Chemical reaction networks in a Laplacian framework

Extended fractional-polynomial generalizations of diffusion and Fisher-KPP equations on directed networks: Modeling neurodegenerative progression

“…To prove that all solutions starting in R M ≥ 0 or R M ≤ 0 converge to equilibria, we will utilize a modified version of the "free-energy" function of the Becker-Döring model, describing the evolution of coagulation and fragmentation of clusters [74,75]. Buhagiar [74] observed that this free-energy function is a Lyapunov function for the Becker-Döring cluster equations, and recently, it has been demonstrated that it is also a Lyapunov function for any system of differential equations generated by chemical reaction networks whose components are strongly connected [32]. Due to the fact that it is possible to derive a system of differential equations similar to Eq.…”

“…However, similar to [30], it did not discuss the existence and uniqueness of the solution of the graph diffusion equation with initial conditions ρ i (0) ≥ 0 in a more general sense. Recently, Veerman et al [32] propounded a nonlinear Laplacian framework for CRNs, which leads to the polynomial system of differential equations , where S is a non-negative matrix with no zero rows, L out is an out-degree Laplacian matrix, and each element of vector ψ is a monomial. It has been demonstrated that for a componentwise strongly connected network when KerS ∩ ImL T = {0}, the proposed polynomial system has exactly one positive equilibrium x* in a specified invariant set X z .…”

“…In the literature, the standard graph Laplacian has been developed in a variety of ways to study spectral information, such as eigenvalues, eigenvectors, and Cheeger constants, for graphs and hypergraphs [11][12][13][14][15][16][17][18]. Aside from the extensions regarding spectral graph theory, several generalizations of the standard graph Laplacian exist particularly for exploring anomalous and nonlinear diffusion processes, most of which are restricted to undirected graphs [19][20][21][22][23][24][25][26][27][28][29][30][31], and for studying the dynamics of chemical reaction networks (CRNs) [32], which are directed networks [33,34]. Fractional graph Laplacian operators constructed by raising the Laplacian matrix to a real power between 0 and 1 [19][20][21][22], d-path Laplacian operators defined by using path matrices accounting for the existence of shortest paths of length d between two nodes [23], and the Mellin-transformed d-path Laplacian operators ∞ d=1 L d d −s [24][25][26], where s is a nonnegative real number, have been applied to investigate superdiffusion with a linear differential equation.…”