where A represents the magnetic vector potential, is an integral of the hydromagnetic equations. This -integral made it possible to formulate a variational principle for the force-free magnetic fields. The integral expresses the fact that motions cannot transform a given field in an entirely arbitrary different field, if the conductivity of the medium isconsidered infinite. In this paper we shall show that the full set of hydromagnetic equations admit five more integrals, besides the energy integral, if dissipative processes are absent. These integrals, as we shall presently verify, are I2 =fbHvdV, (2)

Proceedings of the National Academy of Sciences

Complex biological systems and cellular networks may underlie most genotype to phenotype relationships. Here, we review basic concepts in network biology, discussing different types of interactome networks and the insights that can come from analyzing them. We elaborate on why interactome networks are important to consider in biology, how they can be mapped and integrated with each other, what global properties are starting to emerge from interactome network models, and how these properties may relate to human disease.

Interactome networks and human disease

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.

Control Principles of Complex Networks

What is discrete math? • The real numbers are continuous in the senses that: * between any two real numbers there is a real number • The integers do not share this property. In this sense the integers are lumpy, or " discrete " So discrete math is the study of mathematical objects that are discrete. " It's all the math that counts " Some discrete mathematical concepts: • Integers: Between two integers there is not another integer. • Propositions: Either true or false, there are no 1/2 truths (in math) • Sets: An item is either in a set or not in a set, never partly in and partly out. • Relations: A pair of items are related or not. • Networks (graphs): Between two terminals of a network connection there are no terminals. Propositional Logic Propositions are statements that are either true or false. Principles: Substituting an equivalent statement. Replacing a logic variable in a tautology. Defn algebraic proof

Discrete math = 離散数学

The protein–protein interaction (PPI) network is crucial for cellular information processing and decision-making. With suitable inputs, PPI networks drive the cells to diverse functional outcomes such as cell proliferation or cell death. Here, we characterize the structural controllability of a large directed human PPI network comprising 6,339 proteins and 34,813 interactions. This network allows us to classify proteins as “indispensable,” “neutral,” or “dispensable,” which correlates to increasing, no effect, or decreasing the number of driver nodes in the network upon removal of that protein. We find that 21% of the proteins in the PPI network are indispensable. Interestingly, these indispensable proteins are the primary targets of disease-causing mutations, human viruses, and drugs, suggesting that altering a network’s control property is critical for the transition between healthy and disease states. Furthermore, analyzing copy number alterations data from 1,547 cancer patients reveals that 56 genes that are frequently amplified or deleted in nine different cancers are indispensable. Among the 56 genes, 46 of them have not been previously associated with cancer. This suggests that controllability analysis is very useful in identifying novel disease genes and potential drug targets.

https://www.pnas.org/content/pnas/113/18/4976.full.pdf

Controllability analysis of the directed human protein interaction network identifies disease genes and drug targets

Minimum dominating set-based methods for analyzing biological networks.

Recent studies have drawn attention to the problem of how complex networks can be controlled through a small number of controller nodes. Here, we develop an algorithmic procedure and mathematical tools to compute and evaluate the critical and redundant nodes in controlling directed and undirected scalefree networks using the minimum dominating set (MDS) approach. Because there are multiple MDS configurations that control the entire network, we can classify the nodes depending on the condition whether a node is part of all (critical), some but not all (intermittent), or does not participate in any (redundant) possible MDS. The presented mathematical analysis predicts the probability of finding a critical node in undirected scale-free networks with k−γ , where k is a node degree, as a function of the scaling exponent γ and its node degree. Critical nodes tend to have high degree and are more abundant in undirected scale-free networks with high γ . In addition, analytical expressions of lower bounds for the number of critical nodes for both undirected and directed networks are also derived. By applying the MDS control model, we find that undirected networks can be controlled with relatively fewer nodes than those engaged in controlling directed networks. On the other hand, our computational experiments also show that the MDS is unimodal with varying the average degree 〈k〉 in both directed and undirected networks. In particular, by increasing 〈k〉, the fraction of nodes engaged in control becomes smaller, which highlights a centralized control mode. The analysis of a set of undirected and directed real-world networks confirms the findings shown in theoretical analysis and simulation experiments.

Analysis of critical and redundant nodes in controlling directed and undirected complex networks using dominating sets

Prospect theory is widely viewed as the best available descriptive model of how people evaluate risk in experimental settings. According to prospect theory, people are typically risk-averse with respect to gains and risk-seeking with respect to losses, known as the “reflection effect”. People are much more sensitive to losses than to gains of the same magnitude, a phenomenon called “loss aversion”. Despite of the fact that prospect theory has been well developed in behavioral economics at the theoretical level, there exist very few large-scale empirical studies and most of the previous studies have been undertaken with micro-panel data. Here we analyze over 28.5 million trades made by 81.3 thousand traders of an online financial trading community over 28 months, aiming to explore the large-scale empirical aspect of prospect theory. By analyzing and comparing the behavior of winning and losing trades and traders, we find clear evidence of the reflection effect and the loss aversion phenomenon, which are essential in prospect theory. This work hence demonstrates an unprecedented large-scale empirical evidence of prospect theory, which has immediate implication in financial trading, e.g., developing new trading strategies by minimizing the impact of the reflection effect and the loss aversion phenomenon. Moreover, we introduce three novel behavioral metrics to differentiate winning and losing traders based on their historical trading behavior. This offers us potential opportunities to augment online social trading where traders are allowed to watch and follow the trading activities of others, by predicting potential winners based on their historical trading behavior.

https://dash.harvard.edu/bitstream/handle/1/13347560/4198126.pdf?sequence=1

Prospect Theory for Online Financial Trading

Prospect theory is widely viewed as the best available descriptive model of how people evaluate risk in experimental settings. According to prospect theory, people are risk-averse with respect to gains and risk-seeking with respect to losses, a phenomenon called "loss aversion". Despite of the fact that prospect theory has been well developed in behavioral economics at the theoretical level, there exist very few large-scale empirical studies and most of them have been undertaken with micro-panel data. Here we analyze over 28.5 million trades made by 81.3 thousand traders of an online financial trading community over 28 months, aiming to explore the large-scale empirical aspect of prospect theory. By analyzing and comparing the behavior of winning and losing trades and traders, we find clear evidence of the loss aversion phenomenon, an essence in prospect theory. This work hence demonstrates an unprecedented large-scale empirical evidence of prospect theory, which has immediate implication in financial trading, e.g., developing new trading strategies by minimizing the effect of loss aversion. Moreover, we introduce three risk-adjusted metrics inspired by prospect theory to differentiate winning and losing traders based on their historical trading behavior. This offers us potential opportunities to augment online social trading, where traders are allowed to watch and follow the trading activities of others, by predicting potential winners statistically based on their historical trading behavior rather than their trading performance at any given point in time.

/pdf/prospect-theory-for-online-financial-trading-2wbvgr1v8t.pdf

Robust control theory has been successfully applied to numerous real-world problems using a small set of devices called controllers However, the real systems represented by networks contain unreliable components and modern robust control engineering has not addressed the problem of structural changes on complex networks including scale-free topologies Here, we introduce the concept of structurally robust control of complex networks and provide a concrete example using an algorithmic framework that is widely applied in engineering The developed analytical tools, computer simulations, and real network analyses lead herein to the discovery that robust control can be achieved in scale-free networks with exactly the same order of controllers required in a standard nonrobust configuration by adjusting only the minimum degree The presented methodology also addresses the probabilistic failure of links in real systems, such as neural synaptic unreliability in Caenorhabditis elegans, and suggests a new direction to pursue in studies of complex networks in which control theory has a role

/pdf/structurally-robust-control-of-complex-networks-2vi6mpwnrp.pdf

Jose C. Nacher

Papers

Minimum dominating set-based methods for analyzing biological networks.

Analysis of critical and redundant nodes in controlling directed and undirected complex networks using dominating sets

Prospect Theory for Online Financial Trading

Prospect Theory for Online Financial Trading

Structurally robust control of complex networks.