1,559 research outputs found
Gradient and Passive Circuit Structure in a Class of Non-linear Dynamics on a Graph
We consider a class of non-linear dynamics on a graph that contains and
generalizes various models from network systems and control and study
convergence to uniform agreement states using gradient methods. In particular,
under the assumption of detailed balance, we provide a method to formulate the
governing ODE system in gradient descent form of sum-separable energy
functions, which thus represent a class of Lyapunov functions; this class
coincides with Csisz\'{a}r's information divergences. Our approach bases on a
transformation of the original problem to a mass-preserving transport problem
and it reflects a little-noticed general structure result for passive network
synthesis obtained by B.D.O. Anderson and P.J. Moylan in 1975. The proposed
gradient formulation extends known gradient results in dynamical systems
obtained recently by M. Erbar and J. Maas in the context of porous medium
equations. Furthermore, we exhibit a novel relationship between inhomogeneous
Markov chains and passive non-linear circuits through gradient systems, and
show that passivity of resistor elements is equivalent to strict convexity of
sum-separable stored energy. Eventually, we discuss our results at the
intersection of Markov chains and network systems under sinusoidal coupling
A network dynamics approach to chemical reaction networks
A crisp survey is given of chemical reaction networks from the perspective of
general nonlinear network dynamics, in particular of consensus dynamics. It is
shown how by starting from the complex-balanced assumption the reaction
dynamics governed by mass action kinetics can be rewritten into a form which
allows for a very simple derivation of a number of key results in chemical
reaction network theory, and which directly relates to the thermodynamics of
the system. Central in this formulation is the definition of a balanced
Laplacian matrix on the graph of chemical complexes together with a resulting
fundamental inequality. This directly leads to the characterization of the set
of equilibria and their stability. Both the form of the dynamics and the
deduced dynamical behavior are very similar to consensus dynamics. The
assumption of complex-balancedness is revisited from the point of view of
Kirchhoff's Matrix Tree theorem, providing a new perspective. Finally, using
the classical idea of extending the graph of chemical complexes by an extra
'zero' complex, a complete steady-state stability analysis of mass action
kinetics reaction networks with constant inflows and mass action outflows is
given.Comment: 18 page
Complex and detailed balancing of chemical reaction networks revisited
The characterization of the notions of complex and detailed balancing for
mass action kinetics chemical reaction networks is revisited from the
perspective of algebraic graph theory, in particular Kirchhoff's Matrix Tree
theorem for directed weighted graphs. This yields an elucidation of previously
obtained results, in particular with respect to the Wegscheider conditions, and
a new necessary and sufficient condition for complex balancing, which can be
verified constructively.Comment: arXiv admin note: substantial text overlap with arXiv:1502.0224
Computing all possible graph structures describing linearly conjugate realizations of kinetic systems
In this paper an algorithm is given to determine all possible structurally different linearly conjugate realizations of a given kinetic polynomial system. The solution is based on the iterative search for constrained dense realizations using linear programming. Since there might exist exponentially many different reaction graph structures, we cannot expect to have a polynomial-time algorithm, but we can organize the computation in such a way that polynomial time is elapsed between displaying any two consecutive realizations. The correctness of the algorithm is proved, and possibilities of a parallel implementation are discussed. The operation of the method is shown on two illustrative examples
Fast solitons on star graphs
We define the Schr\"odinger equation with focusing, cubic nonlinearity on
one-vertex graphs. We prove global well-posedness in the energy domain and
conservation laws for some self-adjoint boundary conditions at the vertex, i.e.
Kirchhoff boundary condition and the so called and boundary
conditions. Moreover, in the same setting we study the collision of a fast
solitary wave with the vertex and we show that it splits in reflected and
transmitted components. The outgoing waves preserve a soliton character over a
time which depends on the logarithm of the velocity of the ingoing solitary
wave. Over the same timescale the reflection and transmission coefficients of
the outgoing waves coincide with the corresponding coefficients of the linear
problem. In the analysis of the problem we follow ideas borrowed from the
seminal paper \cite{[HMZ07]} about scattering of fast solitons by a delta
interaction on the line, by Holmer, Marzuola and Zworski; the present paper
represents an extension of their work to the case of graphs and, as a
byproduct, it shows how to extend the analysis of soliton scattering by other
point interactions on the line, interpreted as a degenerate graph.Comment: Sec. 2 revised; several misprints corrected; added references; 32
page
Kirchhoff Graphs
Kirchhoff\u27s laws are well-studied for electrical networks with voltage and current sources, and edges marked by resistors. Kirchhoff\u27s voltage law states that the sum of voltages around any circuit of the network graph is zero, while Kirchhoff\u27s current law states that the sum of the currents along any cutset of the network graph is zero. Given a network, these requirements may be encoded by the circuit matrix and cutset matrix of the network graph. The columns of these matrices are indexed by the edges of the network graph, and their row spaces are orthogonal complements. For (chemical or electrochemical) reaction networks, one must naturally study the opposite problem, beginning with the stoichiometric matrix rather than the network graph. This leads to the following question: given such a matrix, what is a suitable graphic rendering of a network that properly visualizes the underlying chemical reactions? Although we can not expect uniqueness, the goal is to prove existence of such a graph for any matrix. Specifically, we study Kirchhoff graphs, originally introduced by Fehribach. Mathematically, Kirchhoff graphs represent the orthocomplementarity of the row space and null space of integer-valued matrices. After introducing the definition of Kirchhoff graphs, we will survey Kirchhoff graphs in the context of several diverse branches of mathematics. Beginning with combinatorial group theory, we consider Cayley graphs of the additive group of vector spaces, and resolve the existence problem for matrices over finite fields. Moving to linear algebra, we draw a number of conclusions based on a purely matrix-theoretic definition of Kirchhoff graphs, specifically regarding the number of vector edges. Next we observe a geometric approach, reviewing James Clerk Maxwell\u27s theory of reciprocal figures, and presenting a number of results on Kirchhoff duality. We then turn to algebraic combinatorics, where we study equitable partitions, quotients, and graph automorphisms. In addition to classifying the matrices that are the quotient of an equitable partition, we demonstrate that many Kirchhoff graphs arise from equitable edge-partitions of directed graphs. Finally we study matroids, where we review Tutte\u27s algorithm for determining when a binary matroid is graphic, and extend this method to show that every binary matroid is Kirchhoff. The underlying theme throughout each of these investigations is determining new ways to both recognize and construct Kirchhoff graphs
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