Foundations & Physics  (FP) Session 4

Abstract: The paper discusses two related questions: where to ‘cut’ system definitions and systemic relations based on the perspective of the involved stakeholders. Both are historically related to the genetic historical /-critical, monist approach of psychophysicist Ernst Mach. For the analysis of (causal) interactions in complex systems (Auyang 1998), Simon and Ando (Ando and Simon 1961, see also Shpak et al. 2004) have developed the concept of (near) decomposability, where interactions in systems are separated into groups of interactions according to the strength of interactions between elements of a system. The danger in this assumption is that interactions between groups of variables can be neglected such that microstate variables can be aggregated into macro-state variables. This assumption may work in the short run under normal conditions, but may also fail under longer terms and unusual conditions. From a ‘complexity / non-linear mathematics perspective’ ‘small’ effects may lead under positive feedback to the crossing of thresholds and phase transitions and then may be observed as increased stress, risk and catastrophes in a system’s development (cp. Thom 1989, Jain and Krishna 2002, Sornette 2003). In order to tackle the question of where to ‘cut’ system definition, decomposition and system aggregation, the paper proposes to employ physicist-psychologist-philosopher Ernst Mach’s genetic perspective on the evolution of knowledge based on his research in the history of science (Mach 1888, 1905, 1883). Mach suggests to replace causality with functional relations, which describe the relationship between the elements of the measured item and the standard of measurement (Mach 1905, Heidelberger 2010) as functional dependencies of one appearance on the other. The paper sketches the links between Mach’s and Simon’s approach to derive requirements for ‘tools’ to converse about system definition, decomposition, and aggregation (modularization) interrelated with and dependent on scientists worldviews. Close

Abstract: The black-box behavior of all natural things can be characterized by their input-output response. For example, neural networks can be considered devices that transform sensory inputs to electrical impulses known as spike trains. A goal of quantitative science is then to build mathematical models of such input-output processes; such that they can be used to construct devices that simulate such input-output behavior. In this talk we discuss the simplest models, the ones that can perform such simulations with the lest memory – as measured by the device's internal entropy. Such constructions serve a dual purpose. On the one hand they allow us to engineer devices that replicate the desired operational behaviour with minimal memory. On the other, the memory such a model requires tells us the minimal amount of structure any process exhibiting the same input-output behaviour must possess – and is therefore adopted as a way of quantifying the structural complexity of such processes [1]. Here, we demonstrate that the simplest models of general input-output processes are generally quantum mechanical – even if the inputs, and outputs are described purely by classical information. Specifically, we first review the provably simplest classical devices that exhibit a particular input-output behaviour; known as epsilon transducers [1]. We then outline recent work on modifying these devices to take advantage of quantum information processing; such that they can enact statistically identical input-output behaviour with reduced memory requirements [2]. This opens the potential for quantum information to be relevant in both the simulation of input-output processes, and the study of their structural complexity. [1] Barnett and Crutchfield, Journal of statistical physics, 161, 404 (2015) [2] J. Thompson et. al. Using quantum theory to reduce the complexity of input-output processes. arXiv:1601.05420 (2016) Close

Abstract: An alternative description of complex networks can be given in terms of quivers. These are objects in abstract algebra (they also have a category-theoretic definition). Using this formal machinery, we provide an alternative definition of multiplex networks. Then, identifying the path algebra of a multiplex, we find a gradation in the algebra that leads to the adjacency matrix of the multiplex. In fact, this formulation reveals two types of multiplex networks, the simple-multiplex, where the nodes are as usual, but there is an additional product map in the algebra; and the supra-multiplex, where nodes are replaced by supra-nodes. A supra-node is a collection of nodes belonging to an equivalence class with respect to a given equivalence relation. Though these equivalence classes can themselves be represented as connected graphs, the edges within a supra-node are of a distinct type than those between supra-nodes. By this classification, we identify all the tensorial adjacency matrices, that are usually discussed in the multiplex literature, as corresponding to supra-multiplexes, whereas, the original definition of multiplexes with nodes as basic entities, corresponds to simple-multiplexes. The benefit of an algebraic approach is that it helps in parsing these two types of multiplex networks in a precise way, leading to distinct path algebras and adjacency matrices for each. We show that the adjacency matrix of a simple-multiplex requires the construction of a new color-product map. To the best of our knowledge, this is the first formal derivation of this matrix. Close

Abstract: Our understanding of a variety of phenomena in physics, biology and economics crucially depends on the analysis of multivariate time series. While a wide range of tools and techniques for time series analysis already exist, the increasing availability of massive data structures calls for new approaches for multidimensional signal processing. We present here a non-parametric method to analyse multivariate time series, based on the mapping of a multidimensional time series into a multilayer network, which allows to extract information on a high dimensional dynamical system through the analysis of the structure of the associated multiplex network. The method is simple to implement, general, scalable, does not require ad-hoc phase space partitioning, and is thus suitable for the analysis of large, heterogeneous and non-stationary time series. We show that simple structural descriptors of the associated multiplex networks allow to extract and quantify nontrivial properties of coupled chaotic maps, including the transition between different dynamical phases and the onset of various types of synchronization. As a concrete example we then study financial time series, showing that a multiplex network analysis can efficiently discriminate crises from periods of financial stability, where standard methods based on time-series symbolization often fail. Close