arXiv subjects

Matthew Visomirski

Publications and source records attributed to Matthew Visomirski.

3 recordsLinked to original sources

arXiv2026

New Families and Operations on Integrable Graphs

In this paper, we investigate the integrability of Lotka-Volterra (replicator) systems arising from interaction matrices generated from corresponding graph structures, continuing work started by Visomirski and Griffin [J. Phys. A., 58:015701, 2025] and Evripidou et al. [J. Phys. A., 55:325201, 2022] (among others). In particular, we define a new family of graphs, the spoked graphs, and show that all dynamics generated from this family are integrable. In reference to Evripidou et al. (2022), we define a new anti-cloning operator and show that its action on balanced tournament graphs (with odd vertex count) generates new graphs whose dynamics are integrable. Interestingly, we provide numerical evidence that this anti-cloning operation leads to chaotic behaviour when applied to other graph families (e.g., the directed cycles that generate the classically integrable Volterra lattice). This work completes a taxonomy of all integrable dynamics generated by directed graphs with up to six vertices started by Visomirski and Griffin (2025), and suggests several future directions of study on this topic.

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arXiv2026

Coherent Structures and Travelling Waves in Spatial Replicators from a Biased Volterra Lattice

The Volterra lattice is a well-known integrable family that is also a special class of replicator dynamics and whose members can be put in one-to-one correspondence with the directed cycle graphs. In this paper, we study a variation of the Volterra lattice by introducing a bias term in the replicator interaction matrix. The resulting system can still be put into one-to-one correspondence with the directed cycles, and the dynamics offer one generalisation of the classic rock-paper-scissors evolutionary game. We study the resulting spatial dynamics of this family, showing that travelling wave solutions are present in those dynamics corresponding to the directed 5- and 6-cycles, but not the 4-cycle. Instead, the 4-cycle exhibits a set of stationary solutions that we call `frozen waves' that are similar to but distinct from Turing patterns. This type of solution is also found in the dynamics generated from the directed 6- and 8-cycles. We discuss how these stationary solutions can represent naturally emergent ecological niches in these systems, and offer generalizing conjectures for the existence of both travelling wave solutions and frozen wave solutions in this family of dynamics as a potential program of future investigation.

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arXiv2024

Integrability of Generalised Skew-Symmetric Replicator Equations via Graph Embeddings

It is known that there is a one-to-one mapping between oriented directed graphs and zero-sum replicator dynamics (Lotka-Volterra equations) and that furthermore these dynamics are Hamiltonian in an appropriately defined nonlinear Poisson bracket. In this paper, we investigate the problem of determining whether these dynamics are Liouville-Arnold integrable, building on prior work graph in graph decloning by Evripidou et al. [J. Phys. A., 55:325201, 2022] and graph embedding by Paik and Griffin [Phys. Rev. E. 107(5): L052202, 2024]. Using the embedding procedure from Paik and Griffin, we show (with certain caveats) that when a graph producing integrable dynamics is embedded in another graph producing integrable dynamics, the resulting graph structure also produces integrable dynamics. We also construct a new family of graph structures that produces integrable dynamics that does not arise either from embeddings or decloning. We use these results, along with numerical methods, to classify the dynamics generated by almost all oriented directed graphs on six vertices, with three hold-out graphs that generate integrable dynamics and are not part of a natural taxonomy arising from known families and graph operations. These hold-out graphs suggest more structure is available to be found. Moreover, the work suggests that oriented directed graphs leading to integrable dynamics may be classifiable in an analogous way to the classification of finite simple groups, creating the possibility that there is a deep connection between integrable dynamics and combinatorial structures in graphs.

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