Generic identifiability of subnetworks in a linear dynamic network: the full measurement case

Identifiability conditions for single or multiple modules in a dynamic network specify under which conditions the considered modules can be uniquely recovered from the second-order statistical properties of the measured signals. Conditions for generic identifiability of multiple modules, i.e. a subnetwork, are developed for the situation that all node signals are measured and excitation of the network is provided by both measured excitation signals and unmeasured disturbance inputs. Additionally, the network model set is allowed to contain non-parametrized modules that are fixed, and e.g. reflect modules of which the dynamics are known to the user. The conditions take the form of path-based conditions on the graph of the network model set. Based on these conditions, synthesis results are formulated for allocating external excitation signals to achieve generic identifiability of particular subnetworks. If there are a sufficient number of measured external excitation signals, the formulated results give rise to a generalized indirect type of identification algorithm that requires only the measurement of a subset of the node signals in the network.


Introduction
Due to the increasing complexity of current technological systems, the study of large-scale interconnected dynamic systems receives considerable attention (Cheng & Scherpen, 2021;Ishizaki, Chakrabortty, & Imura, 2018;Mesbahi & Egerstedt, 2010). The resulting dynamic networks can adequately describe a wide class of complex engineering systems appearing in various applications, including multi-robot coordination (Mesbahi & Egerstedt, 2010), power grids (Ishizaki et al., 2018) and gene networks (Adebayo et al., 2012). For data-driven modeling problems in structured dynamic networks, different types of network models have been used. Connecting to prediction-error identification methods, the most popular modeling framework is based on a network of transfer functions, initially introduced in Gonçalves and Warnick (2008) and Van den Hof, Dankers, Heuberger, and Bombois (2013), where vertices represent the internal signals, that are ✩ This project has received funding from the European Research Council (ERC), Advanced Research Grant SYSDYNET, under the European Union's Horizon 2020 research and innovation programme (grant agreement No 694504). The material in this paper was partially presented at the 21st IFAC World Congress (IFAC 2020), July 12-17, 2020, Berlin, Germany. This paper was recommended for publication in revised form by Associate Editor Gianluigi Pillonetto under the direction of Editor Torsten Söderström. and Warnick (2008), the concept of global network identifiability was introduced for an identification setting in Weerts, Dankers, and Van den Hof (2015) and Weerts, Van den Hof, and Dankers (2018a), as a property that reflects the ability to distinguish between network models in a parametrized model set based on measured data. In the literature, there are two notions of network identifiability, namely, global identifiability (van Waarde, Tesi, & Camlibel, 2020;Weerts et al., 2015Weerts et al., , 2018a) that requires all model to be distinguishable from the other models in the model set, 1 and generic identifiability (Bazanella, Gevers, Hendrickx, & Parraga, 2017;Weerts, Van den Hof, & Dankers, 2018c), which means that almost all models can be distinguished from all other models in the model set. It has been shown in Bazanella et al. (2017) and Hendrickx et al. (2019) that by considering generic identifiability, the algebraic conditions for identifiability can be recast into path-based conditions on the topology/graph of the network models, which largely simplifies the analysis.
Network identifiability is typically dependent on several structural properties of the model set, such as the network topology, the modeled correlation structure of process noises, the presence and location of external excitation signals and the availability of measured vertex signals. Conditions for network identifiability have been analyzed for different problem settings. In the setting with full excitation (Bazanella et al., 2017;Hendrickx et al., 2019;van Waarde et al., 2020), all vertices are excited by external excitation signals while only a subset of vertices is measured. In contrast, the full measurement setting in Weerts et al. (2018aWeerts et al. ( , 2018c assumes that all vertex signals are measured, while only a subset of them is excited. Recent contributions in Bazanella, Gevers, and Hendrickx (2019) and Shi, Cheng, and Van den Hof (2021b) also address the combined situation.
In this paper our objective is to derive path-based conditions for generic identifiability of only a subset of modules (subnetwork) in the network, while we assume all internal signals in the network to be available for measurement, the so-called full measurement case. Additionally, we will require the conditions to be suitable for solving the synthesis problem too, i.e. the allocation of a minimum set of external excitation signals, so as to achieve generic identifiability of the subnetwork.
For analyzing this problem, we start from the path-based results for generic identifiability of a subnetwork as presented in Bazanella et al. (2017) and Hendrickx et al. (2019). In these works identifiability is defined as a property of a single network, while disturbance signals are not exploited. In our approach we follow up on the more general model-set type of definition of identifiability as introduced in Weerts et al. (2018aWeerts et al. ( , 2018c, that is more aligned to the use of this concept in an identification setting. Moreover, this step allows to incorporate the following attractive features that are all addressed in the current paper: (a) it allows to include the effect of unmeasured disturbance signals in the network as an information source that can contribute to satisfying identifiability conditions; (b) it allows to include modules in the network that are a priori known to the user and thus do not need to be identified. In this setting we will develop novel analysis tools for generic identifiability of subnetworks that are formulated in terms of disconnecting sets in the graph of the network models, and we will show that this leads to a new and effective synthesis procedure for allocating external excitation signals for achieving identifiability of a subnetwork. This synthesis problem was not addressed in Bazanella et al. (2017), Hendrickx et al. (2019) and Weerts et al. (2018aWeerts et al. ( , 2018c. As a side result, a 1 There are actually two versions of global identifiability, reflecting whether either one particular model in the set can be distinguished or all models in the set (Weerts et al., 2018a). new generalized indirect identification method is described that follows immediately form the identifiability conditions, in the situation that a sufficient number of measured excitation signals is present. This method points to a subset of internal signals that in the considered situation would be sufficient for uniquely identifying the subnetwork.
The synthesis problem as formulated above is important for actually designing experimental setups for subnetwork identification. A related synthesis problem for full network identifiability has been addressed in Cheng, Shi, and Van den Hof (2022), but requires completely different tools for analysis.
The paper proceeds as follows. After introducing preliminaries and the problem in Section 2, algebraic and path-based conditions for generic identifiability are formulated in Sections 3 and 4. Disconnecting-set-based conditions are then derived in Section 5, leading to synthesis approaches in Section 6. Then a generalized indirect identification method that directly follows from the identifiability conditions is presented in Section 7.
Preliminary results of this paper were presented in Shi, Cheng, and Van den Hof (2020). In the current paper comprehensive algebraic and path-based conditions are formalized for a generalized situation, including the step from single modules to subnetworks. Additionally a novel indirect identification method is presented.

Dynamic networks
The dynamic network models the relation among a set of measured scalar internal signals W ≜ {w 1 (t), . . . , w L (t)} with L = |W| denoting the cardinality of W, a set of measured excitation signals R ≜ {r 1 (t), . . . , r K (t)} with K = |R|, and a set of unmeasured disturbance signals {v 1 (t), . . . , v L (t)}. The model is written as where G(q) and R(q) are matrices of rational transfer operators with delay operator q −1 , i.e. q −1 w i (t) = w i (t − 1); w(t), r(t) and v(t) are the column vectors that collect all the internal signals, excitation signals and disturbances, respectively. In addition, v(t) is a vector of zero-mean stationary stochastic processes with power spectrum Φ v (ω), which is modeled as a filtered white noise vector e(t) according to: where e(t) has a covariance matrix Λ. Depending on whether Φ v (ω) is of full rank or not, H(q) can either be square or have more rows than columns (Weerts et al., 2018a). Combining (1) and (2) leads to a dynamic network model that is supposed to satisfy the following Assumption 1.

Assumption 1.
(a) G(q) has zero diagonal elements, and G(q) is proper and stable; (b) The network is well-posed in the sense that all principal minors of lim z→∞ (I − G(z)) are non-zero; (c) [I − G(q)] −1 is stable; (d) R(q) is a proper and stable rational transfer matrix; (e) H(q) is minimum phase and monic if square; for the non- , with H a (q) square, monic and minimum phase (Weerts et al., 2018a); (f) The covariance matrix Λ of e(t) is positive definite.
In the above assumptions, (b) and (c) ensure that every transfer function from external signals to internal signals is proper and stable (Dankers, 2014); the stability of G(q) in (a) guarantees that the noise filter defined later in (3) is also inversely stable, which is typical for the modeling of stationary stochastic processes.
Both the excitation signals in R and the entries (white noises) in e(t) are called external signals, and the set of all external signals is denoted by X . The entries in G(q) are referred to as modules. In addition, the dynamic network model leads to mappings from the excitation signals to internal signals: where When applying common statistical identification methods to (3) that are based on first and second moment information of the measured signals, typically the objects T WR (q) and Φv(ω) can be consistently estimated from measured signals w(t) and r(t), if r(t) is persistently exciting (Ljung, 1987). This motivates to use T WR (q) and Φv(ω) as bases for network identifiability (Weerts et al., 2018a).
Given subsetsW ⊆ W andX ⊆ X , TWX (q) denotes a submatrix of T WX (q) with the rows and columns corresponding to the signals inW andX . IfW contains only one signal w k , TWX (q) is often simply written as T kX (q). The above notation applies similarly to submatrices of other matrices and vectors.

Model sets
Network identifiability will be defined on the basis of a network model set, which is introduced as follows. The network model (1), (2) is completely specified by a quadruple M ≜ (G(q), R(q), H(q), Λ). By parametrizing the entries of the network matrices in a rational form and then collecting the parameters into a parameter vector θ, a parametrized model set can be defined.
Definition 1. Consider a rational parametrization of a network model according to Then a network model set M is defined as M = {M(θ)|θ ∈ Θ ⊆ R n }, where M(θ ) satisfies Assumption 1 for all θ ∈ Θ. In addition, there can be certain entries in G(q, θ), R(q, θ) and H(q, θ) that are fixed/known and thus do not depend on the parameters. These entries are said to be known or fixed, and all models in M contain the same known entries. The entries in the network matrices that depend on the parameters are said to be unknown or parametrized. For example, the absence of an interconnection between internal signals is represented by a fixed 0 in G(q, θ); entries in G(q, θ) may be particularly designed controllers that are fixed and known. Similarly, entries in H(q, θ), R(q, θ) and Λ(θ ) can be fixed, e.g., equal to 1 or 0. In the situation of R(q), this implies that it can be specified upfront on which nodes the r-signals enter the network. Note that the dependency of transfer matrices on q, θ and the dependency of power spectra on ω are sometimes omitted for simplicity of notation.
The structural information of a model set is reflected by a directed graph G = (V, E), where V ≜ W ∪ X is a set of vertices representing both the internal signals and the external signals, and E ⊆ V × V denotes a set of directed edges representing those entries in G(q, θ) and X (q, θ) that are not fixed to zero: a directed edge from w i to w j exists, i.e. (w i , w j ) ∈ E, if G ji is not fixed to zero; Similarly, (e k , w j ) ∈ E and (r p , w j ) ∈ E if H jk and R jp are not fixed to zero, respectively. In this way, any parametrized model set or network model induces a directed graph G. Note that w i now represents both a signal and a vertex, and its dependency on t is sometimes omitted for simplicity of notation.
For deriving the results in the sequel of this paper, we will need an additional concept.
Definition 2. Given a model set M with its graph G, the set G ⋆ is defined as the set of all G(q) matrices that satisfy the following conditions: (i) they meet Assumption 1; (ii) they have the same fixed entries as G(q, θ) in M; (iii) entries are strictly proper if the corresponding entries in G(q, θ) are (parametrized to be) strictly proper.
In Definition 2, G ⋆ is the largest possible set of G(q) matrices that have the same fixed entries and feedthrough structure as G(q, θ) in M. So while the modules G in M can be restricted in order, the order of entries in G ⋆ is not bounded. This implies {G(q, θ)|θ ∈ Θ} ⊆ G ⋆ . In Assumption 4 we will introduce a technical condition on {G(q, θ)|θ ∈ Θ} with respect to G ⋆ for proving necessary conditions for identifiability.

Network identifiability
We focus on the identifiability of a subset of unknown modules in one row of the G matrix, i.e. a subnetwork of modules that share the same output node w j . For this output node w j , we define an important set of signals: • W j : all the internal signals that have unknown directed edges (modules) to w j .
For any subsetW j ⊆ W j , G jW j denotes a row vector containing a subset of unknown modules in the jth row of G. For the identifiability of G jW j , we follow the concept of global network identifiability in Weerts et al. (2018a), and extend it with a generic version introduced in Bazanella et al. (2017) and Hendrickx et al. (2019). In this respect we follow an approach that was suggested in Weerts et al. (2018c).

Definition 3.
Given a network model set M, consider any subsetW j ⊆ W j , a parameter vector θ 0 ∈ Θ and the following implication: for all θ 1 ∈ Θ. Then G jW j (q, θ) is said to be • globally identifiable in M from (w, r) if the implication (4) holds for all θ 0 ∈ Θ; • generically identifiable in M from (w, r) if the implication (4) holds for almost all θ 0 ∈ Θ.
In Definition 3, the notion ''almost all'' excludes a subset of Lebesgue measure zero from Θ. The identifiability concept in Definition 3 concerns whether G jW j is unique given the objects that can typically be identified from the first and second moment information of the measured signals, as motivated following (3). If a model set is not identifiable, any identification method that relies on the first and second moments for estimating the network cannot be guaranteed to (asymptotically) identify a unique network model. In addition, recall that the network matrices in M may contain known entries, and this restriction of M may simplify the conditions under which the implication (4) holds.
To further simplify (4), we use the following assumption on the feedthrough terms in G. Assumption 2 (Weerts et al., 2018a).
(a) either all modules G(q, θ) are parametrized to be strictly proper, or (b) the parametrized network model does not contain any algebraic loops, 2 and This assumption ensures that the spectral factorization of Φv admits a unique spectral factor (I − G) −1 H (Weerts et al., 2018a), i.e., the mapping from the white noises to the internal signals as in (3). Then we can reformulate Definition 3 by considering the uniqueness of network modules given Proposition 1. Given a network model set M that satisfies Assumption 2, consider any subsetW j ⊆ W j , θ 0 ∈ Θ and the following implication: for all θ 1 ∈ Θ. Then G jW j (q, θ) is globally (generically) identifiable in M from (w, r) if and only if the above implication holds for all (almost all) θ 0 ∈ Θ.
Proof. This result is a direct consequence of Propositions 1 and 2 in Weerts et al. (2018a). ■ In contrast to Definition 3, Proposition 1 concerns the mapping T WX from both r and e signals to the internal signals. Therefore, the r and e signals play the same role in the identifiability analysis. This is also a key difference from the identifiability concept in Bazanella et al. (2017) and Hendrickx et al. (2019), where the noise information is not exploited. Proposition 1 also extends trivially to the single module case where G jW j only contains one entry, and to the full network case whereW j = W j and j varies over all j ∈ [1, L].

Problem formulation
The identifiability issue is first illustrated in Example 1.

Example 1.
Consider a network model set whose graph is shown in Fig. 1(a), where there is one white noise e 1 and four measured internal signals. All edges are parametrized except for G 42 which is fixed to G 0 42 . The identifiability of module G 41 is considered, and recall that the dependence of transfer functions on q and θ is omitted for simplicity of notation. Then based on Proposition 1, the identifiability of G 41 concerns the uniqueness 2 There exists an algebraic loop around node w n 1 if there exists a sequence of integers n 1 , . . . , n k such that G ∞ of G 41 given the mapping T {w 1 ,w 2 ,w 3 ,w 4 }{e 1 } from e 1 to all the internal signals. According to Fig. 1(a), the mapping satisfies that 42 and the submatrices of T {w 1 ,w 2 ,w 3 ,w 4 }{e 1 } are given, but H 11 and the other entries in G are unknown. The above equations further lead to which is not sufficient for a unique solution of G 41 due to the two unknowns G 41 and G 43 . Therefore, G 41 is neither globally nor generically identifiable in Fig. 1(a).
However, if an additional signal r 2 is allocated at w 2 as in Fig. 1(b), additional mappings from r 2 to the measured internal signals are given. Then it can be found that which shows that a unique G 41 can be found if the above inverses exist. Recall that the mappings depend on parameter θ, and thus G 41 is globally (generically) identifiable if the inverses exist for all (almost all) θ ∈ Θ.
As shown in the above example, identifiability is related to the uniqueness of modules given the mapping T WX . In addition, it is possible to achieve identifiability by allocating extra excitation signals in the network.
Given a model set and its associated graph, we first develop sufficient and necessary conditions for the generic identifiability of a subnetwork G jW j when all the internal signals are measured. More importantly, to avoid the complex algebraic verification as in Example 1, we aim to develop graphical conditions that can be verified efficiently by only inspecting the network topology. Furthermore, when a subnetwork is not generically identifiable, we aim to develop graphical synthesis approaches that allocate additional excitation signals to automatically achieve the generic identifiability of a subnetwork.

Notations and definitions
The following notations are used in the paper. Matrix TWX (q, θ) is called having generically full rank if it has full rank for almost all θ. More generally, a property that depends on θ is said to hold generically if it holds for almost all θ. A matrix C is called a selection matrix if it consists of a subset of rows of an identity matrix.
In a graph G, for a directed edge (w i , w j ), w i is called an inneighbor of w j , and w j is an out-neighbor of w i . The set N + V contains all out-neighbors of the vertices inV, and the set N − V all in-neighbors. A (directed) path from w i to w j is a sequence of vertices and out-going edges starting from w i to w j without repeating any vertex, and w j is said reachable by w i . A single vertex is also regarded as a directed path to itself. Internal vertices of a path are the vertices excluding the starting and the ending vertices.
Two directed paths are called vertex disjoint if they do not share any vertex, including the starting and ending vertices, otherwise they intersect. Given two subsets of vertices V 1 and V 2 , b V 1 →V 2 denotes the maximum number of vertex disjoint paths it intersects with all paths from V 1 to V 2 , and it is a minimum disconnecting set if it has the minimum cardinally among all V 1 −V 2 disconnecting sets (Schrijver, 2003). Note that D may also include vertices in V 1 ∪ V 2 .
Lemma 1 . For a directed graph and given a V 1 −V 2 disconnecting set D, consider the division of all the vertices V into three disjoint sets S ∪D∪P as follows: set S contains all vertices reachable by V 1 without intersecting D, and P = V \ (D ∪ S). Then it holds that no directed edge exists from S to P.
The duality between vertex disjoint paths and disconnecting sets is explained in Menger's theorem.
Theorem 1 (Schrijver, 2003). Let V 1 , V 2 be two subsets of the vertices in a directed graph. The maximum number of vertex disjoint paths from V 1 to V 2 equals the cardinality of a minimum disconnecting set from V 1 to V 2 .

Algebraic conditions for identifiability
We first investigate under which conditions a subnetwork G jW j is globally and generically identifiable. An algebraic rank condition is first presented in this section which will serve as a basis for deriving a more attractive graphical condition in the next section. Recall (3) which leads to (I − G)T WX = X . It then follows from Proposition 1 that identifiability essentially reflects the unique solutions of modules in G from a given T WX and thus is related to the rank of T WX . We first define the following set related to the output w j : • X j : the set of all the external signals that do not have any unknown edge to w j .
Then the following sufficient conditions for identifiability can be obtained from Theorem V.1 in Hendrickx et al. (2019) analogously.

Lemma 2. Given a model set M that satisfies Assumption 2. Then
the following conditions hold: for all (almost all) θ ∈ Θ.
The above result shows that identifiability of G jW j holds if (i) the mapping from X j to the inputs of the target modules has full row rank, and (ii) the rows in the above mapping and the ones in the mapping from X j to the other inputs of w j are linearly independent. Here, we consider the signals in X j instead of X , because the ones in X \ X j , which have unknown edges to w j , contribute to extra unknowns in the analysis in the form of a parametrized disturbance model for v j , and thus are not helpful for the identifiability analysis. If the corresponding noise model is fixed/known, the corresponding white noise source becomes an element of X j and does act as an excitation source in the identifiability analysis.
The sufficient conditions in Lemma 2 are implied in the proof of Theorem V.1 of Hendrickx et al. (2019). While in that proof they were shown to be also necessary, that necessity proof cannot be applied directly to our current setting, owing to a different definition of generic identifiability: Definition 3 is based on a particular model set, while the identifiability concept in Hendrickx et al. (2019) concerns a single network model and its associated graph. To establish the necessity of the rank conditions in this work, we first introduce two technical conditions on the model set M.

Assumption 3.
All the parametrized entries in M(θ ) are parametrized independently. Then the necessity of the rank conditions for identifiability can be obtained as follows.
Theorem 2. Given a model set M that satisfies Assumption 2, if M also satisfies Assumptions 3 and 4, conditions (5a)-(5b) in Lemma 2 are also necessary conditions for the results of the lemma.
The proof of Theorem 2 is presented in Shi, Cheng, and Van den Hof (2021c) and a more comprehensive analysis can be found in Shi (2021). Theorem 2 extends the identifiability conditions in Weerts et al. (2018aWeerts et al. ( , 2018c) from a full network and a single module to any subnetwork, such that the full network and the single module identifiability become special cases. In addition, with extra technical efforts in the analysis of Theorem 2, Assumption 4 becomes less conservative than the assumption in Weerts et al. (2018aWeerts et al. ( , 2018c for the necessity of identifiability. The rank tests in Theorem 2 may require to invert (I − G) for computing the external-to-internal mappings, which can be computationally expensive when the network is large-scale. The following result shows that the rank of any submatrix of T WX can be also evaluated without inverting the matrix I−G for computing T WX .

Lemma 3. Consider a dynamic network and any subsetsW
] .
Then it holds that rank(TWX (q)) = rank[F (W,X )]+|W|−L, where L is the total number of internal signals.
Proof. Recall that TWX = C (I − G) −1X , whereX = X WX and C is a selection matrix that extracts the rows corresponding toW. Definē then according to van der Woude (1991),F can be written as Thus, rank(F ) = rank(TWX ) + L. Since F is obtained fromF by removing the columns inF that correspond to the nonzero entries of C , which through the structure ofF are linearly independent in F , it follows that rank(F ) = rank(F ) − |W|, which proves the result. ■ With the above result, the rank condition in Theorem 2 can be reformulated as follows.
Proposition 2. Consider a network model set and the matrix F defined in (6). The conditions (5a)-(5b) in Theorem 2 are equivalently reformulated as F (W j , X j ) having full row rank and for all (almost all) θ ∈ Θ.

Path-based conditions for generic identifiability
When analyzing generic identifiability, the results of Lemma 2 and Proposition 2 show that the rank of the parametrized matrices have to be evaluated over almost all θ ∈ Θ. The resulting generic rank of transfer matrices in dynamic networks can be evaluated by determining the number of vertex disjoint paths between particular vertices in the graph induced by the model set, i.e. bX →W = rank TWX (q, θ) for almost all θ ∈ Θ. for the particular situation that X = I, and that all edges in the graph G are parametrized, i.e. without having fixed/known modules in the model set. In this section we will extend the results beyond these limitations.
The issue caused by known transfer functions is explained in the following example.
Example 2. Consider two network model sets in Fig. 2, where the model set in (b) contains known modules. In addition, assume R 11 = R 22 = 1 for simplicity. Then in Fig. 2(a), b {r 1 ,r 2 }→{w 3 ,w 4 } = 2, i.e. there exist maximally two paths r 1 → w 1 → w 3 and r 2 → w 2 → w 4 that are vertex disjoint. This implies that rank(T {w 3 ,w 4 }{r 1 ,r 2 } ) = 2 generically according to (8), which can also be seen from As each module is an analytic function of independent parameters, the determinant in (9) is a non-constant analytic function of θ and thus is non-zero for almost all θ. This matches with (8) However, T {w 3 ,w 4 }{r 1 ,r 2 } is not of full rank in Fig. 2 Then det(T {w 3 ,w 4 }{r 1 ,r 2 } ) = 0 for any θ and thus (8) does not hold. ■ As shown in Example 2, the presence of known modules may cause the equality in (8) to fail. In order to circumvent this, we introduce an additional condition based on the concept of structural rank.
Definition 4 (Steffen, 2005). The structural rank of a matrix is the maximum rank of all matrices with the same nonzero pattern. A matrix has full structural rank if it can be permuted so that the diagonal has no zero entries.
The property of structural full rank depends solely on the sparsity pattern of the matrix and does not rely on the numerical values of the entries. In order to characterize and exclude the situations as presented in Example 2, we introduce the following assumption.
Assumption 5. In model set M, the rank of any submatrix of [(G(q, θ) − I) X (q, θ)], which does not depend on θ, is equal to its structural rank. ■ The above assumption prevents known entries to be chosen in such a specific manner that they cause a rank deficiency of TWX , which is not captured by the number of vertex disjoint paths. Using this assumption, equality (8) can be extended from Hendrickx et al. (2019) to incorporate known transfer operators in the model set, as shown in the following result. The proof of the above result is presented in Appendix. By combining Proposition 3 with Theorem 2, the graphical condition for generic identifiability can be obtained, that follows immediately from the previous results: Theorem 3. Consider a model set M that satisfies Assumptions 2, When M additionally satisfies Assumption 4, the above conditions are also necessary.
The above result shows that generic identifiability of the target modules in G jW j is ensured if (i) the inputsW j of the target modules are excited by X j in the sense that X j has a sufficient number of vertex disjoint paths to these inputs, and (ii) the inputs W j receive excitation that is independent of the excitation for the remaining inputs W j \W j . The result of Theorem 3 is also illustrated in Example 3. An important special case of Theorem 3 is whenW j = W j , i.e. all parametrized modules that map into w j are in the subnetwork under study. In this case, conditions (10a) and (10b) can be simplified into a single condition as b X j →W j = |W j |.
Example 3. The generic identifiability of G 41 in the network model set in Fig. 1(a) can be verified using Theorem 3 as follows.
Compared to Theorem 2 and the analysis in Example 1, the conditions in Theorem 3 are fully graphical and thus can be evaluated more efficiently by only inspecting the network topology. Standard graphical algorithms are available for computing the maximum number of vertex disjoint paths (Schrijver, 2003). Note that the conditions (10a)-(10b) are similar to the ones in Hendrickx et al. (2019), but now extended to a different situation. Besides the incorporation of known modules and the model-set-based identifiability in Definition 3, the set X j contains both measured excitation signals and unmeasured white noises, instead of only r signals as considered in Hendrickx et al. (2019). Thus, to satisfy the graphical conditions in the theorem, the white noises can compensate for a possible lack of excitation signals.

Generic identifiability based on disconnecting sets
The graphic condition in Theorem 3 can be applied to analyzing generic identifiability in a given model set. However, it does not explicitly indicate where to allocate external signals such that a particular set of modules becomes generically identifiable. To solve this synthesis question, a new analytic result is developed in this section by exploring the duality between vertex disjoint paths and disconnecting sets, as also indicated in Hendrickx et al. (2019). In this section, we will exploit this relationship in a novel way, by suitably exploiting it for the situation of subnetwork generic identifiability, while providing a solution to the synthesis question as well. This synthesis question will be addressed in more detail in Section 6.
A set of vertices is called a disconnecting set from vertex set V 1 to V 2 if all paths from V 1 to V 2 pass through the set, see Section 2.5. Its relevance in the analysis of generic identifiability is illustrated in Example 4. Fig. 1(a), if r 2 is allocated at w 2 as in Fig. 1

Example 4. Starting from
becomes generically identifiable in Fig. 1(b) based on Theorem 3.
It is important to note that the vertex w 2 , where the new r 2 is allocated, is a disconnecting set from the target input w 1 to the other inputs of w 4 . ■ It is observed from the above example that when G ji is not generically identifiable, its identifiability can be achieved when the vertices in a disconnecting set from {w i } to the other inneighbors of w j are excited. This observation is generalized to obtain a new identifiability condition based on disconnecting sets.

Theorem 4.
Consider the situation of Theorem 3. The conditions (10a)-(10b) can equivalently be written as either one of the following two equivalent conditions: There exists a set of external signalsX j ⊆ X j and aX j −W j \W j disconnecting set D such that bX j →W j ∪D = |D| + |W j |.
The proof of the above result is presented in Appendix. As visualized in Fig. 3, this result states that the target modules in G jW j are generically identifiable, if the inputs of the target modules in W j and the signals in D are excited by external signals through vertex disjoint paths, where D contains the vertices that block the paths from the target inputs to the other inputs that have unknown edges to w j . In contrast to Theorem 3, the conditions in Theorem 4 explicitly state that the signals inW j ∪ D should be excited. When the target modules are not identifiable, the above result will be exploited in Section 6 to allocate additional excitation signals for achieving identifiability.
In Theorem 4, condition (a) more closely resembles Theorem 3 as it involves all external signals in X j ; on the other hand, the condition (b) considers any subset of signals in X j , and this ability of considering subsets of X j will be exploited for the design of synthesis approaches.

Algebraic interpretation of disconnecting sets
A disconnecting set that disconnects external signals from internal signals in the graph of a network model set, gives rise to an attractive factorization of the external-to-internal mapping for the associated dynamic network.

Theorem 5. Consider a network model M with a disconnecting set
The proof of the above result is presented in Appendix. The decomposition in (12) means that if all paths fromX toW intersect with D, then the signals in D act as auxiliary signals that contain all information from the signals inX that is relevant for the signals inW. The explicit form of the K matrix is specified in (26). The factorization in Theorem 5 will serve as a basis for formulating an indirect identification method for identifying G jW j . This will be addressed in Section 7.

Relation with the parallel path and loop condition
In dynamic network identification, there is an important condition used to select measured internal signals for identifying a single module G ji . The condition was first formulated in Dankers et al. (2016) as the parallel path and loop condition and selects a set of internal signals as follows. Determine a set of internal signals D according to: • Every parallel path from w i to w j , i.e. a path that is not the edge G ji , passes through an internal vertex in D, and • Every loop through w j passes through a vertex in D. If in a network the signals w i , w j and D are retained, and all other internal signals are eliminated (immersed), e.g., because they cannot be measured, then in the immersed network that leaves the measured internal signals invariant, the target module G ji remains invariant too. As a result, this condition has become an important tool for selecting internal signals to be measured for identification of a single module, see e.g. also Ramaswamy and Van den Hof (2021) and Weerts, Linder, Enqvist, and Van den Hof (2020). It can be shown that the above condition has a very strong link to disconnecting sets and Theorem 4. The proof of the above result is presented in Appendix. While the parallel path and loop condition is used to select internal signals for identifying G ji , that same condition now has the role of selecting internal signals to be externally excited for single module identifiability, according to the results of Theorem 4. Moreover, a set of signals, which satisfies the parallel path and loop condition, can now effectively be found by constructing a (minimum) disconnecting set.

Signal allocation for generic identifiability
If G jW j is not generically identifiable in a given network model set, extra excitation signals can be allocated to achieve generic identifiability of G jW j . In this section an algorithm will be developed to address this synthesis problem by exploiting condition (b) in Theorem 4. Consider a network model set with a set of initial external signals X 0 j that do not have unknown directed edges to w j . The algorithms aim to allocate a minimum number of additional excitation signals X a j such that generic identifiability of G jW j is guaranteed. In this synthesis problem, it is assumed that if r k is allocated directly at w j , its corresponding transfer function R jk is known.
Condition (b) of Theorem 4 is based on the existence of ā X j − W j \W j disconnecting set givenX j ; however, the setX j depends on the allocated excitation signals and thus is not available before signal allocation. Therefore, in the synthesis approach we first find a candidate disconnecting set D without relying on the excitation signals, and then constructX j by allocating signals such that (i) D becomes aX j − W j \W j disconnecting set and (ii) D ∪W j are excited byX j to satisfy (11). A candidate set D can be motivated by the following corollary.

Corollary 1. Assume that one of the conditions in Theorem 4 is
satisfied, then D is aX j − W j \W j disconnecting set if and only if it is aW j ∪X j − W j \W j disconnecting set subject to D ∩W j = ∅.
Proof. The ''if'' part is trivial. The ''only if'' part is proved by contradiction. If D is not aW j −W j \W j disconnecting set, i.e. there exists a path fromW j to W j \W j which does not intersect with D, then there is also a path from X j (orX j ) viaW j to W j \W j and the path does not intersect with D, which contradicts that D is a disconnecting set. In addition, following from (11), D ∩W j = ∅ must hold. ■ The above result indicates that aW j − W j \W j disconnecting set D subject to D ∩W j = ∅ can be computed first, which is independent of the allocated external signals. Then extra excitation signals can be allocated such that signals in D ∪W j are excited, and D becomes a disconnecting set from the allocated signals to W j \W j .
As the number of required excitation signals depends on the cardinality of the disconnecting set, a minimum disconnecting set is desired. Additionally, based on Corollary 1, a minimum disconnecting set D subject toW j ∩ D = ∅ needs to be found. As standard graphical algorithms for computing minimum disconnecting sets do not take into account any constraint, we redefine the disconnecting set to make standard algorithms applicable.
Proposition 5. Given a model set M and any subsetX j ⊆ X j , D is a minimum disconnecting set fromW j ∪X j to W j \W j subject tō W j ∩ D = ∅ if and only if D is a minimum disconnecting set from Proof. We will first show that for any vertex set D subject tō and only if it is also a disconnecting set fromW j ∪X j to W j \W j .
The ''only if'' part holds because if D intersects with all paths from N + W j ∪X j to W j \W j , then it also intersects with the paths from W j ∪X j to W j \W j . For the ''if'' part, sinceW j ∩ D = ∅ and D intersects with all the paths fromW j ∪X j to W j \W j , those paths fromW j to W j \W j have to intersect with D at their internal vertices or the ending vertices. Since the first internal vertices of the paths belong to set N + W j , then D is also a disconnecting set Having the above result, the proposition is proved by showing that a minimum disconnecting set D from N + does not containW j , because if it does, it remains a disconnecting set afterW j is excluded, which contradicts the minimality of D. ■ The above result shows that a minimum disconnecting set D from N + W j to W j \W j can be computed for the synthesis problem, which now is an unconstrained problem and thus can be solved by standard graphic algorithms, e.g. the Ford-Fulkerson algorithm (Schrijver, 2003). Having established a choice of D, the following synthesis result can be derived from condition (b) in Theorem 4.

Corollary 2.
Consider a network model set M that satisfies Assumptions 2, 3 and 5. Given any minimum disconnecting set D from N + W j to W j \W j , assigning distinct excitation signals to every vertex in D ∪W j leads to generic identifiability of G jW j in M from (w, r).
Proof. Let X a j denote the set of allocated signals, and X a j ⊆ X j holds in the obtained model set after allocation, i.e. X a j has no unknown directed edge to w j . As these signals are allocated based on Proposition 5. In addition, (11) holds withX j = X a j . Thus, Theorem 4(b) is satisfied withX j = X a j and the given D. ■ The above result provides the basis for a synthesis approach that first computes a disconnecting set D and then allocates external signals to directly excite D ∪W j . This approach is valid even if a non-minimum disconnecting set is considered. However, it does not consider the initially present signals X 0 j , and thus may allocate redundant signals. Additionally, excitation signals do not necessarily have to be directly allocated at the vertices in D ∪W j , but could also reach those vertices through vertex disjoint paths. To make use of X 0 j and to explore the freedom to allocate the signals, a more comprehensive method is introduced in Algorithm 1. Given a network model set with the target modules G jW j and the pre-existing external signals X 0 j , Algorithm 1 computes a minimum disconnecting set D from N + W j ∪ X 0 j to W j \W j first and then removes the vertices in D ∪W j that are already excited by X 0 j via vertex disjoint paths. Then the algorithm allocates additional excitation signals to excite the remaining vertices in D ∪W j through vertex disjoint paths. The validity of Algorithm 1 is shown in the following result.
Theorem 6. Given a network model set M that satisfies Assumptions 2, 3, 5. In the returned model set of Algorithm 1, G jW j is generically identifiable from (w, r).
Proof. From step 1 to 2 in Algorithm 1, by construction, if |P| = |D ∪W j |, (11) holds withX j = X 0 j , and thus the modules are generically identifiable in the original model set M. When 2: Based on Lemma 7, compute a set P that contains the maximum number of vertex disjoint paths from X 0 j to D ∪W j , while the paths are internally vertex disjoint with D ∪W j ; 3: LetD ⊆ D ∪W j denote all the ending vertices of the paths in P; 4: if |P| < |D ∪W j | then 5: Find the largest setW ⊆ W such that D is a disconnecting set fromW to W j \W j ; 6: Build a subgraphḠ ⊆ G by removing all vertices and edges of the paths in P;

7:
Find a set W exp ⊆W such that inḠ, there are |D ∪W j | − |P| vertex disjoint paths from W exp to (D ∪W j ) \D ; 8: In G, assign distinct excitation signals to every vertex in W exp , which leads to a new model set M out with a new graph G out and a new R matrix; 9: Return M out with the graph G out ; 10: else 11: M out ← M and G out ← G;

12:
Return M out with the graph G out . 13: end if |P| < |D ∪W j |, based on Theorem 4, we need to allocate extra |D ∪W j | − |P| signals, such that: (i) there are |D ∪W j | − |P| vertex disjoint paths from these signals to (D ∪W j ) \D, and these paths are also vertex disjoint with P; (ii) D remains a disconnecting set from the added signals to W j \W j . Then we can find in the algorithm, steps 5 guarantees (ii), and steps 3, 6, 7 and 8 guarantee (i). This concludes the proof. ■ Remark 1. In Algorithm 1, the signals do not need to be allocated directly at the vertices in D∪W j . It suffices when they reach these vertices through vertex disjoint paths from excitation locations elsewhere in the network.
Algorithm 1 and the synthesis approach in Corollary 2 guarantee a minimum number of allocated signals when X 0 j = ∅, as a minimum disconnecting set is used. While a minimum number of allocated signals cannot be guaranteed when X 0 j ̸ = ∅, the following bound on the number of additionally required signals can be derived.
Corollary 3. Given a network model set M that satisfies Assumptions 2, 3, 5. Let X 0 j denote the set of initial external signals that have no unknown directed edge to w j . Let D be a minimum disconnecting

and let c denote the number of excitation signals available for allocation. Then the number of additional excitation signals c is sufficient to make G jW j generically
Algorithm 1 is illustrated in the following example.
Example 5. In Fig. 4(a), the problem is to allocate excitation signals such that G 73 becomes generically identifiable. X 0 7 = {e} is the only external signal that is initially present. Firstly, a disconnecting set from X 0 7 ∪ {w 3 } = {w 3 , e} to the other in-neighbors of w 7 through unknown edges, i.e. W 7 \ {w 3 } = {w 4 , w 6 , w 8 }, is constructed as D = {w 4 , w 7 }, indicated in Fig. 4(b). Based on Theorem 4, generic identifiability of G 73 requires three vertex disjoint paths from external signals to D ∪ {w 3 } = {w 3 , w 4 , w 7 }, while D remains a disconnecting set from the external signals to W 7 \ {w 3 }. Following step 2 in Algorithm 1, we find a path e → w 4 from X 0 7 to D ∪ {w 3 } (colored blue in Fig. 4(b)). Thus we only need to allocate extra excitation signals which have two vertex disjoint paths to {w 3 , w 7 }, and the two paths should be vertex disjoint with path e → w 4 . As in step 5, the set of potential locations to allocate excitation signals is given bȳ W = {w 1 , w 2 , w 3 , w 4 , w 5 , w 7 }, which satisfies the requirement that D remains a disconnecting set fromW to W 7 \ {w 3 }. Then we choose W exp = {w 1 , w 5 } ⊆W to be excited by r 1 and r 5 , as shown in Fig. 4(c). Based on Theorem 4, G 73 is indeed generically identifiable in Fig. 4(c): There exists a disconnecting set D = {w 4 , w 7 } from X 7 = {r 1 , r 5 , e} to W − 7 \ {w 3 } = {w 4 , w 6 , w 8 } such that b X 7 →D∪{w 3 } = 3, i.e., the blue paths in Fig. 4(c).
When generic identifiability of {G jW j |j ∈ {1, . . . , L}} is of interest, Algorithm 1 can be applied recursively for each j. If generic identifiability of the full network is concerned, a compact graphical approach for signal allocation is presented in Cheng, Shi, and Van den Hof (2019) and Cheng et al. (2022), where effective use is made of the interdependence of the L different subproblems.

Indirect identification method
In the previous sections, graphical tools have been developed to verify identifiability and achieve identifiability via signal allocation. However, the identifiability results do not point directly to an identification method to actually estimate the modules of interest from data. In this section we will show that, when the identifiability conditions in Theorem 4 are satisfied, in a particular situation the identifiability results directly point to an identification method for estimating G jW j . The particular situation is that the identifiability conditions are satisfied on the basis of a set of external signalsX j that contains r signals only. We will first show how the target modules can be reconstructed on the basis of transfer functions between measured signals only, by combining the identifiability result and Theorem 5. For simplicity, we consider the setting where all the known transfer operators in the network are fixed zeros.

Corollary 4.
Consider a model set M that has all the known transfer operators being zeros and satisfies Assumptions 2, 3. If condition (b) of Theorem 4 is satisfied withX j having no directed edge to w j , then it holds that where (·) † denotes the right inverse and exists for almost all θ, and I |W j | denotes an identity matrix of size |W j |.
Proof. When all the known transfer operators in M are zeros, Assumption 5 is not needed, and W j contains all the internal signals having directed edges to w j . IfX j has no edge to w j , the jth row in (I − G)T WX = X can be reformulated as If condition (b) of Theorem 4 holds, K exists so that T (W j \W j )X j = KT DX j according to Theorem 5. Combining the above decomposition and (14) leads to (13), where the right inverse exists generically due to (11) and (8). ■ The expression (13) shows an immediate opportunity to estimate G jW j based on a selected set of measured internal signals. WhenX j contains only measured excitation signals r, the transfer functions on the right hand side of (13) can all be estimated consistently through a standard identification method, based on the measured signals inX j ∪W j ∪ D ∪ {w j }.
The above method is a generalization of the ''classical'' indirect method of identification, as developed for closed-loop systems (Ljung, 1987) and dynamic networks (Gevers et al., 2018), where all the in-neighbors of w j are measured and directly excited for identifying the target modules. The method based on Corollary 4 generalizes this situation by allowing to measure and excite the internal signals determined by disconnecting sets. This generalization can lead to an experimental setup with fewer actuators and sensors for identifying the target modules than required by the classical method.
Remark 2. WhenX j only contains r signals, it is shown in this section that G jW j can be estimated consistently based on the signals in D∪W j ∪{w j }. It is then straightforward that under these conditions G jW j is generically identifiable in an appropriately chosen model set, without requiring to measure all the internal signals. This establishes a particular generic identifiability result with unmeasured internal signals, as also addressed in Bazanella et al. (2019).
If the setX j contains both r and e-signals, then the reasoning underlying the indirect approach fails, and one can apply a direct method for consistently identifying the target module, see Ramaswamy and Van den Hof (2021).

Duality
It is not hard to extend the results of this work from the full measurement setting to the full excitation setting in Bazanella et al. (2017) and Hendrickx et al. (2019), where all internal signals are excited with a subset unmeasured, and only r signals are used as excitation sources for identifiability. In this case, the goal is to ensure the uniqueness of a subset of out-going edges of w i , given the mapping from the r signals to the measured internal signals. Then the path-based condition as in Theorem V.I of Hendrickx et al. (2019) can be obtained, and a disconnecting-set-based result can be obtained similarly.
However, exploiting the noise signals as excitation sources for identifiability, as done in Proposition 1, is not trivial when there are unmeasured internal signals. This requires to handle the noise information in a submatrix of the power spectrum Φv of the disturbance, instead of the full matrix Φv as in this work. This more complex situation is addressed in the follow up work (Shi et al., 2021b).

Conclusion
In this work, generic identifiability of a subnetwork, i.e. a subset of modules, in linear dynamic networks has been investigated in the setting where all internal signals are measured. Algebraic conditions are obtained for both generic and global identifiability. Then a path-based condition for generic identifiability is obtained by extending the results in Hendrickx et al. (2019), in order to handle the model-set type of identifiability, while including disturbance inputs in the network and allowing network modules to be fixed/known. This path-based condition is then equivalently reformulated into a novel graphical condition based on disconnecting sets. With this new condition, the synthesis problem is tackled, which aims to allocate the minimum number of excitation signals to achieve generic identifiability of the modules. In addition, the new condition allows us to make a further step towards a new indirect identification method for estimating the subnetwork. The next step in this research is the handling of the more complex situation of partial measurement and partial excitation. This topic is addressed in the follow up work (Shi et al., 2021b). The Matlab software for conducting identifiability analysis based on Theorem 3 and synthesis according to Algorithm 1 can be found in Shi, Cheng, and Van den Hof (2021a). matrix whose entries are either zeros or distinct indeterminates: where A is hollow, A and D are of dimensions L × L and m × m, respectively. Here L or m is allowed to be zero. (15) is not structural full rank, then M can be permuted as

Lemma 5. If M in
whereĀ has dimension k 1 × (k 1 − 1) for some k 1 ⩾ 1. on Lemma 1, we can divideV into three disjoint sets D, S and P with |S| + |P| = L + m + 1. Moreover, there is no edge from S to P, where S ⊆W ∪X , P ⊆W ∪Ȳ. Thus, from the definition ofĒ, we can find a permutation of M in the form of (16) with a zero block, whose rows and columns correspond to P and S, respectively. Furthermore, the column dimension ofĀ is computed as L+m−|S| = |P|−1, which completes the proof. ■ Lemma 6. Let the non-zero entries of M in (15) be divided into two disjoint sets P 1 and P 2 . If det(M) does not depend on the entries in where A i is a square submatrix of M that contains only non-zero entries in P 2 .
Proof. When P 1 = ∅, the proof is trivial. Now suppose P 1 ̸ = ∅, Here, the sign ± depends on the permutation. IfÃ orD contains other entries in P 1 , the above analysis can be applied recursively, which proves the lemma. ■ With the previous lemmas, Lemma 4 can be achieved.
Proof. Without loss of generality, consider |W| ⩽ |X |, and thus TWX having full generic rank implies that its square submatrix TWX 1 has full rank generically. ConsiderF defined as (7) Denote the above submatrix ofF as F 1 . ThenF is proved to be generically full rank by showing that the term (18) is non-zero and cannot be canceled by the other terms in det(F ) generically as follows. As (17) contains the maximum number of parametrized entries compared to the other permutations, det(F 1 ) does not depend on the parametrized entries when considered as a polynomial of F 1 's all non-zero entries. Based on Lemma 6 and the fact that C is a selection matrix, (18) can be reformulated as for some l ⩾ 1 and j ∈ {0, 1}, where A i contains only known transfer functions and is square submatrix of F 1 and thus of F (W,X 1 ) defined in (6). As (19) contains the non-zero term (17) after expansion, A i is structural full rank and thus full rank by Assumption 5, for all i. This means that det(F 1 ) is non-zero and thus det(F ) contains at least the non-zero term (18). As each parametrized entry has independent parameters by Assumption 3, the non-zero term (18)  where C is a selection matrix that extracts rows of (I − A) −1 corresponding toX 1 . Then following Lemma 3 similarly, it holds that is full rank, and thus F (W 1 ,X 1 ) defined in (6) is structural full rank. Then based on Lemma 4, TW 1X1 is generically full rank that equals bX →W , and thus rank(TWX ) ⩾ bX →W generically.
For rank(TWX ) ⩽ bX →W generically, a minimumX −W disconnecting set can be considered as in Hendrickx et al. (2019), which leads to permuted network matrices with block zeros as also explored later in (21). Then the proof can be achieved similarly as in Hendrickx et al. (2019) Proof. We prove the lemma by showing that there always exists a P new by modifying the paths in P. Let w i → w j be an arbitrary path in P which contains an internal vertex in V, then we can always replace w i → w j by its subpath which contains a starting vertex in V 1 and an ending vertex in V 2 , while the other vertices in the subpath are not in V. This includes the special case that the obtained subpath has no internal vertex. Applying the above modification to all the paths in P, which contain internal vertices in V, leads to P new . ■ Based on the above result, there always exists a set of the maximum number of vertex disjoint paths from V 1 and V 2 , which are internally vertex disjoint with V 1 ∪ V 2 . We denote such a set by B V 1 →V 2 . With this notation, an instrumental graphical result can be obtained.
Proof. We first prove that (1) holds if and only if (2) holds. If (2) holds, (20) shows that there exist |D| and |V 2 | vertex disjoint paths from V 1 to D and from V 1 toV 2 , respectively. Then according to Lemma 7, there exist B V 1 →V 2 with cardinality |V 2 | and B V 1 →D with cardinality of |D|, and the paths in the above two sets are vertex disjoint. A new set B D→V 2 \V 2 can also be introduced and is vertex disjoint with both B V 1 →V 2 and B V 1 →D , because if not vertex disjoint, a path from V 1 to V 2 \V 2 will exist and do not intersect with D, contradicting D as a disconnecting set. Then by linking a subset of paths in B V 1 →D and all paths in B D→V 2 \V 2 , we can obtain a set of vertex disjoint paths from V 1 to V 2 \V 2 . Since D is a V 1 −V 2 \V 2 disconnecting set and |B V 1 →D | = |D|, the obtained set of paths forms a set B V 1 →V 2 \V 2 with the maximum number of vertex disjoint paths, and its cardinality equals |B D→V 2 \V 2 |.
In addition, the obtained B V 1 →V 2 \V 2 is also vertex disjoint with B V 1 →V 2 . Since b V 1 →V 2 ⩽ b V 1 →V 2 + b V 1 →V 2 \V 2 always holds, B V 1 →V 2 with cardinality |V 2 | and B V 1 →V 2 \V 2 together forms a set of maximum number of vertex disjoint paths from V 1 to V 2 , which proves (1).
If (1) holds, let D be a minimum V 1 − V 2 \V 2 disconnecting set, and we have B V 1 →V 2 with cardinality |V 2 | and B V 1 →D with cardinality |D| which are vertex disjoint. Thus the above two sets together form a set of vertex disjoint paths from V 1 toV 2 ∪ D, which leads to b V 1 →V 2 ∪D ⩾ |D| + |V 2 |. As b V 1 →V 2 ∪D is upper bounded by |D| + |V 2 |, it then holds that b V 1 →V 2 ∪D = |D| + |V 2 |, which leads to (1) ⇐⇒ (2).

A.3. Proof of Proposition 4
Before proving the proposition, we first prove that there exists a directed path from w i to W j \ {w i } if and only if there exist a parallel path from w i to w j or a cycle around the output w j . Note that due to model (1), G is a simple graph, i.e. there is no self-loop such as (w i , w i ), and no parallel directed edges from one vertex to another vertex. For ''if'' part, if there exists a parallel path from w i to w j , this parallel path has to intersect with W j \ {w i }. Then we can find a directed path from w i to one vertex in W j \ {w i } as a subpath of the parallel path. If a cycle around w j exists, it will also intersect with W j \ {w i }, and thus the cycle contains a subpath from w j to one vertex in W j \ {w i }. Linking this subpath and the edge (w i , w j ) leads to a path from w i to W j \ {w i }. For ''only if'' part, for any directed path from w i to w k ∈ W j \ {w i }, if the path does not contain edge (w i , w j ), then combining the path and the edge (w k , w j ) will create a parallel path. If the path contains (w i , w j ), then combining the path and the edge (w k , w j ) while excluding (w i , w j ) will lead to a cycle around w j . This concludes the relationship between the parallel paths, the cycles around the output and the paths from w i to W j \ {w i }.
Then based on the above result, the ''only if'' of the proposition is straightforward. For the ''if'' part, if we collect an internal vertex from each parallel path and a vertex from cycle around the output into D, D then must disconnect from w i to W j \ {w i }.

A.4. Proof of Theorem 5
According to Lemma 1, the disconnecting set separates all the vertices in the graph into three disjoint sets as V = S ∪ D ∪ P, while there is no directed edge from S to P. In addition, each set may contain both internal signals and external signals, i.e. S = S x ∪ S w , D = D x ∪ D w and P = P x ∪ P w , and it holds thatX ⊆ S x ∪ D x andW ⊆ D w ∪ P w . Algebraically, the above statements mean that G can be permuted and partitioned as X and T WX can be permuted and partitioned similarly with block X Pw Sx = 0, where, for example, T Sw Dx denotes the mapping from D x to S w . Based on the above partition and the equation (I − G)T WX = X , our goal is to find a proper matrix K such that TWX = KT DX .
Firstly, considering the division of the sets D andX , the mapping T DX can be re-written as whereX S =X ∩ S x andX D =W ∩ D x , the identity matrix is the mapping TX DXD . Note that the rows of the bottom block matrices in (22) correspond to the vertices in D x \X D . In addition, TWX can be written as whereW P =W ∩ P w andW D =W ∩ D w . Thus, it is clear that where C is a selection matrix that extracts the rows of The formulation of the above K matrix can be further simplified.
Based on the permuted matrices and the equation (I − G)T WX = X , it holds (I − G Pw Pw )T Pw Dx − G Pw Dw T Dw Dx = X Pw Dx . Thus, we can conclude that whereK is defined in (25); C is defined in (24) and its rows correspond toW D ; the columns of the last block column in (26) correspond to D x \X D . Note that certain blocks in K may disappear depending on if the corresponding set of signals is empty. ■