Abstract
We define what appears to be a new construction. Given a graph G and a positive integer k , the reduced k th power of G , denoted G ( k ) , is the configuration space in which k indistinguishable tokens are placed on the vertices of G , so that any vertex can hold up to k tokens. Two configurations are adjacent if one can be transformed to the other by moving a single token along an edge to an adjacent vertex. We present propositions related to the structural properties of reduced graph powers and, most significantly, provide a construction of minimum cycle bases of G ( k ) . The minimum cycle basis construction is an interesting combinatorial problem that is also useful in applications involving configuration spaces. For example, if G is the state-transition graph of a Markov chain model of a stochastic automaton, the reduced power G ( k ) is the state-transition graph for k identical (but not necessarily independent) automata. We show how the minimum cycle basis construction of G ( k ) may be used to confirm that state-dependent coupling of automata does not violate the principle of microscopic reversibility, as required in physical and chemical applications.
Highlights
Time-homogenous Markov chains [19] are used as a mathematical formalism in applications as diverse as computer systems performance analysis [21], queuing theory in operations research [18], simulation and analysis of stochastic chemical kinetics [12], and biophysical modeling of ion channel gating [10].c b This work is licensed under http://creativecommons.org/licenses/by/3.0/Ars Math
We show how the minimum cycle basis construction of G(k) may be used to confirm that state-dependent coupling of automata does not violate the principle of microscopic reversibility, as required in physical and chemical applications
Symmetry of model composition allows for interactions among stochastic automata, so long as the transition rates qij for i, j ∈ {1, 2, · · ·, v}, i = j are constant or functions of the number of automata n (t) in each state, 0 ≤ n (t) ≤ k, 1 ≤ ≤ v
Summary
Time-homogenous Markov chains [19] are used as a mathematical formalism in applications as diverse as computer systems performance analysis [21], queuing theory in operations research [18], simulation and analysis of stochastic chemical kinetics [12], and biophysical modeling of ion channel gating [10]. Many properties of a Markov chain, such its rate of mixing and its steady-state probability distribution, can be numerically calculated using its transition matrix [24]. The transition matrix for k coupled stochastic automata, each of which can be represented by an v-state Markov chain, has η = vk states and requires algorithms of O(v3k) complexity. Redundancy in master Markov chains for interacting stochastic automata can often be eliminated without approximation Both lumpability at the level of individual automata and model composition have been extensively researched, though the latter reduces the state space in a manner that eliminates Kronecker structure [4, 6, 13].
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