-shrinking solutions to curvature flows

We show that (a) any entire graphic self-shrinking solution to the Lagrangian mean curvature flow in ${\mathbb C}^{m}$ with the Euclidean metric is flat; (b) any space-like entire graphic self-shrinking solution to the Lagrangian mean curvature flow in ${\mathbb C}^{m}$ with the pseudo-Euclidean metric is flat if the Hessian of the potential is bounded below quadratically; and (c) the Hermitian counterpart of (b) for the K\"ahler Ricci flow.Comment: 10 page

Pseudolocality for the Ricci flow and applications

In \cite{P1}, Perelman established a differential Li-Yau-Hamilton (LYH) type inequality for fundamental solutions of the conjugate heat equation corresponding to the Ricci flow on compact manifolds (also see \cite{N2}). As an application of the LYH inequality, Perelman proved a pseudolocality result for the Ricci flow on compact manifolds. In this article we provide the details for the proofs of these results in the case of a complete non-compact Riemannian manifold. Using these results we prove that under certain conditions, a finite time singularity of the Ricci flow must form within a compact set. We also prove a long time existence result for the \KRF flow on complete non-negatively curved \K manifolds.Comment: 44 pages; added Corollary to Theorem 1.1; correction to Theorem 8.

Exploring Evolving Plants as Interacting Particles in a Randomly Generated Heterogeneous Environment

We model evolution of plants in a world, made up of different locations, with multiple environments (mutually exclusive and collectively exhaustive subsets of locations). Each environment (landmass) has temperature, rainfall, and other attributes that directly affect plant growth and reproduction. Each plant has preferences for environment attributes. Depending on how suitable the environment is to the plants, seeds are released or death occurs. With every reproductive cycle, genetic mutations occur. To model competition, plants in compete for survival, and success is stochastically dependent on environmental fitness. Our model determines whether and how evolution occurs, and how the attributes of plants change and possibly converge over time in relation to the attributes of the environment

TRP Channels in Cardiovascular Disease

Transient receptor potential (TRP) channels are evolutionarily conserved ion channels that have been implicated in a wide range of physiological and pathophysiological responses. As versatile ion channels that are permeable to calcium, the exact nature of these ion channels have been furiously studied since its initial discovery in Drosophila. Many TRP channels are thought to be gated by PIP2, a well-known membrane signaling molecule. Here we show that membrane potential alters PIP2 in such a way to reduce TRPM7 activity, presumably through PIP2 depletion. This novel mechanism of TRPM7 regulation gives us a clearer picture into a complicated protein that has been implicated in processes ranging from embryonic development to cancer. Furthermore, we show how TRP channels are involved in the development and progression of various cardiovascular diseases. Our research implicates the oxidative stress activated channel TRPM2 in the progression of atherosclerosis, with data pointing to its role in driving inflammation by increasing circulating myeloid cell populations. We also show that the bifunctional channel-enzyme TRPM7 plays a deleterious role in the cardiac fibrogenesis cascade in hypertensive heart failure, and deletion of Trpm7 specifically in the cardiac fibroblast is protective against negative cardiac remodeling. My research demonstrates not only how TRP channels are important mediators of cardiovascular disease, but also how they are regulated at a basic molecular level

Towards Physics-informed Deep Learning for Turbulent Flow Prediction

While deep learning has shown tremendous success in a wide range of domains, it remains a grand challenge to incorporate physical principles in a systematic manner to the design, training, and inference of such models. In this paper, we aim to predict turbulent flow by learning its highly nonlinear dynamics from spatiotemporal velocity fields of large-scale fluid flow simulations of relevance to turbulence modeling and climate modeling. We adopt a hybrid approach by marrying two well-established turbulent flow simulation techniques with deep learning. Specifically, we introduce trainable spectral filters in a coupled model of Reynolds-averaged Navier-Stokes (RANS) and Large Eddy Simulation (LES), followed by a specialized U-net for prediction. Our approach, which we call turbulent-Flow Net (TF-Net), is grounded in a principled physics model, yet offers the flexibility of learned representations. We compare our model, TF-Net, with state-of-the-art baselines and observe significant reductions in error for predictions 60 frames ahead. Most importantly, our method predicts physical fields that obey desirable physical characteristics, such as conservation of mass, whilst faithfully emulating the turbulent kinetic energy field and spectrum, which are critical for accurate prediction of turbulent flows

Control principles of metabolic networks

Deciphering the control principles of metabolism and its interaction with other cellular functions is central to biomedicine and biotechnology. Yet, understanding the efficient control of metabolic fluxes remains elusive for large-scale metabolic networks. Existing methods either require specifying a cellular objective or are limited to small networks due to computational complexity. Here we develop an efficient computational framework for flux control by introducing a complete set of flux coupling relations. We analyze 23 metabolic networks from all kingdoms of life, and identify the driver reactions facilitating their control on a large scale. We find that most unicellular organisms require less extensive control than multicellular organisms. The identified driver reactions are under strong transcriptional regulation in Escherichia coli. In human cancer cells driver reactions play pivotal roles in tumor development, representing potential therapeutic targets. The proposed framework helps us unravel the regulatory principles of complex diseases and design novel engineering strategies at the interface of gene regulation, signaling, and metabolism.Comment: 24 pages, 5 figures, 1 tabl

Real-Coded Chemical Reaction Optimization with Different Perturbation Functions

Chemical Reaction Optimization (CRO) is a powerful metaheuristic which mimics the interactions of molecules in chemical reactions to search for the global optimum. The perturbation function greatly influences the performance of CRO on solving different continuous problems. In this paper, we study four different probability distributions, namely, the Gaussian distribution, the Cauchy distribution, the exponential distribution, and a modified Rayleigh distribution, for the perturbation function of CRO. Different distributions have different impacts on the solutions. The distributions are tested by a set of well-known benchmark functions and simulation results show that problems with different characteristics have different preference on the distribution function. Our study gives guidelines to design CRO for different types of optimization problems

Fundamental limitations of network reconstruction

Network reconstruction is the first step towards understanding, diagnosing and controlling the dynamics of complex networked systems. It allows us to infer properties of the interaction matrix, which characterizes how nodes in a system directly interact with each other. Despite a decade of extensive studies, network reconstruction remains an outstanding challenge. The fundamental limitations governing which properties of the interaction matrix (e.g., adjacency pattern, sign pattern and degree sequence) can be inferred from given temporal data of individual nodes remain unknown. Here we rigorously derive necessary conditions to reconstruct any property of the interaction matrix. These conditions characterize how uncertain can we be about the coupling functions that characterize the interactions between nodes, and how informative does the measured temporal data need to be; rendering two classes of fundamental limitations of network reconstruction. Counterintuitively, we find that reconstructing any property of the interaction matrix is generically as difficult as reconstructing the interaction matrix itself, requiring equally informative temporal data. Revealing these fundamental limitations shed light on the design of better network reconstruction algorithms, which offer practical improvements over existing methods.Comment: 11 pages, 3 figure