Implications of network topology on stability.

Kinkhabwala A - PLoS ONE (2015)

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pone.0122150.g005: 2-cycle networks.(A) All possible 2-cycle network influence topologies. (B) Rotation network influence topology.
Mentions: All possible 2-cycle networks are schematically represented in the single graph shown in Fig. 5 (the degeneracy of these networks will be addressed further below). After cycle compaction (defining q0 = r1r2) and temporal scaling ( and ), it is clear that b1 = 0 and b2 = −c2 = ∓1. For all of these topologies, it is obvious that Δ1 = −b1 = 0 and Δ2 = −b1b2+b0b3 = 0 due to the absence of 1-cycles in the network. That all Routh-Hurwitz conditions are equal to 0 implies that no information can be obtained from first-order perturbations about the steady state; higher order perturbations must be assessed to establish the stability of a given steady state. An important example of a 2-cycle network is:x˙1=−k1x2(45)x˙2=k2x1,(46)which, for k1 = k2, corresponds to constant rotational motion at a fixed radius determined by the initial values (boundary conditions). We can rewrite the rotation network in a more general way as:x˙1=−V21(47)x˙2=V12.(48)In the above, I again employ the shorthand notation for the reaction functions explained above, with corresponding to reaction k with positive monotonic dependence on species i. The principal minors for this generalized network are b1 = 0 and b2 = −c2 = 1, which, as for the general case, leads to Δ1 = 0 and Δ2 = 0 and no information about steady state stability obtainable at first order. For the original rotation network (Equations 45 and 46), the linearity of the reactions implies that all higher order perturbations are trivially 0. The different solutions of this network depend on the initial conditions and foliate the x1-x2 phase space as circles of each possible radius centered on the origin. Inclusion of non-zero constant terms (V0 terms) would merely shift the origin of these foliated circular trajectories.

Bottom Line: Visualization of the Hurwitz determinants over the reduced parameters defines the network's stability phase space, delimiting the range of its dynamics (specifically, the possible numbers of unstable roots at each steady state solution).Any further explicit algebraic specification of the network will project onto this stability phase space.Stability analysis via this hierarchical approach is demonstrated on classical networks from multiple fields.

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Affiliation: Department of Systemic Cell Biology, Max Planck Institute of Molecular Physiology, Dortmund, Germany.

ABSTRACT
In analogy to chemical reaction networks, I demonstrate the utility of expressing the governing equations of an arbitrary dynamical system (interaction network) as sums of real functions (generalized reactions) multiplied by real scalars (generalized stoichiometries) for analysis of its stability. The reaction stoichiometries and first derivatives define the network's "influence topology", a signed directed bipartite graph. Parameter reduction of the influence topology permits simplified expression of the principal minors (sums of products of non-overlapping bipartite cycles) and Hurwitz determinants (sums of products of the principal minors or the bipartite cycles directly) for assessing the network's steady state stability. Visualization of the Hurwitz determinants over the reduced parameters defines the network's stability phase space, delimiting the range of its dynamics (specifically, the possible numbers of unstable roots at each steady state solution). Any further explicit algebraic specification of the network will project onto this stability phase space. Stability analysis via this hierarchical approach is demonstrated on classical networks from multiple fields.

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