On the Trade-off Between Controllability and Robustness in Networks
of Diffusively Coupled Agents
In this paper, we study the relationship between two crucial properties in linear dynamical networks of diffusively coupled agents – controllability and robustness to noise and structural changes in the network. In particular, for any given network size and diameter, we identify networks that are maximally robust and then analyze their strong structural controllability. We do so by determining the minimum number of leaders to make such networks completely controllable with arbitrary coupling weights between agents. Similarly, we design networks with the same given parameters that are completely controllable independent of coupling weights through a minimum number of leaders, and then also analyze their robustness. We utilize the notion of Kirchhoff index to measure network robustness to noise and structural changes. Our controllability analysis is based on novel graph-theoretic methods that offer insights on the important connection between network robustness and strong structural controllability in such networks.
Network controllability, network robustness, graph-theoretic methods, network structure.
In a networked control system, controllability and robustness to noise and structural changes in the network are two of the most crucial attributes. Controllability describes the ability to manipulate and drive the network to a desired state through external inputs, whereas, network robustness expresses the ability of the network to maintain its structure in the event of device or link failures. Another aspect of robustness is the ability to function correctly in the presence of noisy information. Network controllability and robustness are both needed to design networks that achieve desired goals and objectives in practical scenarios. However, it is often observed that networks easier to control exhibit lesser robustness and vice versa, for instance see . Thus, exploiting trade-offs between network controllability and robustness can have a far reaching impact on the overall network design.
In this paper, we study the relationship between controllability and robustness in diffusively coupled leader-follower networks by focusing on finding extremal networks for these properties. In particular, for given parameters, we obtain networks with maximal robustness and then analyze their controllability. Similarly, we design networks with maximal controllability, and then evaluate their robustness. To characterize network robustness, we utilize a widely used metric Kirchhoff index (), that captures both aspects of robustness – the effect of structural changes in the network as well as the effect of noise on the overall dynamics (for instance, see [2, 3, 4]). To quantify control performance, we consider the minimum number of inputs (leaders) needed to make the network strong structurally controllable, that is, completely controllable irrespective of the coupling weights between nodes (e.g., see [5, 6, 7]). Accordingly, a network that requires fewer leaders for strong structural controllability is preferred over the one requiring many leaders.
Our approach is primarily graph-theoretic, and turns out to be effective in exploiting the relationship between network controllability and robustness. Our main contributions are:
For any given number of nodes and diameter , we identify networks with maximum robustness and provide a detailed analysis of their controllability, that is, the number of leaders that are necessary and sufficient to completely control such networks with arbitrary coupling weights between nodes.
For any number of nodes and diameter , we design networks that are strong structurally controllable with the minimum number of leaders. For this, we first provide a sharp upper bound on the minimum number of leaders for strong structural controllability with arbitrary and .
We also evaluate the robustness of maximally controllable networks and compare it with the robustness of maximally robust graphs for the same and .
I-a Related Work
Kirchhoff index or equivalently effective graph resistance based measures have been instrumental in quantifying the effect of noise on the expected steady state dispersion in linear dynamical networks, particularly in the ones with the consensus dynamics, for instance see [2, 8, 9]. Furthermore, limits on robustness measures that quantify expected steady-state dispersion due to external stochastic disturbances in linear dynamical networks are also studied in [10, 11]. To maximize robustness in networks by minimizing their Kirchhoff indices, various optimization approaches (e.g., [12, 13]) including graph-theoretic ones  have been proposed. The main objective there is to determine crucial edges that need to be added or maintained to maximize robustness under given constraints .
To quantify controllability, several approaches have been adapted, including determining the minimum number of inputs (leader nodes) needed to (structurally or strong structurally) control a network, determining the worst-case control energy, metrics based on controllability Gramians, and so on (e.g., see [15, 16]). Strong structural controllability, due to its independence on coupling weights between nodes, is a generalized notion of controllability with practical implications. There have been recent studies providing graph-theoretic characterizations of this concept [5, 6, 7]. There are numerous other studies regarding leader selection to optimize network performance measures under various constraints, such as to minimize the deviation from consensus in a noisy environment [17, 1], and to maximize various controllability measures, for instance [18, 19, 20, 21]. Recently, optimization methods are also presented to select leader nodes that exploit submodularity properties of performance measures for network robustness and structural controllability [16, 22].
Very recently in , trade-off between controllability and fragility in complex networks is investigated. Fragility measures the smallest perturbation in edge weights to make the network unstable. Authors in  show that networks that require small control energy, as measured by the eigen values of the controllability Gramian, to drive from one state to another are more fragile and vice versa. In our work, for control performance, we consider minimum leaders for strong structural controllability, which is independent of coupling weights; and for robustness, we utilize the Kirchhoff index which measures robustness to noise as well as to structural changes in the underlying network graph. Moreover, in this work we focus on designing and comparing extremal networks for these properties.
The rest of the paper is organized as follows: Section II describes preliminaries and network dynamics. Section III explains the measures for robustness and controllability, and also outlines the main problems. Section IV presents maximally robust networks for a given and , and also analyzes their controllability. Section V provides a design of maximally controllable networks and also evaluates their robustness. Finally, Section VI concludes the paper.
Let be an undirected graph with a vertex set and edge set . The graphs in this paper are loop-free, that is, no self loops between nodes. A node is a neighbor of if an edge exists between and , which is denoted by an unordered pair . The neighborhood of is denoted by . The distance between nodes and , denoted by , is the number of edges in the shortest path between and . The diameter of , denoted by , is the maximum distance between any two nodes in . A graph is weighted if edges are assigned values (weights) using some weighting function . The adjacency matrix of is defined as
Similarly, the degree matrix of is defined as
The Laplacian of is then defined as
Ii-a Network Dynamics
We consider a network of agents modeled by a graph in which the node set represents agents and the edge set represents inter-connections between agents. Each agent updates its state by the following dynamics
where is the coupling strength between nodes and . Moreover, to control and drive the network as desired, external control inputs are injected through a subset of nodes called leaders. The dynamics of the leader node is,
Let the set of leaders be represented as , where, without loss of generality, the leaders are labeled such that . If the total number of nodes is and the number of leader nodes is , then the overall system level dynamics can be written using the underlying graph’s Laplacian as
where be the state vector, be the control input to the leaders, and be an input matrix with the following entries
Iii Network Measures and Problem Setup
Iii-a Robustness Measure
To measure network robustness, we use the notion of Kirchhoff index of a graph, denoted by , and defined as
where is the number of nodes and are positive eigenvalues of the Laplacian of the graph (weighted or unweighted). A smaller value of indicates higher robustness in networks and vice versa.
Our motivation to use this robustness measure is twofold. First, it is very useful in characterizing the robustness to noise of linear consensus over networks. In fact, as shown in , it is directly related to the norm that measures the expected steady-state dispersion of the nodes under white noise via the relationship . Thus, it characterizes the functional robustness – ability of the network to perform well in the presence of noise that corrupts measurements or information exchange within the network. Other applications of in the study of various control theoretic problems have been surveyed in [8, 24].
Second, of a network captures its structural robustness – the ability of the network to retain its structural attributes in the case of edge or node deletions. It assimilates the effect of not only the number of paths between nodes, but also their quality as determined by the lengths of the paths . For a detailed discussion, we refer the readers to [3, 4, 12].
Iii-B Controllability Measure
A state is reachable if there exists some input that can drive the system in (6) from origin to in a finite amount of time. A set of all reachable states constitutes the controllable subspace, which is the range space of the following matrix.
The dimension of controllable subspace is the rank of , which needs to be for complete controllability. The rank of depends not only on the edge set of the graph but also on the edge weights. In fact, a graph that is completely controllable for one set of edge weights might not remain completely controllable if edge weights are changed. For a given graph and leader nodes (inputs), the minimum rank of for any choice of edge weights is the dimension of strong structurally controllable subspace. A graph is said to be strong structurally controllable with a given set of leaders, if the resulting controllability matrix is full rank with any choice of edge weights. Thus, in a strong structurally controllable network, perturbation in edge weights has no effect on the dimension of controllable subspace, which makes the notion of strong structural controllability quite general and applicable in situations where exact information of edge weights is inscrutable.
As a result, we are interested in finding the minimum number of leaders required to make a network strong structurally controllable.
We are interested in exploring relationships and trade-offs between robustness and controllability (as defined above) in diffusively coupled systems (6). In particular, we focus on extremal cases, and look at the following problems.
For a given number of nodes and diameter , which graphs have the minimum and thus, the maximum robustness?
What is the control performance – in terms of the minimum number of leaders needed to achieve strong structural contrallability – of the maximally robust graphs?
For any and , what is the minimum number of leaders that guarantee strong structural controllability? Furthermore, how can we construct graphs that achieve strong structural controllability with that many leaders.
What is the robustness of graphs in point (3) above?
Iv Maximally Robust Networks and their Controllability
In this section, our goal is to identify maximally robust networks, and then analyze their controllability.
Iv-a Maximally Robust Networks
For a given and , which graphs are maximally robust, that is, have the minimum amongst all such graphs? Another way to state this problem is to consider a complete graph of nodes, denoted by , and obtain a subgraph of that has a diameter and has the minimum amongst all such subgraphs.
For the unweighted case, it has been shown explicitly in  that for any and , optimal graphs having the minimum belong to a special class known as the clique chains, defined below. A clique is a subgraph in which all vertices are pairwise adjacent.
missing(Clique chain ) Let be a set of positive integers and , then a clique chain of nodes and diameter is a graph obtained from a path graph of diameter , that is , by replacing each node with a clique of size such that the vertices in distinct cliques are adjacent if and only if the corresponding original vertices in the path graph are adjacent. We denote such a clique chain by .
An example is illustrated in Figure 1.
In fact, the following result establishes the optimality of clique chains in terms of the minimum .
 For a given number of nodes and , graphs that achieve the minimum are necessarily clique chains of the form where .
Note that the and are always 1 in the optimal clique chains. Now we explicitly consider a weighted case and assume that is a complete graph with edge weights assigned by some weighting function . The question is to obtain a weighted spanning subgraph of that has a diameter and has the minimum . Using the same arguments as in , we get the following.
If is a weighted complete graph, then among all the subgraphs of with nodes and diameter , the graph that has the minimum is a clique chain where .
Proof – Let be an optimal subgraph with nodes and diameter , and is not a clique chain. Then, must be a subgraph of some clique chain, say (by Theorem 4 in ). It means there are some edges in that are not in . Adding edges strictly reduces the , and hence is not the optimal subgraph, which is a contradiction.
Thus, for a given and , maximally robust graphs (both for the weighted and unweighted cases) are clique chains of the form .
Iv-B Controllability of Clique Chains
Next, we analyze the strong structural controllability of the maximally robust graphs, that is, clique chains. The main result of this section is stated below.
Let be a clique chain with diameter , and be the number of leaders needed for the strong structural controllability of , then
We prove this result in Section IV-C by the graph-theoretic tools for the controllability of networked systems. In particular, we utilize the notions of
the notion of distance-to-leaders vectors and pseudo-monotonically increasing sequences (PMI) that we introduced in  to get the upper bound.
We explain these concepts with examples as well as relevant results in Appendix for completeness and clarity.
To obtain the lower bound in (10), we first note that the maximal LIEEP consisting of only singleton cells is a necessary condition for complete controllability (Theorem 0.1 in Appendix). Next, we determine the minimum number of leaders to have such a maximal LIEEP, which directly gives the minimum number of leaders for strong structural controllability. For the upper bound in (10), we determine the minimum number of leaders such that the graph has a full PMI sequence (see Appendix), which in turn would imply that the network is strong structurally controllable with that many leaders (Theorem 0.2). A detailed proof is given below.
Iv-C Proof of Theorem 4.3
We first prove the lower, and then the upper bound in (10).
Iv-C1 Lower Bound
The following result simply states that in the maximal LIEEP of a clique chain, all the non-leader nodes of a clique will be in the same cell.
Let be a clique chain and be its maximal LIEEP. If are non-leader nodes in the same clique , then they belong to the same cell of .
Proof – Assume belong to two different cells and of . Since and belong to the same clique, their neighborhoods are exactly same, which implies , . This means, we can combine and into one cell, and have a LIEEP with one lesser cell, which contradicts that is optimal.
Next, we show in the following result that in the maximal LIEEP of clique chain, a cell that contains non-leader nodes of a clique with a leader(s), contains the non-leader nodes of that clique only.
Consider a clique chain with . Let be respectively, a leader and a non-leader node in some clique . Also let be the cell of in the maximal LIEEP of . For any other node , lies in the same clique .
Proof – Proof is by contradiction. Let be the singleton cell containing . Clearly nodes must be neighbors in as otherwise . Assume, without loss of generality, that . If , let node belongs to , and be included in a cell . Note that cannot contain any node that is adjacent to . Since all nodes in the neighborhood of are adjacent to , does not contain any neighbor of . This means that . However, that is in the same cell as , is adjacent to , and thus has , which is not possible in . Thus and are not in the same cell in this case.
If on the other hand, when , consider a node . Since a node (such a node exists because ) is adjacent to and not adjacent to , . By Lemma 4.4 all non-leader nodes in are in and none of the non-leader nodes in are in . Clearly . Hence, and cannot be in the same cell, which is a contradiction.
Let be a clique chain with , then the number of leaders needed to have the maximal LIEEP of in which each node is in a singleton cell, is at least .
Proof – Let be the maximal LIEEP with all nodes in singleton cells. From Lemma 4.4, we know that all the non leader nodes of a clique will be in the same cell in . Moreover, from Lemma 4.5, we deduce that if is a clique with a leader node(s), then all the non-leader nodes of will be in the same cell and that cell does not contain a node of any other clique. Thus, we need at least leaders in the clique to have all of its nodes in singleton cells in . Thus, the minimum number of leaders in is .
For complete controllability, and hence strong structural controllability, maximal LIEEP in which each node is in a singleton cell, is a necessary condition (Theorem 0.1). By Proposition 4.6, we need at least leaders to have such a maximal LIEEP, which gives us a lower bound on the number of leaders as in Theorem 4.3.
Iv-C2 Upper Bound
We first state the following result that uses the notion of PMI sequence explained in the Appendix.
Let be a clique chain with , then leaders are enough to have a full PMI sequence in .
Proof – If we add a node from the first clique to the leader set, then there are at least nodes (not including ) that are at distinct distances from . Save these nodes, and include all the remaining nodes in the graph to the leader set. With such a set of leader nodes, we get a full PMI sequence of distance-to-leaders vectors.
A direct consequence of the above lemma is that leaders are sufficient for the strong structural controllability of clique chains.
V Maximally Controllable Networks and their Robustness
In the previous section, we looked at maximally robust networks, and analyzed their controllability. Here, we obtain graphs that are strong structurally controllable with the minimum leaders and evaluate their robustness.
V-a Maximally Controllable Networks
For any given and , which graphs exhibit strong structural controllability with the minimum number of leaders? To answer this, we first need to study for an arbitrary and , what is the minimum number of leaders needed to guarantee strong structural controllability? One of the main results in this section is as follows:
For any and , there exist graphs that are strong structurally controllable with leaders, where
Remark 1 - The above bound on the number of leaders is tight and cannot be improved for arbitrary and . In other words, there are graph classes for which we need at least leaders for strong structural controllability, for instance path graphs ( and ), cycle graphs ( and ), complete graphs ( and ).
To construct graphs satisfying the conditions in Theorem 5.1, we again use the notion of PMI sequences of distance-to-leaders vectors along with the result in Theorem 0.2. For any and , we construct graphs that give a full PMI sequence wih leaders, thus, graphs with strong structural controllability. Moreover, we want to be as small as possible, and note that for certain and , is as discussed previously. In fact, we first show that if a graph has a full PMI sequence with leaders, then .
Let be a graph with nodes, diameter , and leaders such that has a full PMI sequence, then .
Proof – Without loss of generality let be the maximum size PMI sequence where through are the leader nodes. For a pair of nonnegative integers , we observe that for all leader nodes ,
Further, there always exists at least one leader for which
by the definition of PMI sequence. Let denote the expression . Next, consider the following sequence of integers,
Thus, to have a full PMI sequence, we cannot do better than selecting a minimum of leaders. Next, we show that for any and , we can construct graphs that have full PMI sequences (and hence strong structural controllability) with leaders. Our approach is as follows:
First, for given positive integers and , we construct a sequence of vectors satisfying the PMI property. Each vector in the sequence is -dimensional and contains values from the set .
Second, we construct a graph with nodes and leaders such that the distance-to-leader vectors of nodes are exactly same as the vectors obtained in the above step. Thus, the constructed graph has a full PMI sequence of distance-to-leader vectors. The maximum distance between any leader and non-leader node in such a graph will be .
Third, we densify the above graph, that is, maximally add edges to the graph while ensuring that the distance-to-leader vectors of nodes do not change. Consequently, we get graphs with nodes, diameter and leaders. Adding edges always reduces and hence, improves robustness. The graphs obtained have full PMI sequences of distance-to-leader vectors, and are strong structurally controllable.
To construct sequences, we state the following proposition.
Let define the following set of vectors in :
then the following sequence of vectors in defines a PMI sequence for any positive integers and .
Next, we construct a graph with leaders and nodes whose distance-to-leader vectors are same as in (15). To do so, consider a vertex set
where and . Nodes in are leaders. We connect these vertices as follows:
All leader nodes are pair-wise adjacent and induce a clique.
is adjacent to each and , .
For each , is adjacent to leaders , .
For each , is adjacent to , where .
The above construction is illustrated in Figure 2.
Next, we compute the distance-to-leader vectors of nodes in as follows:
For all , the distance-to-leaders vector of is a vector of all 1’s except at the index , where it is 0. For the node , it is a vector of all 1’s.
For node , where , it is a vector in which all entries are .
For node , where and , the distance-to-leaders vector has first entries equal to and the remaining entries are , that is
where the arrow indicates the element of the vector.
Next, we consider the following sequence of nodes,
If the distance-to-leader vectors of nodes in are arranged in the same order as in (17), we get the same sequence as in (15), which is a PMI sequence of length . Hence, has a full PMI sequence, and is strong structurally controllable.
Example: Consider the graph in Figure 3, with nodes and leaders. For any leader , the maximum distance between and any other node is . A full PMI sequence of distance-to-leaders vectors is given below. Note that for each vector, there is an index (row index of the circled value) such that the corresponding row value of all the subsequent vectors in the sequence is strictly larger than the circled value, thus constituting a full PMI sequence.
Adding Edges to Graph
We note that removing an edge from could change the distance-to-leader vectors of nodes. However, we can add edges to to improve its robustness by lowering the Kirchhoff index. Next, we construct a new graph by maximally adding edges to while preserving distances between leaders and all other nodes. Consequently, all distance-to-leader vectors and resulting PMI sequence of and are same. We describe the addition of new edges below.
For a fixed , all the nodes in , where induce a clique.
Each is adjacent to .
For a fixed , each , where , is adjacent to , .
An example of obtained from for , , and is shown in Figure 4.
For a fixed and , the graph is maximal in the sense that adding any new edge would change the distance-to-leader vector of some node.
Proof: We classify edges that can be added to into four types, and will rule them out one by one.
Edge where : such an edge would reduce distance .
Edge where : such an edge would reduce distance .
Edge where : will reduce distance .
Edge where : will reduce distance .
There is only one other edge , and clearly we cannot add it without changing the distance between and .
Next, we state the following:
If is the maximum distance between a leader node and some other node in , then is the diameter of constructed from .
Proof: Nodes make a clique for all , and is a path of length . Therefore for all such pairs of nodes. Since all distance-to-leader vectors are preserved in due to Proposition 5.4, farthest node from each leader is still at distance . Thus the graph has diameter .
Remark 3 - So far, we have assumed that for some integer . However, we can obtain the desired graph for any by modifying . Let be the actual number of nodes, and be the desired diameter, then we construct a graph with nodes where . We need at least that many leaders to have a graph with a full PMI sequence (Theorem 5.2). Since , we need to delete nodes from . We delete the required number of nodes in the following order: first, we delete the nodes (in the same order) , then , and so on until the total number of nodes in the remaining graph is . Note that the nodes , where are not deleted to preserve the diameter . In fact, it is easy to verify that as a result of nodes deletion, the distance-to-leaders vectors of nodes in the remaining graph remain the same as in the original graph, and hence the longest PMI sequence of distance-to-leaders vectors of the nodes in the remaining graph has a length (full PMI sequence). Thus, we can state the following proposition.
For any and , there exist graphs that have full PMI sequences with leaders.
V-B Robustness of Maximally Controllable Networks
Here, we compare the robustness of maximally controllable graphs for a given and as obtained above with the the robustness of maximally robust graphs, that is clique chains. Although we know that for given and , maximally robust graphs belong to where ; we don’t know the exact values of ’s in general and need to compute them numerically. In Table I, we choose the same values of and as in Table 1 in , wherein the of optimal (unweighted) clique chains corresponding to the selected and are given. We compare these values with the of the maximally controllable graphs (unweighted) for the same and . It is seen that the of maximally controllable graphs is roughly the double of the of the corresponding clique chain, especially for the larger values.
Similarly, in Table II, we select and the number of leaders and then generate optimal clique chains (through exhaustive search) with , and also maximally controllable graphs (as in Section V-A) with the same , , and . We then compare the of both the clique chains and , and again observe that clique chains are roughly twice as robust as the corresponding , especially for larger values.
Networks that exhibit higher robustness to noise and structural changes typically require many leader nodes (inputs) to be completely controllable. For a fixed number of nodes , complete graphs are maximally robust but require leaders for complete controllability. At the same time, path graphs require only one leader for complete controllability, however, such graphs are minimally robust. We observed a similar relationship between controllability and robustness if we also fix the diameter of a graph along with vertices. Clique chains are optimal from the robustness perspective for a given and . However, they require a large number of leaders, either or , for strong structural controllability. On the other hand, for arbitrary and , we can construct graphs that are strong structurally controllable with at most leaders, which is a sharp bound. However, such graphs are much less robust compared to optimal clique chains with the same and . To exploit the controllability and robustness trade-off, graph-theoretic tools for network controllability including equitable partitions and distances of nodes to leaders are particularly useful. In the future, we aim to explore graph operations that maximally improve one of the two properties while deteriorating the other one minimally.
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A. Maximal Leader-Invariant External Equitable Partition
Let be a leader-follower network, whose nodes are partitioned into cells such that . Let be two distinct cells and , then the node to cell degree of to is , and is denoted by . A partition is a leader-invariant external equitable partition (LIEEP), denoted by , if the following conditions are satisfied.
Each leader node is in a singleton cell, that is, if is a leader and it is in a cell , then .
For any cell , let , then
A partition is maximal LIEEP, denoted by , if it is LIEEP and has the minimum number of cells among all LIEEPs. We note that the maximal LIEEP of a graph is unique. Moreover, if a graph is completely controllable, then the number of cells in is same as the number of nodes in a graph, that is, each node is in a singleton cell. An example of maximal LIEEP is illustrated in Figure 5.
An important result that relates the notion of maximal LIEEP to complete controllability in leader-follower networks is following.
B. Pseudo-Monotonically Increasing (PMI) Sequence
Let be a sequence of vectors where , . Moreover, we denote the entry of by . is a PMI sequence if for each , there exists an index such that
In our context, we are interested in finding the longest PMI sequence of distance-to-leaders vectors of nodes in a leader follower graph as defined below.
Let be a leader follower graph with leader nodes . For each node , we define a distance-to-leaders vector such that the entry of is the distance of node with the leader , that is,
An illustration of the distance-to-leaders vectors is shown in Figure 6. A PMI sequence of distance-to-leaders vectors is,
Note that for each vector, there is an index – of the circled value – such that the values of all the subsequent vectors at the corresponding index are strictly greater than the circled value. For instance, the value at the first index is circled in the vector , and the values at the first indices of all the subsequent vectors are greater than 0. We also note that if multiple nodes have identical distance-to-leaders vectors, for instance in the below example, we can include it only once in a PMI sequence.
As is shown in , PMI sequences of distance-to-leaders vectors in leader-follower networks are particularly useful in studying their strong structural controllability. We use the following result in our work.
 The dimension of the controllable subspace in the sense of strong structural controllability is at least equal to the length of the longest PMI sequence of distance-to-leaders vectors.
If the longest PMI sequence of distance-to-leaders vectors in a graph has a length equal to the number of nodes in a graph, we say that the graph has a full PMI sequence. Hence, if a network graph has a full PMI sequence, then it is strong structurally controllable.