Dynamical Stability of Threshold Networks over Undirected Signed Graphs

In this paper we study the dynamic behavior of threshold networks on undirected signed graphs. While much attention has been given to the convergence and long-term behavior of this model, an open question remains: How does the underlying graph structure influence network dynamics? While similar papers have been carried out for threshold networks (as well as for other networks) these have largely focused on unsigned networks. However, the signed graph model finds applications in various real-world domains like gene regulation and social networks. By studying a graph parameter that we call "stability index," we search to establish a connection between the structure and the dynamics of threshold network. Interestingly, this parameter is related to the concepts of frustration and balance in signed graphs. We show that graphs that present negative stability index exhibit stable dynamics, meaning that the dynamics converges to fixed points regardless of threshold parameters. Conversely, if at least one subgraph has positive stability index, oscillations in long term behavior may appear. Finally, we generalize the analysis to network dynamics under periodic update schemes and we explore the case in which the stability index is positive for some subgraph finding that attractors with superpolynomial period on the size of the network may appear

Game Theory Relaunched

The game is on. Do you know how to play? Game theory sets out to explore what can be said about making decisions which go beyond accepting the rules of a game. Since 1942, a well elaborated mathematical apparatus has been developed to do so; but there is more. During the last three decades game theoretic reasoning has popped up in many other fields as well - from engineering to biology and psychology. New simulation tools and network analysis have made game theory omnipresent these days. This book collects recent research papers in game theory, which come from diverse scientific communities all across the world; they combine many different fields like economics, politics, history, engineering, mathematics, physics, and psychology. All of them have as a common denominator some method of game theory. Enjoy

A complex systems approach to education in Switzerland

The insights gained from the study of complex systems in biological, social, and engineered systems enables us not only to observe and understand, but also to actively design systems which will be capable of successfully coping with complex and dynamically changing situations. The methods and mindset required for this approach have been applied to educational systems with their diverse levels of scale and complexity. Based on the general case made by Yaneer Bar-Yam, this paper applies the complex systems approach to the educational system in Switzerland. It confirms that the complex systems approach is valid. Indeed, many recommendations made for the general case have already been implemented in the Swiss education system. To address existing problems and difficulties, further steps are recommended. This paper contributes to the further establishment complex systems approach by shedding light on an area which concerns us all, which is a frequent topic of discussion and dispute among politicians and the public, where billions of dollars have been spent without achieving the desired results, and where it is difficult to directly derive consequences from actions taken. The analysis of the education system's different levels, their complexity and scale will clarify how such a dynamic system should be approached, and how it can be guided towards the desired performance

Common metrics for cellular automata models of complex systems

The creation and use of models is critical not only to the scientific process, but also to life in general. Selected features of a system are abstracted into a model that can then be used to gain knowledge of the workings of the observed system and even anticipate its future behaviour. A key feature of the modelling process is the identification of commonality. This allows previous experience of one model to be used in a new or unfamiliar situation. This recognition of commonality between models allows standards to be formed, especially in areas such as measurement. How everyday physical objects are measured is built on an ingrained acceptance of their underlying commonality. Complex systems, often with their layers of interwoven interactions, are harder to model and, therefore, to measure and predict. Indeed, the inability to compute and model a complex system, except at a localised and temporal level, can be seen as one of its defining attributes. The establishing of commonality between complex systems provides the opportunity to find common metrics. This work looks at two dimensional cellular automata, which are widely used as a simple modelling tool for a variety of systems. This has led to a very diverse range of systems using a common modelling environment based on a lattice of cells. This provides a possible common link between systems using cellular automata that could be exploited to find a common metric that provided information on a diverse range of systems. An enhancement of a categorisation of cellular automata model types used for biological studies is proposed and expanded to include other disciplines. The thesis outlines a new metric, the C-Value, created by the author. This metric, based on the connectedness of the active elements on the cellular automata grid, is then tested with three models built to represent three of the four categories of cellular automata model types. The results show that the new C-Value provides a good indicator of the gathering of active cells on a grid into a single, compact cluster and of indicating, when correlated with the mean density of active cells on the lattice, that their distribution is random. This provides a range to define the disordered and ordered state of a grid. The use of the C-Value in a localised context shows potential for identifying patterns of clusters on the grid

Higher-order interactions in single-cell gene expression: towards a cybergenetic semantics of cell state

Finding and understanding patterns in gene expression guides our understanding of living organisms, their development, and diseases, but is a challenging and high-dimensional problem as there are many molecules involved. One way to learn about the structure of a gene regulatory network is by studying the interdependencies among its constituents in transcriptomic data sets. These interdependencies could be arbitrarily complex, but almost all current models of gene regulation contain pairwise interactions only, despite experimental evidence existing for higher-order regulation that cannot be decomposed into pairwise mechanisms. I set out to capture these higher-order dependencies in single-cell RNA-seq data using two different approaches. First, I fitted maximum entropy (or Ising) models to expression data by training restricted Boltzmann machines (RBMs). On simulated data, RBMs faithfully reproduced both pairwise and third-order interactions. I then trained RBMs on 37 genes from a scRNA-seq data set of 70k astrocytes from an embryonic mouse. While pairwise and third-order interactions were revealed, the estimates contained a strong omitted variable bias, and there was no statistically sound and tractable way to quantify the uncertainty in the estimates. As a result I next adopted a model-free approach. Estimating model-free interactions (MFIs) in single-cell gene expression data required a quasi-causal graph of conditional dependencies among the genes, which I inferred with an MCMC graph-optimisation algorithm on an initial estimate found by the Peter-Clark algorithm. As the estimates are model-free, MFIs can be interpreted either as mechanistic relationships between the genes, or as substructures in the cell population. On simulated data, MFIs revealed synergy and higher-order mechanisms in various logical and causal dynamics more accurately than any correlation- or information-based quantities. I then estimated MFIs among 1,000 genes, at up to seventh-order, in 20k neurons and 20k astrocytes from two different mouse brain scRNA-seq data sets: one developmental, and one adolescent. I found strong evidence for up to fifth-order interactions, and the MFIs mostly disambiguated direct from indirect regulation by preferentially coupling causally connected genes, whereas correlations persisted across causal chains. Validating the predicted interactions against the Pathway Commons database, gene ontology annotations, and semantic similarity, I found that pairwise MFIs contained different but a similar amount of mechanistic information relative to networks based on correlation. Furthermore, third-order interactions provided evidence of combinatorial regulation by transcription factors and immediate early genes. I then switched focus from mechanism to population structure. Each significant MFI can be assigned a set of single cells that most influence its value. Hierarchical clustering of the MFIs by cell assignment revealed substructures in the cell population corresponding to diverse cell states. This offered a new, purely data-driven view on cell states because the inferred states are not required to localise in gene expression space. Across the four data sets, I found 69 significant and biologically interpretable cell states, where only 9 could be obtained by standard approaches. I identified immature neurons among developing astrocytes and radial glial cells, D1 and D2 medium spiny neurons, D1 MSN subtypes, and cell-cycle related states present across four data sets. I further found evidence for states defined by genes associated to neuropeptide signalling, neuronal activity, myelin metabolism, and genomic imprinting. MFIs thus provide a new, statistically sound method to detect substructure in single-cell gene expression data, identifying cell types, subtypes, or states that can be delocalised in gene expression space and whose hierarchical structure provides a new view on the semantics of cell state. The estimation of the quasi-causal graph, the MFIs, and inference of the associated states is implemented as a publicly available Nextflow pipeline called Stator