Determinacy and Persistence in Dynamic Networks

Conceptual map

V-09.01Original dynamic structure
V-09.02Signed witnesses
V-09.03Root-count repair
V-09.04Repeated-pole witnesses
V-09.05Observable representations

1. Signed dynamics and comparison bounds answer different questions

Absolute-coefficient systems can certify a sufficient contraction bound, while the signed system determines actual eigenvalues, cancellations, and responses.

For $x_{t+1}=Ax_{t}$ , asymptotic stability in finite dimensions is equivalent to spectral radius $\rho(A)<1$ . Let

\[A=\begin{bmatrix}0.6 & 0.6 \\ -0.6 & 0.6\end{bmatrix}.\](1)

Its eigenvalues are $0.6\pm 0.6i$ , with modulus $\sqrt{0.72}\approx0.848528$ , so the system is stable. The entrywise absolute matrix has eigenvalues 1.2 and 0, hence spectral radius 1.2. The absolute comparison fails to certify this stable signed system because it discards rotational cancellation.

Definition 1 · Determinacy audit

For a linear rational-expectations system, the audit records the regular matrix pencil, generalized roots relative to the unit circle, predetermined and forward-looking dimensions, expectational-error restrictions, and boundary roots. A stability calculation for a backward recursion is not a determinacy proof for a forward-looking model.

2. Root counts require the original system

Under the regularity conditions of the standard linear rational-expectations solution, uniqueness requires the number of roots assigned to unstable directions to match the number of forward-looking variables, together with the relevant rank conditions. Too few unstable roots produce indeterminacy; too many generally prevent a bounded solution for arbitrary predetermined states.

Absolute values can provide comparison inequalities such as $\lvert A x\rvert \le \lvert A\rvert \lvert x\rvert$ componentwise. They do not preserve generalized eigenvalues or the count used by a determinacy theorem. Singular pencils, roots exactly on the unit circle, and coincident roots need separate arguments.

3. Repeated roots add polynomial persistence

Let $$J=rI+N$$ , where $N=\begin{bmatrix}0 & 1 \\ 0 & 0\end{bmatrix}$ and $N^{2}=0$ . Repeated multiplication yields a generalized mode.

Proposition 1 · Power of a size-two Jordan block

For integer $t\ge1$ and scalar $$r$$ , $J^{t}=r^{t}I+t r^{t-1}N$ . At $$r=0$$ , the coefficient of $$N$$ is one when $$t=1$$ and zero when $t\ge2$ .

Proof. Expand $(rI+N)^{t}$ by the binomial theorem. Every term containing $N^{k}$ for $k\ge2$ vanishes, leaving the $$k=0$$ and $$k=1$$ terms. ∎

At $$r=0.8$$ and $$t=5$$ , the diagonal factor is $0.8^{5}=0.32768$ and the generalized off-diagonal factor is $5\times 0.8^{4}=2.048$ . The system still converges when $\lvert r\rvert<1$ , while finite-horizon persistence can be much larger than $r^{t}$ .

4. State roots need not appear in an observable response

Failure case · State eigenvalue labeled an observed pole

For transfer $C(zI-A)^{-1}B$ , a mode absent from the input or killed by the output can cancel. Minimality must be checked before interpreting state roots as observed persistence.

A repeated eigenvalue can be diagonalizable, in which case the polynomial factor is absent. At $\lvert r\rvert=1$ , the Jordan factor grows linearly and stability fails. At $$r=0$$ , powers vanish after the nilpotent index. Finite horizons can conceal slow modes, and numerical root counts need a declared unit-circle tolerance. Network sparsity does not guarantee observability or controllability.

5. Implementation, exercises, and sources

Construct the signed transition or matrix pencil from model equations, retain variable timing and predetermined status, and compute generalized roots with balancing and residual checks. Separate sufficient comparison bounds from exact roots. Inspect Jordan or Schur structure near repeated roots, then test controllability, observability, and pole cancellation for each reported response.

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Exercises

Verify both spectral radii in (1).
Compute the size-two Jordan power at $$t=5$$ .
Give $$B$$ and $$C$$ that make one state mode absent from the transfer function.

Partial solutions

1. The signed characteristic roots have squared modulus $0.6^{2}+0.6^{2}$ ; the absolute matrix has equal rows. 2. Substitute into the proposition to obtain diagonal 0.32768 and upper-right 2.048. 3. For diagonal $$A$$ , choose $$B$$ or $$C$$ orthogonal to an eigenvector; that pole then has zero residue.

Olivier Jean Blanchard and Charles M. Kahn (1980), “The Solution of Linear Difference Models under Rational Expectations,” Econometrica 48, 1305–1311.Root counts and determinacy.
Roger A. Horn and Charles R. Johnson, Matrix Analysis.Spectral radius and Jordan powers.
Thomas Kailath, Linear Systems.Minimal realizations and pole cancellation.

6. Audit checkpoint

Signed equations, entrywise comparison, state timing, regular pencil, generalized roots, unit-circle tolerance, predetermined and jump dimensions, rank conditions, singular and coincident roots, Jordan size, finite horizon, initial state, stability versus determinacy, input loading, output map, controllability, observability, minimality, pole cancellation, and root residuals.

7. Scope boundary

The chapter audits finite-dimensional linear dynamics. Nonlinear determinacy, time-varying networks, and infinite-dimensional evolution equations require additional theory.