Local Identifiability Analysis

• Local identifiability analysis is the study of whether model parameters in dynamical networks can be uniquely determined from local input–output data.
• It employs algebraic reformulation and Jacobian rank tests to assess parameter recovery and address discrete ambiguities.
• Monte Carlo-based computational algorithms efficiently decide which network transfer functions are locally identifiable under varying excitation and measurement conditions.

Local identifiability analysis is the study of whether the parameters of a mathematical model—such as transfer functions in a dynamical network—can be uniquely (or finitely) determined, up to discrete ambiguities, from perfect input–output data, within a small neighborhood of a reference parameter set. In the context of dynamical networks with partial excitation and measurement, local identifiability provides criteria for determining which internal transfer functions can be recovered given external excitation and measurement constraints, and how these conditions can be efficiently and generically decided via algebraic and matrix rank-based methods (Legat et al., 2020).

1. Formal Definition and Setting

Consider a network of $n$ node signals interconnected by transfer functions $G_{ij}(z)$, with external inputs injected at a subset $\mathcal{B}$ and outputs measured at a subset $\mathcal{C}$. The network dynamics (in the frequency domain) can be written as

$w = G\,w + B\,r, \qquad y = C\,w$

where $G$ is an $n\times n$ matrix (with sparsity specified by the network topology), $B$ and $C$ are selection matrices for excitations and measurements, $w$ is the vector of all node signals, $r$ is the input vector, and $y$ the output.

Local identifiability of edge $G_{ij}$ at a nominal value $G$ means there exists $\varepsilon>0$ so that, for all $G'$ with the same zero/known structure as $G$ and $\|G'-G\|<\varepsilon$,

$$C\,(I-G')^{-1} B = C\,(I-G)^{-1} B \implies G_{ij}' = G_{ij}.$$

The whole network is locally identifiable if this holds for every $G_{ij}$ (Legat et al., 2020).

Distinct from global identifiability, which requires uniqueness over the entire parameter space, local identifiability asserts uniqueness only in a small neighborhood, allowing for discrete ambiguities elsewhere.

2. Algebraic Reformulation and Jacobian Analysis

The identifiability question is formulated via the input–output map: $$h(x) = \mathrm{vec}\left[ C (I-G(x))^{-1} B \right]$$ where $x$ collects all unknown $G_{ij}$. Local identifiability of an edge corresponds to coordinate-injectivity of the map $h$ in $x_e$ for $x$ in a neighborhood.

Key Jacobian: The critical analytic tool is the Jacobian matrix

$K(x) = \left(B^T T(x)^T \otimes C T(x)\right) I_G$

with $T(x) = (I-G(x))^{-1}$, and $I_G$ selecting columns associated to free parameters.

Main rank condition: For almost all $x$ (i.e., generically),

• $G_{ij}$ is locally identifiable $\iff$ the $e$th coordinate is free in the kernel of $K(x)$,
• The full network is locally identifiable $\iff$ $\mathrm{rank}\,K(x) = |E|$ (number of unknown edges) (Legat et al., 2020).

Thus, local identifiability reduces to a generic Jacobian rank test.

3. Computational Algorithms and Genericity

Because the identifiability property is generic (holds except on a set of measure zero), a randomized (Monte Carlo) numerical procedure suffices to decide identifiability almost surely:

• Algorithm:

1. For $N$ trials, sample a random $G$ compatible with the network structure.
2. Compute $K$ and its rank, as well as the kernel.
3. For each $G_{ij}$, if in every trial all kernel vectors are zero in the $e$th coordinate, mark $G_{ij}$ as identifiable.

With $N$ moderate, this selects the generically locally identifiable set of transfer functions with probability $1$ (Legat et al., 2020).

4. Relation to Coordinate Injectivity and Discrete Ambiguity

In local identifiability, if the map $h$ is not globally injective, any remaining ambiguity after restricting to a sufficiently small neighborhood can only be a discrete set of "mirror" solutions. This is a direct consequence of the real-analytic structure of $h$. That is, although several $G$ may lead to identical input–output maps globally, within a small enough neighborhood only finitely many solutions (often just one) exist. Hence local identifiability ensures recovery "up to a discrete ambiguity" (Legat et al., 2020).

The key lemma shows that injectivity in the $e$th coordinate of $h$ is determined by whether the $e$th standard basis vector is orthogonal to the kernel of $K(x)$ for generic $x$: $$\ker K(x) \perp e_e \quad \text{generically} \implies G_{ij} \text{ locally identifiable}.$$

5. Examples and Graphical Visualization

The method allows for graphical output directly on the network:

• Identifiable edges: solid green
• Non-identifiable edges: dashed red
• Excited nodes: circled blue
• Measured nodes: shaded gray

Illustrative applications confirm that changes in where excitation or measurement occurs can dramatically affect which edges are locally identifiable:

• For an 11-node network, partial measurement and excitation sufficed for global identifiability of all edges, under the generic condition.
• In another network, adding an excitation at a particular node can restore identifiability of otherwise non-identifiable edges, regardless of their topological distance (Legat et al., 2020).

6. Open Problems and Research Directions

The local identifiability framework has raised several open questions:

• Combinatorial criteria: While Jacobian-based algebraic tests are fully rigorous and generic, finding purely combinatorial (graph-theoretic) criteria for local (and global) identifiability remains unsolved.
• Local vs. global equivalence: It is not generally settled when local and global identifiability coincide for arbitrary dynamical networks, though they do coincide on separable networks and empirical evidence suggests frequent equivalence.
• Discrepancy structure: The nature and classification of the discrete ambiguity sets in non-globally identifiable cases are not fully understood.

Further work in these directions is suggested (Legat et al., 2020).

7. Summary Table: Core Components of Local Identifiability Analysis

Concept	Description	Paper Section
Local identifiability	Uniqueness/finiteness of solution in local neighborhood	(2)
Input–output map $h$	Encodes measurable I/O as analytic map of free params	(3)
Jacobian matrix $K(x)$	Derivative of $h$; generic full rank $\implies$ ID	(4,5)
Rank test / MC algorithm	Probability-1 method for deciding identifiability	(6)
Graphical visualization	Edge-level output to illustrate ID properties	(7)

The framework unifies algebraic, analytic, and computational perspectives, providing necessary and sufficient rank-based conditions for local identifiability of network parameters, and enables practical, scalable assessments that guide both theoretical investigation and applied data-driven network modeling (Legat et al., 2020).