Mathew Zuparic

Fixed points and stability in the two-network frustrated Kuramoto model

We examine a modification of the Kuramoto model for phase oscillators coupled on a network. Here, two populations of oscillators are considered, each with different network topologies, internal and cross-network couplings and frequencies. Additionally, frustration parameters for the interactions of the cross-network phases are introduced. This may be regarded as a model of competing populations: internal to any one network phase synchronisation is a target state, while externally one or both populations seek to frequency synchronise to a phase in relation to the competitor. We conduct fixed point analyses for two regimes: one, where internal phase synchronisation occurs for each population with the potential for instability in the phase of one population in relation to the other; the second where one part of a population remains fixed in phase in relation to the other population, but where instability may occur within the first population leading to ‘fragmentation’. We compare analytic results to numerical solutions for the system at various critical thresholds.

Gaussian noise and the two-network frustrated Kuramoto model

Abstract We examine analytically and numerically a variant of the stochastic Kuramoto model for phase oscillators coupled on a general network. Two populations of phased oscillators are considered, labelled ‘Blue’ and ‘Red’, each with their respective networks, internal and external couplings, natural frequencies, and frustration parameters in the dynamical interactions of the phases. We disentangle the different ways that additive Gaussian noise may influence the dynamics by applying it separately on zero modes or normal modes corresponding to a Laplacian decomposition for the sub-graphs for Blue and Red. Under the linearisation ansatz that the oscillators of each respective network remain relatively phase-synchronised centroids or clusters, we are able to obtain simple closed-form expressions using the Fokker–Planck approach for the dynamics of the average angle of the two centroids. In some cases, this leads to subtle effects of metastability that we may analytically describe using the theory of ratchet potentials. These considerations are extended to a regime where one of the populations has fragmented in two. The analytic expressions we derive largely predict the dynamics of the non-linear system seen in numerical simulation. In particular, we find that noise acting on a more tightly coupled population allows for improved synchronisation of the other population where deterministically it is fragmented.

Noise-driven current reversal and stabilization in the tilted ratchet potential subject to tempered stable Lévy noise

We consider motion of a particle in a one-dimensional tilted ratchet potential subject to two-sided tempered stable Lévy noise characterized by strength Ω, fractional index α, skew θ, and tempering λ. We derive analytic solutions to the corresponding Fokker-Planck Lévy equations for the probability density. Due to the periodicity of the potential, we carry out reduction to a compact domain and solve for the analog of steady-state solutions which we represent as wrapped probability density functions. By solving for the expected value of the current associated with the particle motion, we are able to determine thresholds for metastability of the system, namely when the particle stabilizes in a well of the potential and when the particle is in motion, for example as a consequence of the tilt of the potential. Because the noise may be asymmetric, we examine the relationship between skew of the noise and the tilt of the potential. With tempering, we find two remarkable regimes where the current may be reversed in a direction opposite to the tilt or where the particle may be stabilized in a well in circumstances where deterministically it should flow with the tilt.

Complex socio-organisational phenomena and ill-defined problem spaces: a multi-method approach to the rescue

This paper reports on a multi-method (also referred to as mixed method) approach for studying complex socio-organisational phenomena where the problem space is ill-defined, and its utility in providing meaningful decision support to a branch within a military headquarters. The study comprised two phases: diagnostic (I) and intervention (II). The Phase I instruments were an organisation and team effectiveness survey and workforce modelling. Phase II involved a series of workshops situated in sensemaking and complexity theory. Combining the modelling results with qualitative and quantitative data provided by the survey resulted in: capitalisation on the strengths of individual methods while compensating for their weaknesses, a better delineation of issues and understanding of the branch processes, and provision of a solid foundation for design of the intervention workshops. This research design integrates multiple organisational measures to develop instruments suitable for application in other settings where problem definitions are nebulous.

Quantifying the impact of communication on performance in multi-agent teams

In this work, we relate the extent and quality of inter-agent communication and the overall performance in teams of multiple agents. Specifically, we examine the RoboCup Soccer Simulation 2D League, and carry out multiple simulation experiments against two evenly matched teams. For each simulated run (a 2D soccer simulation game), we generate the communication efficiencies (i.e., communications sent/communications received) for each agent pair. Applying linear regression and principal component analyses, we then correlate these efficiencies with measures of performance (i.e., goals scored and goals conceded), enabling the construction of inter-agent communication networks. Analysis of these networks highlights the microscopic player-to-player and macroscopic role-to-role communications correlated with performance. The approach determines the salient pathways within inter-agent communications which globally affect the coordination and the overall performance in multi-agent teams.