In this case, u could e. To provide an immediate graphical illustration of the power of the method, we set to stabilize two distinct fixed points.
In the first example see Fig 1 , upper left panel the control is designated so as to enhance the degree of activity of the peripheral nodes of the tree. These latter are characterized by a similar value of the activity, apart for slight randomly superposed fluctuations. Similarly, the nodes that define the bulk of the tree display a shared degree except for tiny stochastic modulation of residual activity.
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In the upper right panel of Fig 1 , the dual pattern is instead obtained and stabilized: the peripheral nodes are now being silenced and the activity concerns the nodes that fall in the center of the tree. Lower panel of Fig 1 displays the root locus diagram relative to the situation reported in the second panel.
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For each chosen fixed point that is being stabilized the zeros z k can be selected so as to make the asymptote intercept the horizontal axis in the left-half of the plane. Upper left panel: the control is modulated so as to enhance the activity of the peripheral nodes of the tree, as compared to the inner ones.
Upper right panel: the control makes now the bulk nodes more active as compared to the peripheral ones. Lower panel: the root locus diagram relative to the situation displayed in the second panel is plotted. These are randomly assigned and kept unchanged for all tests performed. The figure on the right is a zoom of the plot displayed on the left. As a second application of the proposed control strategy, we set to study the dynamics of the gut microbiota [ 6 ].
The intestinal microbiota is a microbial ecosystem of paramount importance to human health [ 39 ]. Efforts are currently aimed at understanding the microbiota ability to resist to enteric pathogens and assess the response to antibiotics cure of intestinal infections. Recent advances in DNA sequencing and metagenomics make it possible to quantitatively characterize the networks of interactions that rule the dynamics of the microbiota ecosystem.
This was for instance achieved in [ 40 ] by analyzing available data on mice [ 41 ] with an innovative approach which combines classical Lotka-Volterra model and regression techniques. Eleven species were identified and thoroughly analyzed in terms of self and mutual dynamics.
In the following we shall apply the method here developed to control the dynamics of the whole microbioma [ 40 ] or a limited sub-portion of it. The constants r i and s i are provided in [ 40 ] and follow from direct measurements. The weighted matrix of connections A presents both positive and negative entries, assigned according to [ 40 ]. The results of the analysis are organized under different headings that reflect the three distinct control strategies explored. Consider the system of 11 species, as defined in [ 40 ] see SI for a discussion on the bacterial species involved.
For illustrative purposes, we will restrict the analysis to all sub-systems that combine 5 out of the 11 species analyzed in [ 40 ]. The fixed points for the obtained 5 species systems are calculated. Those displaying positive concentrations are then retained for subsequent analysis.
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The stability of each selected fixed point is established upon evaluation of the spectrum of the Jacobian of the reduced dynamics. Starting from this setting, we will introduce a suitably shaped controller, following the above discussed guidelines, in order to stabilize a slightly perturbed version of an originally unstable fixed point, see pie charts in Fig 2 a. More specifically, one eigenvalue exhibits a positive real part, so flagging the instability that one aims to control. At variance, the crosses in Fig 2 b stand for the roots z k of , and fall in the left side of the complex plane. Panel a : a reduced 5 species subsystem of the microbiota is considered case A and all possible fixed points computed.
Only those displaying non-negative concentrations are retained and their stability assessed. The two pie charts refer to the initially unstable fixed point upper chart and the stabilized equilibrium lower chart. Panel b : the root loci diagram relative to the case discussed in panel a , is shown. The vertical red line is the asymptote that eventually attracts the two residual eigenvalues. Panel c : the goal is here to reduce the concentration of the pathogen species, C.
The concentration of C. The insertion of the species of uncl. Panel d : we now modify a stable fixed point, by driving to extinction one of the existing population, the pathogen C. The obtained concentrations are reported in the left graph pluses and confronted with the initial unperturbed solution diamonds. As anticipated. Select now a stable fixed point, mixture of five distinct species. One of them is Clostridium difficile , a species of Gram-positive spore-forming bacteria that may opportunistically dominate the gut flora, as an adverse effect of antibiotic therapy.
As controller we shall here employ one of the other 6 species that compose the microbioma [ 42 , 43 ]. The aim is to drive the system towards another equilibrium, stable to linear perturbations, which displays a decreased pathogen concentration. The equilibrium solution that can be attained by the controlled system is determined as , and clearly depends on the species used as controller.
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The only meaningful solutions are those displaying non negative components. In the example depicted in Fig 2 c only three solutions can be retained, namely the ones obtained by using uncl. Lachnospiraceae, uncl. Mollicutes and Enterococcus as respective control. In one of the inspected cases, the amount of C. The asymptotic concentration that is eventually attained is sensibly lower than the one initially displayed.
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The pie charts in Fig 2 c represent, respectively, the initial fixed point and the final stationary equilibrium, as shaped by the control in the most beneficial case, i. The root locus plot obtained for this specific case is reported in the SI. Importantly, the discussed scheme can be straightforwardly modified so as to account for a generic nonlinear self-reaction dynamics for the control species, e.
As an additional example, we wish to modify a stable fixed point of the dynamics, by silencing one of the existing populations with an indirect control. In other words we shall introduce and stabilize a novel fixed point, that displays a negligible residual concentration of the undesired species, by acting on the other species of the collection. This is for instance relevant when aiming at, e. With this in mind, we consider a reduced ecosystem consisting of 6 species, selected among the 11 that define the microbiota. A stable fixed point exists black diamonds in Fig 2 d , left panel which displays a significant concentration of C.
Assign to this latter species the index 6. We now insert a controller which cannot directly interfere with C. We then require the concentration of the C. This latter condition translates into a constraint that should be matched by the other 5 species, namely. A possible solution of the problem is reported in Fig 2 d : in the left panel plus symbols the components of the fixed point stabilized by the control are shown.
As anticipated, the concentration of C. Notice also that the control scheme here developed could be in principle exploited to drive the system towards a stable fixed point of the unperturbed dynamics, starting from out-of-equilibrium initial conditions. Such parameters could hence be chosen so as to reflect the specificity of the target system. In the annexed SI we demonstrate this intriguing possibility. Summing up, we have here proposed and tested a method to control the dynamics of multidimensional systems on a complex graph.
Enabling Controlling Complex Networks with Local Topological Information
The original system is made up of N interacting populations obeying a set of general equations, which bear attributes of universality. One additional species, here referred to as the controller, is inserted and made interact with the existing constellation of species. By tuning the strength of the couplings or equivalently the composition of the inserted controller , we can drive the system towards a desired equilibrium.
The stability of the achieved solution is enforced by adjusting the parameters that ultimately govern the rate of change of the controller.