Asymptotics and Estimates for Spectral Minimal Partitions of Metric Graphs

We study properties of spectral minimal partitions of metric graphs within the framework recently introduced in Kennedy et al. (Calc Var 60:6, 2021). We provide sharp lower and upper estimates for minimal partition energies in different classes of partitions; while the lower bounds are reminiscent of the classic isoperimetric inequalities for metric graphs, the upper bounds are more involved and mirror the combinatorial structure of the metric graph as well. Combining them, we deduce that these spectral minimal energies also satisfy a Weyl-type asymptotic law similar to the well-known one for eigenvalues of quantum graph Laplacians with various vertex conditions. Drawing on two examples we show that in general no second term in the asymptotic expansion for minimal partition energies can exist, but show that various kinds of behaviour are possible. We also study certain aspects of the asymptotic behaviour of the minimal partitions themselves.

Systemic States of Spreading Activation in Describing Associative Knowledge Networks II: Generalisations with Fractional Graph Laplacians and q-Adjacency Kernels

Associative knowledge networks are often explored by using the so-called spreading activation model to find their key items and their rankings. The spreading activation model is based on the idea of diffusion- or random walk -like spreading of activation in the network. Here, we propose a generalisation, which relaxes an assumption of simple Brownian-like random walk (or equally, ordinary diffusion process) and takes into account nonlocal jump processes, typical for superdiffusive processes, by using fractional graph Laplacian. In addition, the model allows a nonlinearity of the diffusion process. These generalizations provide a dynamic equation that is analogous to fractional porous medium diffusion equation in a continuum case. A solution of the generalized equation is obtained in the form of a recently proposed q-generalized matrix transformation, the so-called q-adjacency kernel, which can be adopted as a systemic state describing spreading activation. Based on the systemic state, a new centrality measure called activity centrality is introduced for ranking the importance of items (nodes) in spreading activation. To demonstrate the viability of analysis based on systemic states, we use empirical data from a recently reported case of a university students’ associative knowledge network about the history of science. It is shown that, while a choice of model does not alter rankings of the items with the highest rank, rankings of nodes with lower ranks depend essentially on the diffusion model.

A note on the degree conjecture for separability of multipartite quantum states

The degree conjecture for bipartite quantum states which are normalized graph Laplacians was first put forward by Braunstein et al. [Phys. Rev. A 73 (2006) 012320]. The degree criterion, which is equivalent to PPT criterion, is simpler and more efficient to detect the separability of quantum states associated with graphs. Hassan et al. settled the degree conjecture for the separability of multipartite quantum states in [J. Math. Phys. 49 (2008) 0121105]. It is proved that the conjecture is true for pure multipartite quantum states. However, the degree condition is only necessary for separability of a class of quantum mixed states. It does not apply to all mixed states. In this paper, we show that the degree conjecture holds for the mixed quantum states of nearest point graph. As a byproduct, the degree criterion is necessary and sufficient for multipartite separability of [Formula: see text]-qubit quantum states associated with graphs.

Simulation of bimolecular reactions: Numerical challenges with the graph Laplacian

An important framework for modelling and simulation of chemical reactions is a Markov process sometimes known as a master equation. Explicit solutions of master equations are rare; in general the explicit solution of the governing master equation for a bimolecular reaction remains an open question. We show that a solution is possible in special cases. One method of solution is diagonalization. The crucial class of matrices that describe this family of models are non-symmetric graph Laplacians. We illustrate how standard numerical algorithms for finding eigenvalues fail for the non-symmetric graph Laplacians that arise in master equations for models of chemical kinetics. We propose a novel way to explore the pseudospectra of the non-symmetric graph Laplacians that arise in this class of applications, and illustrate our proposal by Monte Carlo. Finally, we apply the Magnus expansion, which provides a method of simulation when rates change in time. Again the graph Laplacian structure presents some unique issues: standard numerical methods of more than second-order fail to preserve positivity. We therefore propose a method that achieves fourth-order accuracy, and maintain positivity. References A. Basak, E. Paquette, and O. Zeitouni. Regularization of non-normal matrices by gaussian noise–-the banded toeplitz and twisted toeplitz cases. In Forum of Mathematics, Sigma, volume 7. Cambridge University Press, 2019. doi:10.1017/fms.2018.29. S. Blanes, F. Casas, J. A. Oteo, and J. Ros. The magnus expansion and some of its applications. Phys. Rep., 470(5-6):151–238, 2009. doi:10.1016/j.physrep.2008.11.001. B. A. Earnshaw and J. P. Keener. Invariant manifolds of binomial-like nonautonomous master equations. SIAM J. Appl. Dyn. Sys., 9(2):568–588, 2010. doi10.1137/090759689. J. Gunawardena. A linear framework for time-scale separation in nonlinear biochemical systems. PloS One, 7(5):e36321, 2012. doi:10.1371/journal.pone.0036321. A. Iserles and S. MacNamara. Applications of magnus expansions and pseudospectra to markov processes. Euro. J. Appl. Math., 30(2):400–425, 2019. doi:10.1017/S0956792518000177. S. MacNamara. Cauchy integrals for computational solutions of master equations. ANZIAM Journal, 56:32–51, 2015. doi:10.21914/anziamj.v56i0.9345. S. MacNamara, A. M. Bersani, K. Burrage, and R. B. Sidje. Stochastic chemical kinetics and the total quasi-steady-state assumption: Application to the stochastic simulation algorithm and chemical master equation. J. Chem. Phys., 129:095105, 2008. doi:10.1063/1.2971036. S. MacNamara and K. Burrage. Stochastic modeling of naive T cell homeostasis for competing clonotypes via the master equation. SIAM Multiscale Model. Sim., 8(4):1325–1347, 2010. S. MacNamara, K. Burrage, and R. B. Sidje. Multiscale modeling of chemical kinetics via the master equation. SIAM Multiscale Model. and Sim., 6(4):1146–1168, 2008. doi:10.1137/060678154. S. MacNamara, Wi. McLean, and K. Burrage. Wider contours and adaptive contours, pages 79–98. Springer International Publishing, 2019. doi:10.1007/978-3-030-04161-8_7. M. J. Shon. Trapping and manipulating single molecules of DNA. PhD thesis, Harvard University, 2014. http://nrs.harvard.edu/urn-3:HUL.InstRepos:11744428. M. J. Shon and A. E. Cohen. Mass action at the single-molecule level. J. Am. Chem. Soc., 134(35):14618–14623, 2012. doi:10.1021/ja3062425. C. Timm. Random transition-rate matrices for the master equation. Phys. Rev. E, 80(2):021140, 2009. doi:10.1103/PhysRevE.80.021140. L. N. Trefethen and M. Embree. Spectra and pseudospectra: The behavior of nonnormal matrices and operators. Princeton University Press, 2005. https://press.princeton.edu/books/hardcover/9780691119465/spectra-and-pseudospectra.

Non-local network dynamics via fractional graph Laplacians

We introduce non-local dynamics on directed networks through the construction of a fractional version of a non-symmetric Laplacian for weighted directed graphs. Furthermore, we provide an analytic treatment of fractional dynamics for both directed and undirected graphs, showing the possibility of exploring the network employing random walks with jumps of arbitrary length. We also provide some examples of the applicability of the proposed dynamics, including consensus over multi-agent systems described by directed networks.