Limits...
Implications of network topology on stability.

Kinkhabwala A - PLoS ONE (2015)

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.

View Article: PubMed Central - PubMed

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.

No MeSH data available.


Related in: MedlinePlus

Analysis of general networks.(A) Network consisting of two distinct levels of overlapping cycles. (B) Species trajectories (χ1, cyan; χ2, magenta; χ3, black) of an explicit algebraic version (Equations 158–160) of a network having the influence topology shown in A (see §6 for details). (C) Similar two-level network as in A, but with the levels swapped. (D) Species trajectories (χ1, cyan; χ2, magenta; χ3, black) of an explicit algebraic version (Equations 161–163) of a network having the influence topology shown in C (see §6 for details). (E) Example of a more complicated multilevel network.
© Copyright Policy
Related In: Results  -  Collection

License
getmorefigures.php?uid=PMC4380337&req=5

pone.0122150.g012: Analysis of general networks.(A) Network consisting of two distinct levels of overlapping cycles. (B) Species trajectories (χ1, cyan; χ2, magenta; χ3, black) of an explicit algebraic version (Equations 158–160) of a network having the influence topology shown in A (see §6 for details). (C) Similar two-level network as in A, but with the levels swapped. (D) Species trajectories (χ1, cyan; χ2, magenta; χ3, black) of an explicit algebraic version (Equations 161–163) of a network having the influence topology shown in C (see §6 for details). (E) Example of a more complicated multilevel network.

Mentions: Consider the simple example shown in Fig. 12A of a network consisting of two such levels. A specific algebraic form for this network is:V11V1232V23V34x˙1=x1−(1+x3)x1x2(158)x˙2=(1+x3)x1x2−x2,(159)x˙3=−0.1x3.(160)Trajectories of the species in this network for the specific choice of initial conditions of x1(0) = 1, x2(0) = 3, and x3(0) = 1 is shown in Fig. 12B. Here, x1 and x2 exhibit Lotka-Volterra-like oscillations that grow in an asymptotic fashion towards a fixed amplitude, with x3 decaying to 0. Starting from Level 1, the 1-cycle is negative and therefore stable (see §4), with x3 in the vicinity of the steady state asymptotically approaching a steady value (in this case, to the value of 0, but any other fixed value would produce the same result). Species x3 then serves effectively as a constant species input to Level 2, in the same manner as an orphan species (see §3), meaning we can then proceed to consideration of Level 2. In the vicinity of the steady state, Level 2 can be considered as an (asymptotically) autonomous subnetwork. This level has an influence topology identical to the Lotka-Volterra network, permitting the possibility for instability.


Implications of network topology on stability.

Kinkhabwala A - PLoS ONE (2015)

Analysis of general networks.(A) Network consisting of two distinct levels of overlapping cycles. (B) Species trajectories (χ1, cyan; χ2, magenta; χ3, black) of an explicit algebraic version (Equations 158–160) of a network having the influence topology shown in A (see §6 for details). (C) Similar two-level network as in A, but with the levels swapped. (D) Species trajectories (χ1, cyan; χ2, magenta; χ3, black) of an explicit algebraic version (Equations 161–163) of a network having the influence topology shown in C (see §6 for details). (E) Example of a more complicated multilevel network.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4380337&req=5

pone.0122150.g012: Analysis of general networks.(A) Network consisting of two distinct levels of overlapping cycles. (B) Species trajectories (χ1, cyan; χ2, magenta; χ3, black) of an explicit algebraic version (Equations 158–160) of a network having the influence topology shown in A (see §6 for details). (C) Similar two-level network as in A, but with the levels swapped. (D) Species trajectories (χ1, cyan; χ2, magenta; χ3, black) of an explicit algebraic version (Equations 161–163) of a network having the influence topology shown in C (see §6 for details). (E) Example of a more complicated multilevel network.
Mentions: Consider the simple example shown in Fig. 12A of a network consisting of two such levels. A specific algebraic form for this network is:V11V1232V23V34x˙1=x1−(1+x3)x1x2(158)x˙2=(1+x3)x1x2−x2,(159)x˙3=−0.1x3.(160)Trajectories of the species in this network for the specific choice of initial conditions of x1(0) = 1, x2(0) = 3, and x3(0) = 1 is shown in Fig. 12B. Here, x1 and x2 exhibit Lotka-Volterra-like oscillations that grow in an asymptotic fashion towards a fixed amplitude, with x3 decaying to 0. Starting from Level 1, the 1-cycle is negative and therefore stable (see §4), with x3 in the vicinity of the steady state asymptotically approaching a steady value (in this case, to the value of 0, but any other fixed value would produce the same result). Species x3 then serves effectively as a constant species input to Level 2, in the same manner as an orphan species (see §3), meaning we can then proceed to consideration of Level 2. In the vicinity of the steady state, Level 2 can be considered as an (asymptotically) autonomous subnetwork. This level has an influence topology identical to the Lotka-Volterra network, permitting the possibility for instability.

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.

View Article: PubMed Central - PubMed

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.

No MeSH data available.


Related in: MedlinePlus