Edge Augmentation with Controllability Constraints in
Directed Laplacian Networks

Each agent $u$ in the network has a state $x_{u}\in\mathbb{R}$, and the overall state of the network is $x\in\mathbb{R}^{n}$, where $n=|V|$. The network dynamics are given by the following equation:

where $L_{w}$ is the weighted Laplacian matrix of $G$ and defined as $L_{w}=(\text{Deg}-A_{w})$. Here, $A_{w}\in\mathbb{R}^{n\times n}$ is the weighted adjacency matrix of $G$ whose $uv^{th}$ entry is

and $\text{Deg}\in\mathbb{R}^{n\times n}$ is the degree matrix defined as

In (2), $B\in\mathbb{R}^{n\times m}$ is the input matrix, where $m$ is the number of inputs, which is equal to the number of leader nodes. If $V_{\ell}=\{\ell_{1},\ell_{2},\cdots,\ell_{m}\}\subset V$ is the set of leader nodes, then

A state $x^{\prime}\in\mathbb{R}^{n}$ is reachable if there is an input $u$ that can drive the network in (2) from origin (initial state) to $x^{\prime}$ in a finite amount of time. A network $G=(V,E)$ with edge weights defined by $w$ and leader set $V_{\ell}$ is completely controllable, that is every point in $\mathbb{R}^{n}$ is reachable, if and only if the following controllability matrix is full rank.

The rank of $\Gamma(L_{w},V_{\ell})$ defines the dimension of the controllable subspace consisting of all the reachable states. A Laplacian network $G=(V,E)$ with a given set of leader nodes is called strong structurally controllable (SSC) if it is completely controllable for any choice of $w$ as in (1). At the same time, the dimension of strong structurally controllable subspace (SSCS), denoted by $\gamma(G,V_{\ell})$, is the minimum rank of the controllablility matrix $\Gamma(L_{w},B)$ over all feasible $w$ (edge weights), i.e.,

The main objective in the paper is to identify the maximum number of missing edges in a given network such that the dimension of SSCS of the network is preserved even after adding those edges. Since computing the dimension of SSCS is computationally challenging, we consider its lower bounds, including the zero forcing (ZF) and distance-based bounds (explained in Sections III and IV, respectively). These bounds are tight and have numerous applications [10]. If $\delta$ is a (ZF-based or distance-based) lower bound on the dimension of SSCS of the network, then the goal is to maximally densify the graph while maintaining the dimension of SSCS to be at least $\delta$. Formally, we state the problem below.

Problem Let $G=(V,E)$ be a directed network of agents in which $V_{\ell}\subseteq V$ be a set of leaders and the network dynamics are defined by (2). Let the dimension of SSCS of the network be at least $\delta$. Then, find the maximum size edge set $E^{\prime}$ such that $E\subseteq E^{\prime}$ and the dimension of SSCS of the network $G^{\prime}=(V,E^{\prime})$ with the same set of leaders $V_{\ell}$ is also at least $\delta$, i.e., $\delta\leq\gamma(G^{\prime},V_{\ell})$.

First, we explain the notion of Zero Forcing process and its relation to the dimension of SSCS [5, 6, 7].

• Definition

  (Zero Forcing Process) Consider a directed graph $G=(V,E)$ such that each node $v\in V$ is initially assigned either a white or black color. The following coloring defines the zero forcing process: if a black colored node $v\in V$ has exactly one white in-neighbor $u$, then change the color of $u$ to black. We say that $v$ infected $u$.¹¹1Since an edge $(u,v)$ indicates that the state of node $u$ is influenced by the state of $v$ in our system model (as in (2) and (3)), we use in-neighbors in the ZF process for consistency. If an edge $(u,v)$ indicates that node $u$ influences node $v$’s state, then out-neighbors should be used in the ZF process, as done in some other works.

• Definition

  (Derived Set) Consider a directed graph $G=(V,E)$ where $V^{\prime}\subseteq V$ is an initial set of black nodes (also called the input set), and apply the zero forcing process until no further color changes are possible. The resulting set of black nodes is the derived set, denoted by $\text{dset}(G,V^{\prime})\subseteq V$. For a given set of input nodes, the derived set is unique [14]. Moreover, an input set $V^{\prime}$ is called a zero forcing set (ZFS) if $\text{dset}(G,V^{\prime})=V$.

As an example, consider $G$ in Figure 1, where $V^{\prime}=\{v_{1},v_{2}\}$ is the set of input nodes. As a result of ZF process, $v_{1}$ infects $v_{3}$, $v_{2}$ infects $v_{4}$ and the resulting derived set is $\{v_{1},v_{2},v_{3},v_{4}\}$.

In the context of SSC, the cardinality of the derived set is significant as it provides a lower bound on the dimension of SSCS, as stated in the following result.

We provide an algorithm to add edges to a directed graph with a given set of leaders. The algorithm ensures that the derived set of the graph remains the same after adding edges, thus, preserving the ZF-based bound on the dimension of SSCS. The proposed algorithm is a modification of the ZF process. In summary, we look for a BLACK node with a single WHITE in-neighbor, add edges incident to the BLACK node that do not change change the size of the derived set, change the WHITE node’s color, and then repeat the same procedure until there is no BLACK node with a single WHITE in-neighbor. When this process concludes, we add any extra edges that can be added while preserving the derived set. The algorithm is outlined below. We denote the color of the node $v$ by COLOR($v$).

Figure 2(b) illustrates an example of edge augmentation due to Algorithm 1. The derived set in $G$ and $G^{\prime}$ is same.

Next, we show that Algorithm 1 is optimal and adds the maximum number of edges while preserving the size of the derived set returned by the ZF process.

We can count the number of edges in the directed graph $G^{\prime}=(V,E^{\prime})$ returned by Algorithm 1 in terms of $|V|$, $|V_{\ell}|$ and the size of the derived set.

The time complexity of Algorithm 1 is $O(n^{2})$ where $n$ is the number of nodes in the graph. Note that the main task in the algorithm is the addition of edges to the edge set $E^{\prime}$. If the edges are kept in an adjacency matrix, each addition takes a constant amount of time. We do not add an edge twice to this set, so the time complexity is bounded by the maximum number of possible edges in a directed graph. We also observe that the time complexity is:

When the size of derived set is at least $n/2$, the first term is $\Omega(n^{2})$, and when it is less than $n/2$, the second term is $\Omega(n^{2})$. Thus, the time complexity of Algorithm 1 is $\Theta(n^{2})$.

Given a network with $m$ leaders $V_{\ell}=\{\ell_{1},\cdots,\ell_{m}\}$, we define the distance-to-leaders (DL) vector of each $v_{i}\in V$ as

The $j^{th}$ component of $D_{i}$, denoted by $[D_{i}]_{j}$, is equal to the distance of $v_{i}$ to $\ell_{j}$. Next, we provide the definition of pseudo-monotonically increasing sequences of DL vectors.

• Definition

  (Pseudo-monotonically Increasing (PMI) Sequence) A sequence of distance-to-leaders vectors $\mathcal{D}$ is PMI if for every $i^{th}$ vector in the sequence, denoted by $\mathcal{D}_{i}$, there exists some $\pi(i)\in\{1,2,\cdots,m\}$ such that

  $$[\mathcal{D}_{i}]_{\pi(i)}<[\mathcal{D}_{j}]_{\pi(i)},\;\;\forall j>i.$$

  (8)

  In other words, (8) needs to be satisfied for all the subsequent distance-to-leader vectors $\mathcal{D}_{j}$ appearing after $\mathcal{D}_{i}$ in the sequence. We say that $\mathcal{D}_{i}$ satisfies the PMI property at coordinate $\pi(i)$ whenever $[\mathcal{D}_{i}]_{\pi(i)}<[\mathcal{D}_{j}]_{\pi(i)},\;\forall j>i$.

An example of DL vectors is illustrated in Fig. 3. A PMI sequence of length five can be constructed as

Indices of bold values in (9) are the coordinates, $\pi(i)$, at which the corresponding distance-to-leaders vectors are satisfying the PMI property.

The longest PMI sequence of DL vectors is related to the dimension of SSCS as stated in the following result.

Let $G=(V,E)$ be a graph with a leader set $V_{\ell}$, $\mathcal{D}$ be a PMI sequence of length $\delta$, and $\tilde{V}\subseteq V$ be the set of nodes whose DL vectors are included in $\mathcal{D}$. If we add edges in $G$ to obtain a new graph $G^{\prime}=(V,E^{\prime})$ such that the DL vectors of nodes in $\tilde{V}$ remain the same in $G^{\prime}$, then $\mathcal{D}$ will also be a PMI sequence of $G^{\prime}$ and $\delta\leq\gamma(G^{\prime},V_{\ell})$. Therefore, one approach to augment edges in a graph while preserving a bound on the dimension of SSCS is to ensure that the distances from a certain set of nodes to leaders do not change due to edge additions. In this direction, we first need to study the maximal edge augmentation in a graph while preserving the distance from a given node $a$ to another node $b$.

• Definition

  (Distance Preserving Edge Augmentation (DPEA) Problem) Given a directed graph $G=(V,E)$ and nodes $a,b\in V$ such that $d_{G}(a,b)=k$, find a graph $G^{\prime}(V,E^{\prime})$ with the (same) node set $V$ and an edge set $E^{\prime}\supseteq E$ such that $d_{G^{\prime}}(a,b)=k$ and $|E^{\prime}|$ is maximized.

Next, we characterize optimal solutions of the DPEA problem for a given node pair $(a,b)$ in $G$. We show that the optimal solution belongs to a special class of graphs, which is obtained by the union of clique chains and modified clique chains described below.

• Definition

  (Directed clique chain) A directed graph $\mathcal{C}_{k}=(V,E)$ is a directed clique chain if the node set $V$ can be partitioned into sets $V_{0},V_{1},V_{2},\ldots,V_{k}$, such that there is an edge from every node in $V_{i}$ to every node in $V_{i-1}\cup V_{i}\cup V_{i+1}$ for all $1\leq i\leq k-1$. Moreover, nodes in each of $V_{0}$ and $V_{k}$ induce cliques.²²2In a clique, there is an edge between every pair of nodes.

• Definition

  (Directed modified clique chain) A directed graph $\mathcal{M}_{k}=(V,E)$ is a directed modified clique chain if the node set $V$ can be partitioned into sets $V_{0},V_{1},V_{2},\ldots,V_{k}$, such that there is an edge from every node in $V_{i}$ to every node in $V_{j}$ for all $j\leq i$.

Examples of directed clique chain and directed modified clique chain are shown in Figure 4.

From Theorem 4.3, we obtain a simple way to greedily construct a maximally dense graph that preserves the distance from node $a$ to node $b$ in a given graph $G$ as follows:

An example of the DPEA is shown in Figure 5 in which edges are augmented to preserve the distance from node $a=v_{1}$ to node $b=v_{5}$.

We note that the time complexity of Algorithm 2 is $O(n^{2}\log n)$ because we can implement this using runs of Dijkstra’s shortest path algorithm. We can use DPEA to add edges in a graph $G(V,E)$ with a leader set $V_{\ell}$ while preserving a bound on the dimension of SSCS. Let $\mathcal{D}$ be a PMI sequence of length $\delta$ and $\tilde{V}\subseteq V$ be the set of nodes whose DL vectors are included in $\mathcal{D}$. By solving the DPEA problem for the node pair $(v,l)$, where $v\in\tilde{V}$ and $l\in V_{\ell}$, we can obtain edges, say $E_{v,l}$, whose addition to the graph will preserve the distance from node $v$ to leader $l$. By solving DPEA problem for all node pairs $(v,l)$, where $v\in\tilde{V}$ and $l\in V_{\ell}$, we can obtain edges that are common in all solutions, that is, $E_{comm}=\cap_{v\in\tilde{V},l\in V_{\ell}}E_{v,l}$. By adding these common edges in the given graph $G$, we obtain a new graph $G^{\prime}=(V,E\cup E_{comm})$ such that $d_{G}(v,l)=d_{G^{\prime}}(v,l)$, $\forall v\in\tilde{V}$ and $\forall l\in V_{\ell}$. Consequently, $\mathcal{D}$ will also be PMI sequence of $G^{\prime}$, which means that the dimension of SSCS in the augmented graph $G^{\prime}$ will also be at least $\delta$. Thus, by solving multiple instances of DPEA problem, we can determine edges whose addition to the graph will preserve the distance-based bound on the dimension of SSCS.

Since the problem of finding an optimal set of edges while preserving the distance-based bound is computationally challenging, in the next subsection, we present and analyze a randomized edge augmentation algorithm that is simple to understand, easy to implement, and offers good numerical results in our experiments.

First, we discuss that for a given $G$, if $\mathcal{D}$ is a PMI sequence of length $\delta$ containing DL vectors of nodes $\tilde{V}\subseteq V$, then we can add edges in $G$ to obtain an augmented graph $G^{\prime}$ with the same leader set $V_{\ell}$. Moreover $G^{\prime}$ will also have a PMI sequence of length $\delta$ even if $d_{G}(v,l)\neq d_{G^{\prime}}(v,l)$, $\forall v\in\tilde{V},\forall l\in V_{\ell}$. In particular, for every node pair $(v,l)$, where $v\in\tilde{V}$, $l\in V_{\ell}$, there exists an integer $\epsilon_{v,l}\in\mathbb{Z}$ such that if $\epsilon_{v,l}<d_{G^{\prime}}(v,l)\leq d_{G}(v,l)$, then $G^{\prime}$ will have a PMI sequence consisting of DL vectors of nodes in $\tilde{V}$ and having a length $\delta$. We will then provide a randomized edge augmentation algorithm satisfying conditions to preserve the distance-based bound on the dimension of SSCS. Finally, we will analyze the performance of the algorithm.

Let $\mathcal{D}_{i}$ be the $i^{th}$ vector in the PMI sequence $\mathcal{D}$ of the given graph $G$. Then by the definition of PMI sequence, for each element $[\mathcal{D}_{i}]_{j}$, there is an integer, say $\epsilon_{i,j}$, such that $[\mathcal{D}_{i}]_{j}>\epsilon_{i,j}$. We denote the maximum possible value of $\epsilon_{i,j}$ by $\epsilon^{\ast}_{i,j}$ and define $\epsilon^{\ast}_{i}:=\left[\begin{array}[]{rrrr}\epsilon_{i,1}^{\ast}&\epsilon_{i,2}^{\ast}&\cdots&\epsilon_{i,|V_{\ell}|}^{\ast}\end{array}\right]^{T}$. For instance, consider the PMI sequence in (9). The vectors $\mathcal{D}_{i}$ in the sequence and the corresponding $\epsilon^{\ast}_{i}$ are given below.

We can think of $\epsilon^{\ast}_{i,j}$ as a strict lower bound on $[\mathcal{D}_{i}]_{j}$. From a given PMI sequence of length $\delta$, we can always obtain $\epsilon^{\ast}_{i}$ for all $i\in\{1,2,\cdots,\delta\}$.

For instance, consider the sequence $\bar{\mathcal{D}}$ below, which is obtained from $\mathcal{D}$ in (9) by replacing $[\mathcal{D}_{1}]_{2}=3$ and $[\mathcal{D}_{3}]_{2}=4$ by $[\bar{\mathcal{D}}_{1}]_{2}=1$ and $[\bar{\mathcal{D}}_{3}]_{2}=1$, respectively. Since $\epsilon^{\ast}_{1,2}=-1$ and $\epsilon^{\ast}_{3,2}=0$, the resulting sequence $\bar{\mathcal{D}}$ is a PMI sequence.

Next, we present a randomized algorithm to add edges in a graph $G$ with a leader set $V_{\ell}$. $G$ has a PMI sequence $\mathcal{D}$ of length $\delta$ consisting of DL vectors of nodes $\tilde{V}\subseteq V$. Moreover, let $S=\left[\begin{array}[]{llllll}s_{1}&s_{2}&\cdots&s_{\delta}\end{array}\right]$, where $s_{i}$ is the index of the node whose DL vector is the $i^{th}$ element (vector) in $\mathcal{D}$. For instance, $\mathcal{D}$ in (9) is a PMI sequence of the graph in Figure 3. The corresponding $\tilde{V}$ is $\{v_{1},v_{2},\cdots,v_{5}\}$ and $S$ is $\left[\begin{array}[]{llllll}1&2&3&4&5\end{array}\right]$. For every $i^{th}$ vector in the sequence, the algorithm first computes the corresponding vector $\epsilon_{i}^{\ast}$, which basically provides minimum distances that need to be maintained from node $v_{s_{i}}$ to all leaders during the edge augmentation. The algorithm then selects a missing edge randomly and augments it to the graph if its addition does not violate distance conditions (line 8 in Algorithm 3), otherwise discards it. This process is repeated until all missing edges from the original graph are either added to the graph, or discarded. The details are outlined in Algorithm 3.

Figure 6(a) illustrates an example of edge augmentation due to Algorithm 3 with the graph $G$ in Figure 3 and a PMI sequence of length $4$ as inputs. The red edges indicate the augmented edges. We observe that the graph $G^{\prime}$ in Figure 6(a) contains a total of 29 edges compared to the graph in Figure 2(b)(b), which contains a total of 25 edges after the ZF-based edge augmentation.

Next, we analyze the performance of Algorithm 3 by the following result.

We omit the proof of this proposition, because it follows the same arguments given in [11, Proposition 4.2] for a similar result on the undirected graphs. Moreover, the time complexity of Algorithm 3 is $O(|E|\times|E^{c}|\times m\log n)$, where $|E^{c}|$ is the number of edges not in the input graph. The term $O(|E|\times m\log n)$ is the cost of running Dijkstra’s shortest path algorithm for all leader nodes and we do it for all of the missing edges $E^{c}$ in the randomized algorithm.