Foundations & Physics (FP) Session 4
Time and Date: 16:00  17:20 on 22nd Sep 2016
Room: L  Grote Zaal
Chair: Vlatko Vedral
353  A Machian Functional Relations Perspective on Complex Systems
[abstract] 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 macrostate 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 / nonlinear 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 physicistpsychologistphilosopher 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.

Carl Henning Reschke 
86  Using quantum mechanics to simplify inputoutput processes
[abstract] Abstract: The blackbox behavior of all natural things can be characterized by their inputoutput 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 inputoutput processes; such that they can be used to construct devices that simulate such inputoutput 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 inputoutput 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 inputoutput 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 inputoutput 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 inputoutput behaviour with reduced memory requirements [2]. This opens the potential for quantum information to be relevant in both the simulation of inputoutput 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 inputoutput processes. arXiv:1601.05420 (2016)

Jayne Thompson, Andrew Garner, Vlatko Vedral and Mile Gu 
450  An Algebraic Formulation of Quivers, Networks and Multiplexes
[abstract] Abstract: An alternative description of complex networks can be given in terms of quivers. These are objects in abstract algebra (they also have a categorytheoretic 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 simplemultiplex, where the nodes are as usual, but there is an additional product map in the algebra; and the supramultiplex, where nodes are replaced by supranodes. A supranode 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 supranode are of a distinct type than those between supranodes. By this classification, we identify all the tensorial adjacency matrices, that are usually discussed in the multiplex literature, as corresponding to supramultiplexes, whereas, the original definition of multiplexes with nodes as basic entities, corresponds to simplemultiplexes. 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 simplemultiplex requires the construction of a new colorproduct map. To the best of our knowledge, this is the first formal derivation of this matrix.

Xerxes Arsiwalla, Ricardo Garcia and Paul Verschure 
326  Network structure of multivariate time series
[abstract] 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 nonparametric 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
adhoc phase space partitioning, and is thus suitable for the analysis
of large, heterogeneous and nonstationary 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 timeseries symbolization
often fail.

Vincenzo Nicosia, Lucas Lacasa and Vito Latora 