supplementary material for conditional random matrix ensembles

Supplementary Material forConditional Random Matrix Ensembles and The

Stability of Dynamical Systems

P. Kirk, D.M.Y. Rolando, A.L. MacLean & M.P.H. Stumpf

April 12, 2015

1 Defining meaningful stability probabilities

In the previous section, we demonstrated the importance of accounting for thestructure present in Jacobians derived from real ODE models when calculatingstability probabilities. For similar reasons, it is also important to account forother properties such as the feasibility of equilibrium points (i.e. whether or notthey are physically meaningful). Since arbitrary choices of the matrix p.d.f. h willlead to arbitrary stability probabilities (further illustrated in Fig. 1), it is vitalthat we instead consider meaningful choices for h that are conditioned on suchproperties. As in other studies (outlined in the main manuscript), here we do soby defining random matrix ensembles (RMEs) for specific models via distributionsover model parameters.

1.1 Defining alternative RMEs

The estimated stability probability defined in the main paper is the probabilityof stability, conditional on a given system architecture (i.e. conditional on thestructure and dependency in the Jacobian that arises from a particular model).

To study the consequences of neglecting or incompletely capturing this struc-ture, we first consider two random matrix distributions constructed by permutationof the entries of our original RME. To allow us to probe further the effects of RMEchoice on estimated stability probabilities, we moreover consider some RMEs forwhich the marginal distributions of the Jacobian entries are constrained to haveparticular parametric forms. Finally, we consider an RME in which we make someattempt to capture the dependency between the entries of the Jacobian.

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1

2

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Figure 1: How the choice of RME can impact the probability of stability and usualmistakes made because of it. The grey ellipse represents Mn(R), the blue areasare the stable areas of the system. Each black ellipse represents the space thatcould potentially be reached by sampling from an RME inferred from a specificODE system. 1. The first error is to forget that one Jacobian can correspond totwo (or more) different real dynamical systems, as illustrated by these two systemsoverlapping. 2. The second error is to forget that the biophysically feasible area,here represented with red stripes, can be different from the overall mathematicallyfeasible area and have very different stability probability. 3. A third mistake isto think that any result obtained for one specific system with m links (or anycharacteristic) can be generalised to any system with m links. Here for instancethe small circle represents the area of the space covered by a specific system withm links, the bigger circle on the other hand is the space covered by all the systemswith m links. It is clear that probability of stability on each space is very differentand thus generalising would be misleading.

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1.1.1 The independent ensemble

First, we form a new matrix ensemble, K∗(1), . . . , K∗(N), in which the dependency

between entries is broken. For each ` ∈ {1, . . . , N} and (i, j) ∈ {1, . . . , p} ×{1, . . . , p} we set

(K∗(`)

)ij

=(J∗θ(q)

)ij

, with q drawn uniformly at random from

{1, . . . , N}. In this way, the marginal distribution of the ij-entries across theensemble of K∗ matrices is the same as the marginal distribution of ij-entriesacross the ensemble of J∗θ matrices. Maintaining the marginal distributions ensuresthat the dependency between entries is the only quantity that we are altering: inparticular, the location of zeros in the matrix and the magnitudes of interactionstrengths are maintained.

1.1.2 The i.i.d. ensemble

We construct a further RME, L∗(1), . . . , L∗(N), where for each ` ∈ {1, . . . , N} and

(i, j) ∈ {1, . . . , p}×{1, . . . , p}, we set(L∗(`)

)ij

=(J∗θ(q)

)rs

, with q drawn uniformly

at random from {1, . . . , N}, and r and s (independently) drawn uniformly atrandom from {1, . . . p}. Now, the location of zeros in the matrix is no longer fixed;although the probability of an entry being zero is the same for the L∗ matrices asfor the J∗θ ’s and K∗’s. Moreover, each entry of the L∗ matrices is i.i.d..

1.1.3 The independent normal ensemble

For each (i, j), we fit an independent normal distribution to the ij-entries of thesampled Jacobians, J∗

θ(1) , . . . , J∗θ(N) . That is, for each (i, j), we calculate the mean,

µind(i,j) =

1

N

N∑q=1

(J∗θ(q)

)ij,

and standard deviation,

σind(i,j) = s.d.

{(J∗θ(q)

)ij

}N

q=1.

We then construct another RME,M ind(1) , . . . ,M

ind(N), where for each ` ∈ {1, . . . , N}

and (i, j) ∈ {1, . . . , p} × {1, . . . , p}, we set(M ind

(`)

)ij

to be a sample drawn from

the univariate normal distribution with mean µind(i,j) and standard deviation σind

(i,j).By construction, the mean and standard deviation of the ij-entries across the en-semble of M ind matrices are the same as the mean and standard deviation of theij-entries across the ensemble of J∗θ matrices (the FCS ensemble) and across theensemble of K∗ matrices (the independent ensemble).

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1.1.4 The independent Pearson ensemble

As in the independent normal case (Appendix 1.1.3), except that rather than justcapturing the mean and standard deviation, we also capture the skewness andkurtosis of the ij-entries of the J∗θ matrices. That is, in addition to µind

(i,j) and σind(i,j)

defined earlier, we also calculate skewness

γind(i,j) = skewness{(J∗θ(q)

)ij

}N

q=1.

and kurtosis,

κind(i,j) = kurtosis{(J∗θ(q)

)ij

}N

q=1.

We then construct an RME, Mpear(1) , . . . ,Mpear

(N) , where for each ` ∈ {1, . . . , N}and (i, j) ∈ {1, . . . , p}× {1, . . . , p}, we set

(M(`)

)pearij

to be a sample drawn from a

univariate Pearson distribution with mean µind(i,j), standard deviation σind

(i,j), skewnessγ(i,j), and kurtosis κ(i,j). This RME thus shares many of the properties of themarginal distributions of ij-entries across the ensemble of J∗θ matrices, but doesnot capture the dependencies between them.

1.1.5 The i.i.d. normal ensemble

As in the independent normal case (Appendix 1.1.3), except that rather thanfitting to the ij-entries of the J∗θ matrices, we instead fit to the ij-entries of theL∗ matrices (i.e. those from the i.i.d. ensemble).

1.1.6 The multivariate normal RME

Finally, we construct an RME that attempts to capture some of the dependenciesbetween the entries of the J∗θ matrices. We define c(M) to be the vector ob-tained by concatenating the columns of the matrix M (and further define c−1

be the inverse operation, so that, for example, c−1(c(M)) = M). Applyingc(·) to the matrices from our FCS RME, we obtain N vectors of length p × p,namely: c(J∗

θ(1)), . . . , c(J∗θ(N)). To these, we fit (by maximum likelihood) a (p× p)-

variate normal distribution. We then sample N vectors, v1, . . . , vN , of lengthp× p from this distribution, and form a new ensemble Mmvn

(1) , . . . ,Mmvn(N) by setting

Mmvn(q) = c−1(vq).

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2 Exemplar models

2.1 Model 1: The Lorenz system

We start by considering the Lorenz system, merely because it is simple and widelyknown.

We define it in the same way as it is in Lorenz’s paper:

x = σ(y − x)

y = x(r − z)− yz = xy − bz.

We consider values of parameters for which the following equilibrium point exists:

[x, y, z]> =[√

b(r − 1),√b(r − 1), r − 1

]>,

and consider the probability of stability for this point.

2.2 Model 2: A model of the cell division cycle

We use the model as defined in the phase plane analysis of the Tyson paper:

u = k4(w − u)(k′4k4

+ u2)− k6u

v = (k1[aa]/[CT ])− k2(v − w)− k6uw = k3[CT ](1− w)(v − w)− k6uy = (k1[aa]/[CT ])− k2(v − w)− k7(y − v)

This system has only one fixed point, details of which can be found in theMatlab code (available upon request from the authors). We assess the stabilityprobability for this point.

2.3 Model 3: The Nowak and Bangham model

As a second example to study, we consider a model of viral dynamics proposedby Nowak and Bangham (1996). This model describes the interactions betweenuninfected cells, x, infected cells, y, and free virus particles, v:

x = λ− dx− βxvy = βxv − ayv = ky − uv.

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This system has two fixed points. We assess the stability probability for the moreinteresting of these, namely,

[x, y, v]> =

[au

βk,λβk − dau

βka,λβk − dau

βku

].

2.4 Models 4, 5 and 6: The SEIR and extended SEIRmodels

2.4.1 Presentation and background

We consider two different extended versions of the SEIR model in which we alloweither the Exposed population or the Infective population to have a collection ofsubpopulations.

Recall the standard SEIR model:

S = µ− βSI − µSE = βSI − (µ+ α)E

I = αE − (µ+ γ)I

Here, S is the proportion of the population that is “Susceptible”, E is theproportion of the population that is “Exposed” (infected, but not yet infective),and I is the proportion of the population that is “Infective”. We omit (explicitly)modelling the proportion of the population that is “Recovered”, making use of thefact that we must have

Susceptible + Exposed + Infective + Recovered = 1.

The parameters of the system are:

• µ: the birth rate, which we assume is equal to the death rate;

• 1/γ: the mean infective period;

• β: the contact rate;

• 1/α: the mean latent period of the disease.

This system has two fixed points. The first one corresponds to the extinctionof the infection, i.e. the whole population is in the recovered state. The secondfixed point is more interesting because the infection survives, its details can befound in the Matlab code (available upon request from the authors). We assessthe stability probability for this second, more interesting, fixed point.

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2.4.2 First extended model: the S(nE)IR model

We now introduce n subpopulations of the “Exposed” population, representing(for example) different age groups. We suppose that the mean latent period ofthe disease varies between these subpopulations. This model will henceforth benamed ‘S(nE)IR’.

This leads to the following model:

S = µ−n∑

j=1

βjSI − µS

Ei = βiSI − (µ+ αi)Ei, for i = 1, . . . , n.

I =n∑

j=1

αjEj − (µ+ γ)I

Here, 1/αi is the mean latent period of the disease in the i-th Exposed subpop-ulation, and β =

∑nj=1 βj is the overall contact rate between “Susceptible” and

“Infective” individuals. Note that for n = 1, we recover the standard SEIR model.As in the SEIR model, this system has two fixed points. The first one cor-

responds to the extinction of the infection, i.e. the whole population is in therecovered state. The second fixed point is more interesting because the infectionsurvives, its details can be found in the Matlab code (available upon request fromthe authors). We assess the stability probability for this second, more interesting,fixed point.

2.4.3 Second extended model: the SE(nI)R model

In the other model we introduce n subpopulations of the “Infective” population.We suppose that the mean infective period of the disease varies between thesesubpopulations. This model will henceforth be named ‘SE(nI)R’.

This leads to the following model:

S = µ−n∑

j=1

βSIj − µS

E =n∑

j=1

βSIj − (µ+n∑

j=1

αj)E

Ii = αiE − (µ+ γi)Ii, for i = 1, . . . , n.

Here, 1/ci is the mean infective period of the disease in the i-th Infectivesubpopulation, and a =

∑nj=1 αj is the overall latent period of the disease. Here

again, for n = 1, we recover the standard SEIR model.

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As in the SEIR model, this system has two fixed points. The first one cor-responds to the extinction of the infection, i.e. the whole population is in therecovered state. The second fixed point is more interesting because the infectionsurvives, its details can be found in the Matlab code (available upon request fromthe authors). We assess the stability probability for this second, more interesting,fixed point.

3 Additional Results

3.1 Stability probabilities of the SE(nI)R and S(nE)IR mod-els

Obtained using the analytical method with 100000 samples. Results are shown inTables 1 and 2.

Table 1: Stability Probabilities of SnEIR models

SEIR S2EIR S3EIR S4EIR S5EIR S6EIRFCS 1 1 1 1 1 1Independent 0.56483 0.54848 0.52312 0.50472 0.49197 0.47679i.i.d. 0.1371 0.04142 0.01193 0.00356 0.00075 0.00014

Table 2: Stability Probabilities of SEnIR models

SEIR SE2IR SE3IR SE4IR SE5IR SE6IRFCS 1 1 1 1 1 1Independent 0.56483 0.64252 0.649 0.65373 0.65538 0.65933i.i.d. 0.1371 0.04791 0.01554 0.00483 0.00105 0.00025

These results are further illustrated in Fig. 2.

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Stab

ility

prob

abilit

y

2 4 6 8−0.5

0

0.5

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1.5

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Max

imum

real

par

t of e

igen

valu

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Den

sity

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1020

50100

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sity

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50100

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sity

Maximum real partof eigenvalues

15

1020

50100

Number of subpopulations

A

B

C

D

FCS

independent

i.i.d.

FCS independent i.i.d

S E

R

β

μ

μμ

μ

μ

I1

In

αn

α1

γ1

γn

Figure 2: How stability changes with system size depends on the random matrixdistribution. A. An extension of the SEIR model in which we model n infectivepopulations, I1, . . . , In. B. Plot showing for each of the random matrix distri-butions how estimated stability probability changes as we increase the numberof exposed populations. Bars denote ±2 s.d. Monte Carlo error bars. C. Plotshowing median (filled circle) and interquartile range (bars) for the distributionsof leading eigenvalues. D. Density estimates for the distributions of leading eigen-values.

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3.2 Additional ranges

In order to ensure that our results our not dependent on the range of the parametersconsidered we have considered different ranges for some of the models.

3.2.1 In the Lorenz model

We will call Lorenzi the Lorenz model with parameters sampled from uniformdistributions with the following ranges:

β ∈ [0, i]

ρ ∈ [1, i+ 1]

σ ∈ [0, i]

We took i = 1, 10, 100, 1000, 10000 and computed the probability of stability usingthe analytical method with 100000 samples. Results are shown in Table 3.

Table 3: Stability Probabilities of Lorenz model for different ranges

Lorenz1 Lorenz10 Lorenz100 Lorenz1000 Lorenz10000

FCS 1 0.99916 0.96825 0.96322 0.96271Independent 0.7225 0.84059 0.81014 0.80335 0.80262i.i.d. 0.10419 0.10348 0.10427 0.10426 0.10436

These results are further illustrated in Fig. 3.

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Figure 3: Stability results for additional ranges for the Lorenz model. A. Theeigenspectra for each range and random matrix distribution, shown as scatterplots.B. The eigenvalue distributions visualised using heat maps (to aid visualisation,we omit pure imaginary eigenvalues). C. The distributions of maximal eigenvaluestogether with the estimated stability probabilities.

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3.2.2 In the S(nE)IR and SE(nI)R systems

We will call S(nE)IRi and SE(nI)Ri the S(nE)IR and SE(nI)R models with pa-rameters sampled from uniform distributions with the following ranges:

µ ∈ [0, 1]

γ ∈ [0, i]

β ∈ [0, 1]

α ∈ [0, i]

We took i = 1, 10, 100 and computed the probability of stability using theanalytical method with 100000 samples. Results are shown in Tables 4–8.

Table 4: Stability Probabilities of S2EIR model for different ranges

SE2IR1 SE2IR10 SE2IR100

FCS 1 1 1Independent 0.64252 0.68502 0.69726i.i.d. 0.04791 0.0496 0.04897







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3.3 Numerical S(nE)IR and SE(nI)R systems

The stability probabilities were estimated using 1000 samples, the error bars pro-vided indicate plus/minus one standard deviation. Results are shown in Tables 9and10.

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Table 9: Stability Probabilities of numerical SEnIR models

numSE1IR numSE2IR numSE3IRFCS 1± 0 1± 0 1± 0Independent 0.574± 0.023 0.606± 0.02 0.613± 0.024i.i.d. 0.136± 0.015 0.0559± 0.0099 0.0118± 0.0051

numSE4IR numSE5IR numSE6IRFCS 1± 0 1± 0 1± 0Independent 0.584± 0.023 0.596± 0.023 0.628± 0.021i.i.d. 0.0048± 0.0026 0.00092± 0.0013 0± 0

numSE7IR numSE8IR numSE9IRFCS 1± 0 1± 0 1± 0Independent 0.623± 0.021 0.614± 0.02 0.633± 0.019i.i.d. 0± 0 0± 0 0± 0




numSE100IRFCS 1± 0Independent 0.675± 0.022i.i.d. 0± 0

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Table 10: Stability Probabilities of numerical SnEIR models

numS1EIR numS2EIR numS3EIRFCS 1± 0 1± 0 1± 0Independent 0.551± 0.026 0.516± 0.021 0.549± 0.02i.i.d. 0.139± 0.015 0.0315± 0.0089 0.0135± 0.0049

numS4EIR numS5EIR numS6EIRFCS 1± 0 1± 0 1± 0Independent 0.543± 0.02 0.559± 0.024 0.57± 0.02i.i.d. 0.00192± 0.0018 0.00216± 0.002 0± 0

numS7EIR numS8EIR numS9EIRFCS 1± 0 1± 0 1± 0Independent 0.553± 0.022 0.588± 0.023 0.552± 0.023i.i.d. 0± 0 0± 0 0± 0




numS100EIRFCS 1± 0Independent 0.771± 0.021i.i.d. 0± 0

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1 2 3 4 5 6 7 8 9 100.45

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10 30 50 70 90

Number of infectious subpopulations

TCSmultivariate normalindependentindependent normalindependent Pearsoni.i.d. normali.i.d.

A

Zoom of A

Number of infectious subpopulations

Figure 4: Stability probability of SE(nI)R models with different number of nodes,evaluated using different RMEs.

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1 2 3 4 5 6 7 8 9 100.4

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TCSmultivariate normalindependentindependent normalindependent Pearsoni.i.d. normali.i.d.

A

Zoom of A

Figure 5: Stability probability of S(nE)IR models with different number of nodes,evaluated using different RMEs.

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supplementary material for conditional random matrix ensembles

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