Vol. 3, No. 3 (2014): Regular Issue

special issue of the Journal “Discontinuity, Nonlinearity, and Complexity” contains the proceedings of the joint workshop “Mathematics of Complex Systems” that took place at the Center for Interdisciplinary Research of the University of Bielefeld on the 7-9 October, 2013 supported by the Work program of FP7: FET Proactive Intiative of the European Comission: “Dynamics of Multi-Level Complex Systems (DyM-CS)”.

With online publication and social media taking the main role in dissemination of news, and with the decline of traditional printed media, it has become necessary to devise ways to automatically extract meaningful information from the plethora of sources available and to make that information readily available to interested parties. In this paper we present a method of automated analysis of the underlying structure of online newspapers based onQ-analysis and modularity optimisation. We show how the combination of the two strategies allows for the identification of well defined news clusters that are free of noise (unrelated stories) and provide automated clustering of information on trending topics on news published online.

The Structural Dynamic Economic Model SDEM-2, a follow-up of the model SDEM developed earlier, is essentially an actor-based, systemdynamic model of a closed economy evolving under conditions of conflict of interests of two powerful aggregated actors: entrepreneurs and wageearners. We derive the model equations applicable to both the balanced and unbalanced growth paths, and then study the balanced growth (with neither idle physical capital nor unemployment). Wefirst consider an inflexible control strategy of entrepreneurs for deterministic and stochastic cases, and then turn to a more sophisticated nonlinear control strategy. We also solve a simple optimization problem by calculating the (time-independent) value of model control parameter maximizing the discounted dividend of entrepreneurs. In view of simplicity of model equations, exact analytical solutions can be obtained in many cases, other cases being studied semianalytically. Even the simplest versions of SDEM-2 are able to produce rather versatile trajectories of the economy, dependent on the values of model parameters and initial conditions.

The process of data interpretation is always based on the implicit introduction of equivalence relations on the set of walks over the database. Every equivalence relation on the set of walks specifies a Markov chain describing the transitions of a discrete time random walk. In order to geometrize and interpret the data, we propose the new distance between data units defined as a “Feynman path integral”, in which all possible paths between any two nodes in a graph model of the data are taken into account, although some paths are more preferable than others. Such a path integral distance approach to the analysis of databases has proven its efficiency and success, especially on multivariate strongly correlated data where other methods fail to detect structural components (urban planning, historical language phylogenies, music, street fashion traits analysis, etc. ). We believe that it would become an invaluable tool for the intelligent complexity reduction and big data interpretation.

This paper describes a formalization of agent-based models (ABMs) as random walks on regular graphs and relates the symmetry group of those graphs to a coarse-graining of the ABM that is still Markovian. An ABM in which Nagents can be inδdifferent states leads to a Markov chain with δN states. In ABMs with a sequential update scheme by which one agent is chosen to update its state at a time, transitions are only allowed between system configurations that differ with respect to a single agent. This characterizes ABMs as random walks on regular graphs. The non-trivial automorphisms of those graphs make visible the dynamical symmetries that an ABM gives rise to because sets of micro configurations can be interchanged without changing the probability structure of the random walk. This allows for a systematic loss-less reduction of the state space of the model.

This work combines generic complexity of economic system and economic agents with methodologies of multi-agent system analysis and development. This combination results in an integrative framework that serves as communication protocol for delivering and transmitting agent-based model for economic system. The integrative framework provides guidance for analyzing economic system in micro-level, which embeds with properties of complexity in structure, heterogeneity in agents’ beliefs, and interactions among agents’ behaviors. It provides routines on developing standardized agent-based model for economic system that can be used and reused among interdisciplinary research.

The actor-based system-dynamic approach to macroeconomic modeling is illustrated for a simple model hierarchy consisting of a basic twodimensional model with several alternative three-dimensional extensions. The hierarchy is based on an out-of-equilibrium approach: market clearing is not assumed, supply is not equal to demand, and there exists a stock of unsold goods. Depending on actor behaviour, the models exhibit stable exponential growth or instabilities leading to nonlinear oscillations or economic collapse. In most cases, the simplicity and tractability of the models enables analytical solutions. The examples serve as illustration of more realistic models developed within the Multi-Actor Dynamic Integrated Assessment Model System (MADIAMS) to assess the long-term impacts of climate mitigation policies.

We compare phase transition and critical phenomena of bond percolation on Euclidean lattices, nonamenable graphs, and complex networks. On a Euclidean lattice, percolation showsa phase transition between the nonpercolating phase and percolating phase at the critical point. The critical point is stretched to a finite region, called the critical phase, on nonamenable graphs. To investigate the critical phase, we introduce a fractal exponent, which characterizes a subextensive order of the system. We perform the Monte Carlo simulations for percolation on two nonamenable graphs – the binary tree and the enhanced binary tree. The former shows the nonpercolating phase and the critical phase, whereas the latter shows all three phases. We also examine the possibility of critical phase in complex networks. Our conjecture is that networks with a growth mechanism have only the critical phase and the percolating phase. We study percolation on a stochastically growing network with and without a preferential attachment mechanism, and a deterministically growing network, called the decorated flower, to show that the critical phase appears in those models. We provide afinite-size scaling by using the fractal exponent, which would be a powerful method for numerical analysis of the phase transition involving the critical phase.

We consider a network of interacting phase oscillators endowed with adaptive mechanisms, leading the collective motion to a consensual dynamical state. Specifically, for a given network topology (i.e. an adjacency matrix) governing the mutual interactions, the adaptive mechanisms enable all oscillators to ultimately adopt a consensual frequency. Once reached, the consensual frequency subsists even if interactions between the oscillators are switched off. For the class of models we consider, the consensual frequency is independent of the network topology. Even though this independence might suggest that extension totime-dependent networks is straightforward, this is not true here. For time-dependent networks and spectra of the underlying Laplacian matrices, one may observe the emergence of more complex dynamics. Due to their high degree of complexity, these dynamics generally offer little hope for analytical tractability. In this paper, we focus on connected time-dependent networks with circulant adjacency matrices. The simple spectral structures and commutativity properties enjoyed by circulant matrices enable an analytical stability analysis of the consensus state. Ultimately, we are able to reduce the stability analysis to a dissipative harmonic oscillator with parametric pumping.