Controllability of Bandlimited Graph Processes Over Random Time-Varying Graphs

Controllability of Bandlimited Graph Processes Over Random Time-Varying Graphs

Fer nando Gama, Elvin Isufi, Alejandro Ribeiro and Geert Leus Part of this works is presented in [1]. Work in this paper is supported by NSF CCF 1717120, ARO W911NF1710438, ARL DCIST CRA W911NF-17-2-0181, ISTC-WAS and Intel DevCloud. F. Gama, E. Isufi and A. Ribeiro are with the Dept. of Electrical and Systems Eng., Univ. of Pennsylvania, and G. Leus is with the Dept. of Microelectronics, Delft Univ. of Technology. E-mails: {fgama, eisufi, aribeiro}@seas.upenn.edu, g.j.t.leus@tudelft.nl.
Abstract

Controllability of complex networks arises in many technological problems involving social, financial, road, communication, and smart grid networks. In many practical situations, the underlying topology might change randomly with time, due to link failures such as changing friendships, road blocks or sensor malfunctions. Thus leading to poorly controlled dynamics if the randomness is not properly accounted for. We consider the problem of driving a random time-varying network to a desired state that is graph-bandlimited with respect to the underlying topology, i.e., it has a sparse representation in the graph spectral domain. We first formulate the problem of sparse controllability on deterministic networks and exploit graph signal processing to obtain conditions on the number of required controlled nodes and the final control time. We then move on to random time-varying networks and develop the concept of controllability in the mean to control the expected system towards the desired state. A detailed mean squared analysis is performed to quantify the statistical deviation of a particular random realization of the network. Finally, different sparse control strategies are proposed and their effectiveness is evaluated on synthetic network models and social networks.

Graph signal processing, random graphs, network controllability, sparse control, graph signals, graph process, linear systems on graphs

I Introduction

The controllability of complex networks has shown to play a fundamental role in our understanding of natural and technological systems. Relevant examples involve the control of social, biological, financial, road, communication, and smart grid networks. Several attempts have highlighted the importance of the network topology on the controllability of a system defined on that network [2, 3, 4] and that of considering sparse control signals [5, 6, 7]. Specifically, particular focus is put on designing adequate control signals as well as determining the minimal set of driving nodes [5]. Other approaches consider the spectral features of the transition matrix [6] and characterize the tradeoff between the number of nodes and the energy of the driving signals [7]. The obtained results have shown that sparse inhomogeneous networks are the most challenging to control [5], while clustered networks allow for an easier sparse control [7].

Although the above works provide seminal contributions on the controllability of complex networks, they ignore the nature of the desired state with respect to the underlying topology. Recent evidence from graph signal processing (GSP) [8, 9, 10] has shown that the coupling between the network topology and the signal (state) (a.k.a. the graph signal) can bring substantial benefits in graph signal sampling [11], interpolation [12, 13], adaptive reconstruction [14], and observability of network diffusion processes [15]. A common point that unifies [9, 10, 11, 12, 13, 14, 15] is the so-called graph Fourier transform (GFT). The GFT expresses the graph signal as a linear combination of the graph oscillating modes (obtained as the eigenvectors of the graph description matrix, such as the graph Laplacian).

A particular class of graph signals is that of bandlimited graph signals, i.e., signals that enjoy a sparse representation in the graph Fourier domain. Said differently, these are signals that can be written as a combination of only a few graph oscillating modes. Bandlimited graph signals are commonly observed in practice, such as in networks that exhibit a clustering behavior, i.e., where the signal has similar values within a cluster, but can have arbitrarily different inter-cluster values; or when the graph signal varies smoothly in adjacent vertices, such as temperature measurements in sensor networks.

Sparse network control in the GSP framework is considered in [16] to drive the network state towards a bandlimited graph signal. This is achieved by controlling the input on a fixed set of nodes and then percolating it through the graph by graph filters. The authors first determine a tradeoff between the final control time and the number of control vertices and then provide closed-form conditions that enable full network control, where the latter relate the desired signal bandwidth and the graph filter coefficients.

Differently, [17] studied the practical challenge of driving the network towards a bandlimited state using control inputs of limited energy. Their main result is the established tradeoff between the number of controlled nodes and the control signal energy. However, conditions for full network control in this limited energy setting and relations with the desired signal bandwidth are not provided.

Along these lines, in parallel with the partial version of this paper [1], the GSP framework was also used in [18] to reformulate the specific linear quadratic controller over networks as an autoregressive moving average (ARMA) graph filter [19]. These graph filters allow then for the network control in the graph spectral domain, i.e., each graph oscillating mode is controlled independently and in parallel. However, sparse controllability in the vertex domain is not addressed.

Altogether, the above works concern network controllability on time-invariant topologies. However, in practice, the network structure may change randomly over time due to link losses, or nodes that disappear with a given probability. This might be for instance the case for a communication link that due to channel noise is random, or when in smart grids, a power line/bus goes down due to local failures. In such situations, the network controllability derivations performed in a deterministic setting are not valid anymore and may drive the network to a completely different (undesired) state.

Motivated by the above observations, we study the possibilities to perform open loop control over random time-varying networks. By exploring GSP tools similar to [16, 17, 18], we propose a novel framework that involves the graph randomness into the analysis. More specifically, the main contributions are:

  1. We study the problem of sparse controllability on deterministic networks and characterize the effect that the subset of driving nodes has on the graph frequency content of the controlled signal (Section III). We provide conditions on network controllability that relate the minimum number of driving nodes, the signal bandwidth, and the final control time. This result encompasses the three strategies proposed earlier in [16].

  2. We formulate the problem of sparse network controllability over random time-varying graphs. We develop the concept of controllability in the mean and control the expected system evolution towards a bandlimited state w.r.t. the expected graph (Section IV-A). We then extend the network controllability conditions derived in the deterministic case to the stochastic setting.

  3. We perform a detailed mean squared analysis to quantify the statistical deviation of a particular realization w.r.t. the desired signal (Section IV-B). This analysis illustrates the role played by the graph statistics, the signal bandwidth, and the control signal.

  4. We propose two sparse control strategies to drive the expected state to the desired value with the minimum mean squared error deviation (Section V).

  5. We evaluate the effectiveness of the developed framework and study its performance in different settings through examples from synthetic (Erdős-Rényi and geometric graphs) and real-world social networks (Facebook subnetwork and Zachary’s Karate Club) (Section VI).

To the best of our knowledge, this is the first contribution that approaches the controllability of network dynamics over random time-varying topologies. Although our findings are carried out from a GSP view angle, they can be extended to the general field of network control theory and can be further enriched from the latter bulk of literature.

The remaining part of the paper proceeds as follows: Section II sets down the preliminary concepts and Section III contains our formulation of sparse controllability on deterministic graphs. Section IV formulates the framework of sparse controllability on random graphs. Section V contains the developed control strategies, while Section VI presents the numerical experiments. Finally, Section VII provides the concluding remarks. All proofs are in the appendix.

Notation. Normal case letters (or ) are used for scalars, bold lowercase letters for vectors, and bold uppercase letters for matrices. The th entry of a vector is denoted as while the th entry of a matrix is . Superscripts and denote transpose and Hermitian, respectively. The null vector is while is the vector of all ones. The identity matrix of size is . The diagonal operator denoted as is defined such that with , and is a diagonal matrix with vector on the main diagonal. The expectation operator is denoted as , the trace operator as , the rank of as , the Kronecker product as , and the element-wise Hadamard product as . The vector -norm as well as the corresponding matrix norm are denoted as . The ceiling operator is denoted as and the minimum and maximum operators as and , respectively. If not otherwise stated, calligraphic letters indicate sets and the set cardinality is denoted as .

Ii Diffusion Processes on Graphs

In this work, we consider the task of controlling a diffusion process on random time-varying graphs towards a desired state111To preserve the analogy with the control of linear systems, we will often interchange a desired value with a desired state.. To achieve this, we model diffusion processes in the framework of graph signal processing (GSP). We introduce the basic concepts of GSP in Section II-A, define the random time-varying graph model in Section II-B, and discuss diffusion processes on graphs in Section II-C.

Ii-a Graph signal processing (GSP)

Let denote a graph with the set of vertices, the edge set, and an edge weight function. This graph serves as a mathematical representation of the network, and it is concisely captured by the graph shift operator (GSO) matrix . The th element of , , is nonzero only if or if , so that respects the sparsity of . Standard choices for are the weighted graph adjacency matrix [20, 21], the graph Laplacian matrix [9], or their respective generalizations [22]. In what follows, we consider that is a normal matrix which implies that it has an eigendecomposition , where collects the orthonormal eigenvectors and contains the associated eigenvalues. This holds for every undirected graph (on which the graph Laplacian can be defined) and also for the adjacency matrix of some directed graphs [21, 23].

A graph signal is defined as a mapping from the vertex set to the field of real numbers, i.e., , for . The load charge on a smart grid network is one example of such graph signals [24]. We collect all node signals in the vector with being the th node value [9].

The graph Fourier transform (GFT) is defined as the projection of the graph signal on the eigenbasis and is denoted by [8, 9]. The elements denote the graph Fourier coefficients of , whereas the eigenvectors form the frequency basis. Likewise, the inverse GFT is defined as , i.e., it writes as a linear combination of the frequency basis functions weighted by the frequency coefficients.

A graph signal is said to be bandlimited if it has a sparse support in the graph frequency domain (i.e., it has only a few nonzero frequency coefficients). Without loss of generality, assume that the first elements of are nonzero, so we can write where , . Then, is written in the compact form

(1)

where is the respective column-trimmed eigenvector matrix. The sparse GFT of writes as . This representation has allowed signal reconstruction from few sampled measurements [25, 11, 26, 14, 13] —a property that we will also exploit in Section IV to control the network from few nodes.

Ii-B Random time-varying graphs

We consider the following random graph model over time.

Definition 1 ( graph model [27]).

Given an underlying graph , a random edge sampling (RES) graph realization of consists of the same set of nodes and assumes the edge is sampled at time (i.e., ) with a probability . The edges are sampled independently over both the graph and the temporal dimension and are considered mutually independent from the graph signal if the latter has a stochastic nature.

In other words, Def. 1 states that is a graph realization drawn from the underlying graph , where the instantaneous edge set is generated via an independent Bernoulli process with probability . Let us from now on denote with , , and the adjacency matrix, the degree matrix, and the graph Laplacian matrix of and with , , and the respective matrices of . Furthermore, to ease the exposition let denote the expected graph with respective representation matrices , , and . Under the model it holds that , , and .

We further assume the following.

Assumption 1.

The GSO of the underlying graph has an upper bounded spectral norm as for some .

This assumption is generally met in practice and implies that the graphs of interest have finite dimension and finite edge weights.

We finally remark that more complex models than the can also be found in literature. In particular, for the model in which each edge is sampled independently with a different probability , many of the results presented in this work carry over as well. However, for clarity of exposition, we will focus only on the model.

Ii-C Diffusion on graphs from a GSP perspective

The continuous-time diffusion of a signal on a graph with Laplacian matrix is described by the differential equation [28, 29]

(2)

This equation can be further discretized as [30]

(3)

which is stable as long as satisfies . Alternatively, a diffusion on a graph can be interpreted as the discrete-time shift of through the graph edges [20]

(4)

Models (3) and (4) concern time-invariant topologies. For random time-varying graphs we have that the transition matrix is also time-varying, i.e., . In general, we consider a diffusion process to be valid if its transition matrix satisfies the following assumption.

Assumption 2.

Let be the time-varying transition matrix of a diffusion process over a random time-varying graph . Then, it is assumed that and share the same eigenvectors.

That is, we consider diffusions on random graphs such that the eigenvectors of the expected transition matrix and the underlying GSO coincide. The following lemma shows that this is indeed the case for the diffusion models (3) and (4) on graph realizations.

Lemma 1.

Let be a graph satisfying Assumption 1 and let be a realization of it. For the diffusion models

  1. and [cf. (3)],

  2. and [cf. (4)],

Assumption 2, as well as , and hold.

Other models satisfying these conditions are the wave equation on graphs and graph-based ARMA models, see [15].

Iii Sparse Controllability on Deterministic Graphs

Consider the -state linear system

(5)

where denotes the state value on all nodes at time , is the control signal injected on nodes, and and are the state transition and control input matrix, respectively. System (5) captures the above graph diffusion models and will be used throughout the paper to sparsely control the network dynamics. In this context, we define sparse controllability as follows.

Definition 2 (Sparse network controllability).

An -state system on a graph is sparsely controllable from nodes if for any initial state and some final time there exists a sequence of control signals operating on nodes that drive the network state to any desired value .

Put simply, system (5) is controllable if and only if the controllability matrix

(6)

has full rank [31]. While the full rank of guarantees converging to any desired , we here focus on bringing the network state towards an arbitrary bandlimited graph state222As mentioned in the introduction section and shown by several of the cited works, the family of bandlimited graph signals is large. with . Lead by the promising results of bandlimited graph signal reconstruction by sampling a few nodes [25, 11, 26, 14, 13], we aim to transfer through a fixed, time-invariant, set of nodes of cardinality . For the considered scenario, let denote the selection matrix, where is the binary matrix that selects the nodes in . More formally, belongs to the combinatorial set

(7)

that selects out of different nodes. Observe that and with , such that if and only if node belongs to .

With this in place, we write the linear system on graphs (5) in the GFT domain as

(8)

where holds from Assumption 2. Then, by splitting (8) into its low and high graph frequency bands, we write

(9)

where , and .

Recursion (9) leads to two main observations. First, given a controllable tuple , we can drive to any desired signal through the input signal . Second, the presence of in (9) implies that it is not possible to keep this system evolving within the subspace of bandlimited graph signals, even if we restrict ourselves to bandlimited control signals. Nevertheless, the latter is not an issue, since we can still transfer to a fixed such that and then filter out the spurious frequency content of to obtain the desired signal . In other words, by denoting , we focus only on the dynamics of the Fourier coefficients of interest

(10)

to control the state towards a such that . Then, we use the filter that selects the desired graph frequencies to obtain the controlled bandlimited signal . We note that filter can be implemented locally with a polynomial on the GSO of degree at most [32].

Hereinafter, the design variables are the sampling matrix and the control signals , . Additionally, since the initial state is considered known, we assume without loss of generality that , which is common practice in the control literature [17, 31, 33]. With this set down, we claim our first contribution, which will also be exploited in Section IV for the sparse controllability on random graphs.

Proposition 1.

Consider the linear system (10) describing a process over a deterministic graph . A necessary condition to control the system towards the bandlimited state is that nodes should be selected to inject the control signal.

Proposition 1 provides only a necessary condition on the minimum number of nodes required to control (10) in instants. However, this is not sufficient since the system controllability is affected by the nodes’ position w.r.t. the graph. Still, this condition shows the tradeoff between the cardinality of the sampling set , the signal bandwidth , and the control time . Therefore, for there is the potential to control the network by acting only on one single node. In what follows, we show how this result relates to [16] and [17].

a) Relation with [16]. The result of Proposition 1 encompasses under one single condition the three graph signal reconstruction strategies introduced in [16]. In fact, and covers the multiple node-single time seeding strategy. The case and is a necessary condition for the single node-multiple time seeding strategy to control the graph signal. Finally, covers the more involved multiple node-multiple time seeding approach.

b) Relation with [17]. Differently from this work, [17] focuses on designing the control signal as a tradeoff between sparsity in the vertex domain and the signal energy. Specifically, this problem writes as

(11)
subject to

where the constant trades the energy of the control signal with its sparsity . Even though (11) might sometimes lead to a sparse , it does not lead to a fixed set of control nodes. In this regard, the results of Proposition 1 impose a minimum dimension on such that controllability is possible for a fixed set .

Iv Sparse Controllability on Random Graphs

In this section, we develop our theory for controlling dynamics over random time-varying networks. In Section IV-A we introduce the concept of controllability in the mean and in Section IV-B we perform a mean squared analysis.

Iv-a Mean controllability

The dynamics of a time-varying system on random graphs are given by

(12)

where, under the model in Definition 1), is a set of i.i.d. random matrices with . From Assumption 2, it further holds that . Moreover, the design variables are contained in the second term of (12), which is deterministic, and the state depends on the random system matrices , which are independent from , as well as the deterministic design variables and . Therefore, the mean evolution of (12) writes as

(13)

where . Note that (13) is a deterministic, time-invariant system analogous to (5). We then define the following.

Definition 3 (Sparse network controllability in the mean).

An -state system on random graphs of the form (12) with mean evolution (13) is sparsely controllable in the mean from nodes if for any initial state and some final time , there exists a sequence of control signals that drives the mean network state to any desired value .

Our goal then is to control the mean system to a desired bandlimited graph signal in a finite time from a few nodes, by designing both and the set (through matrix ). The above translates into controlling the mean system (analogous to (10))

(14)

and later applying a (deterministic) linear filter to keep only the desired frequencies such that .

Then, similarly to Proposition 1, we claim the following.

Proposition 2.

Given the linear system (12) describing a process over a sequence of graphs with in-band mean evolution (14). A necessary condition to control system (12) in the mean towards the bandlimited state is that nodes are selected to inject the control signal.

Similar to Proposition 1, the above result establishes a necessary condition for controlling a linear system, now, on random time-varying graphs. As such, the same tradeoff between , , and applies here. The following corollary shows that this result extends to a sufficient condition for particular graphs.

Corollary 1.

Under the conditions of Proposition 2, if the set is built by nodes such that the corresponding rows of are linearly independent vectors, then is a sufficient condition to control system (12) in the mean.

We remark that algorithms for finding a set of linearly independent rows of a matrix are readily available, see [11].

Iv-B Mean squared analysis

Since system (12) is only controlled in the mean, it is paramount to quantify the mean squared error (MSE) of the controlled state to gain statistical insight into how close a specific realization is from the actual desired signal . Towards this end, define as the state transition matrix between time instants . The following theorem then determines the MSE.

Theorem 1.

Let Assumptions 1 and 2 hold and let be a graph process defined over a sequence of graphs described by the linear system (12). Given a set of control nodes characterized by selection matrix and a set of control signals with initial state , the MSE of the particular realization from the actual desired signal is

(15)

which is a quadratic form on with coefficients , and .

We observe that, given and , the MSE (15) holds for any system described by (12), irrespective of controllability. The MSE (15) is a quadratic function in the design variables and the corresponding coefficients , , and depend all on known quantities; specifically: the graph filter , the desired state , and the statistics (first and second order moments) of the underlying support through and . Note that (15) further highlights the impact that the controlled nodes have on the overall performance, and shows its connection with the underlying support and the process bandwidth. More precisely, the coefficient provides the MSE floor if there is no control signal, given by the energy of the desired state; takes into account the similarity between the desired signal and the evolution of the sparsely controlled signal over the mean graph; and accounts for the impact of the variability of the random graph. Finally, we note that the computation of in (15) might be cumbersome for some graph shift operators (or transition matrices ). We thus provide in the appendix two practically useful results that address this issue (first, we provide a general upper bound; second, we show how to exactly compute for undirected graphs following the diffusion models in Lemma 1).

V Sparse Control Strategies

In this section, we propose sparse control strategies (i.e., find and ) for graph processes over random time-varying graphs, where depending on the scenario, one can be preferred over the others. Since we work now in a statistical framework, in Section V-A we propose an unbiased control strategy, while in Section V-B we introduce another control strategy that works in a bias-variance tradeoff mode, minimizing the MSE.

V-a Unbiased controller

The mean state (14) for can be written in the expanded form

(16)

where and . For an unbiased controller, it must hold that at final time . Combining (16) and , we obtain

(17)

with , where we recall that , . For being of full rank (i.e. a controllable system), the linear system (17) has infinite solutions on . Moreover, it often happens that there exists more than one set of nodes that guarantees controllability. We can then select the set of nodes and design the control signals that minimizes the , while guaranteeing that the solution is unbiased. Let be the set of selection matrices that satisfy controllability. The optimal unbiased control strategy can then be posed as

(18)
s. t.

where is given in (15). Oftentimes, we are interested in controlling the system with minimum energy [7, 17]. In such cases, we observe that the minimal energy control signal is [34]

(19)

Then, within the minimal energy control framework, we can select the nodes that minimize the as follows

(20)
s. t.
1:Input: : number of samples, : time horizon
2:: desired frequency response
3:: frequency basis vectors of active frequencies
4:: GFT of transition matrix
5:: function to compute MSE See (15)
6:Output: : selected nodes, : control signals
7:
8:procedure greedy(, , , , , )
9:     Set Selected nodes
10:     Set Remaining nodes
11:     Set
12:     for  do
13:         Set
14:         for  do
15:              Select Choose a remaining node
16:              Compute matrix for
17:              Compute matrix See (17)
18:              if  then
19:                  Solve for
20:                  Compute See (15)
21:                  if  then
22:                       Set
23:                       Set
24:                  end if
25:              end if
26:         end for
27:         Set
28:         Set
29:     end for
30:end procedure
Algorithm 1 Constrained Greedy Approach.

Problem (20) is highly non-convex due to the binary nature of the optimization variable . A heuristic solution to this problem is to follow a constrained greedy approach, as described in Algorithm 1. The objective is to greedily select the nodes that improve the while satisfying the controllability constraint. More specifically, for each new node that we might potentially add to the selected set , we need to check that the rank keeps increasing until we reach controllability as indicated by line 11 [cf. Proposition 2]. Since we are looking for the minimal energy controller, then line 12 entails computing if , and (19) if . While we observe that for such a constrained greedy approach there are no theoretical guarantees [35], in simulations run in Section VI, Algorithm 1 exhibits a very good performance.

V-B Biased controller

In cases where the requirement for an unbiased controller is not strict, we can leverage the bias-variance tradeoff to further reduce the mean squared error of the controlled state, with respect to the desired state. Given a fixed sampling set , the (15) is a quadratic function on the control signals . Therefore, we proceed by expressing as a function of as follows.

First, the derivative of (15) with respect to is

(21)

for with and given in (15). By defining ,

(22)

and setting (21) to zero, we can write

(23)

with and . Note that by construction has rank since . Given then such that , is nonsingular leading to the (parameterized) minimum control signals

(24)
(a) Sweeping time horizon
(b) Sweeping number of selected nodes
Figure 1: Baseline for time-invariant networks on geometric graphs. \subreffixedT Parameter sweeping simulation as a function of time horizon . \subreffixedM Parameter sweeping simulation as a function of number of selected samples . In general, we observe that the biased controller has a similar performance than the min. energy strategy, and they both perform slightly better than the percolation method. The unbiased controller lags behind in terms of MSE. The error bars indicate the estimated standard deviation from the graph realizations.

From the above relation between the control signal and , we consider a two-stage optimization approach [36, Section 4.1.3] to find that minimizes the . This optimization problem writes as

(25)
s. t.

To deal with the non-convexity of (25), similarly to (20), we again rely on a constrained greedy approach analogous to Algorithm 1. More specifically, we replace line 10 by computation of and , condition on line 11 now becomes and line 12 is replaced by (24).

(a) Geometric
(b) Erdős-Rényi
Figure 2: Impact of graph connectivity measured by the average graph degree. \subrefconnectivityGeom Geometric graph: note that when the average degree increases, the connectivity is higher, and as such there are more communication paths through which the signal can flow, and thus is less affected by link losses. \subrefconnectivityER ER graph: the increase in connectivity does not lead to noticeable changes in the MSE since these graphs already have a large average degree (around ). The error bars indicate the estimated standard deviation from the graph realizations.

Vi Numerical Experiments

We evaluate the proposed control strategies on different scenarios to analyze the different tradeoffs between the parameters in controlling the network. We implement the unbiased minimal energy controller (20) and the biased controller (25), and compare them with the control strategies of [16], named Percolation, and [17], named Min. Energy. In the next section, we consider synthetic network models, namely Erdős-Rényi (ER) graphs [37] and geometric graphs, while in Section VI-B we focus on real-world graphs given by Zachary’s Karate Club network [38] and a Facebook subnet [39].

Vi-a Synthetic network models

The ER graph assumes that edges between any two nodes are drawn randomly and independently with probability and has an average degree of . For the geometric model, we first draw nodes uniformly at random in the plane and then compute the Euclidean distance between any pair of nodes, where is the position of node . Subsequently, we assign Gaussian kernel edge weights and finally keep only the nearest neighbors per node. The parameter controls the average degree of the graph. For both models, we only consider realizations that result in connected graphs. To account for the randomness in the generative models as well as in the edge loss, we average the performance over different graphs where for each of them we further accounted for RES realizations.

(a) Geometric
(b) Erdős-Rényi
Figure 3: Impact of the link loss. \subreflinkLossGeom Geometric graph. \subreflinkLossER ER graph. When increases, fewer links are lost and thus the network is easier to control, leading to a lower MSE. The error bars indicate the estimated standard deviation from the graph realizations.
(a) Geometric
(b) Erdős-Rényi
Figure 4: Impact of the control time. \subrefcontrolTimeGeom Geometric graph. \subrefcontrolTimeER ER graph. For the proposed strategies, an increase in control time yields a slightly lower MSE since there is more time to correct for the network evolution. The error bars indicate the estimated standard deviation from the graph realizations.

Unless otherwise specified, we set , and , and the RES link loss probability to . The control time is , and the number of controlled nodes is . The desired state has a bandwidth of with GFT coefficients decaying linearly as for . The controllability performance is measured as the normalized MSE between the bandlimited controlled state and the desired one, i.e., .

Time-invariant network. To set a baseline, we first compare the biased (25) and unbiased (20) controllers with the percolation [16] and min. energy [17] strategies on a fixed time-invariant network. This is the same control scenario that is considered in [16, 17], and is equivalent to setting . Results for geometric graphs can be found in Figure 1, where Figure 0(a) is a parametric simulation as a function of time horizon and Figure 0(b) is a parametric simulation as a function of the number of samples. In general, we observe that the biased estimator has a performance similar to the min. energy strategy, and both are slightly better than the percolation strategy. The unbiased controller lags behind in terms of .

Graph connectivity. In the first random time-varying experiment, we study the impact of the graph connectivity on the controllability performance. We account for the graph connectivity by changing the average degrees, i.e., for the ER graph and in the geometric graph. Figure 2 shows the MSE as the connectivity increases. From Figure 1(a), we observe that the ability to control the geometric network improves with the average degree. This is intuitively satisfying since larger degrees lead to higher connectivity between nodes and thus, render them more robust to the effect of the RES model. Contrarily, for the ER model in Figure 1(b) this behavior is not as much emphasized. We attribute this phenomenon to the large average degree of the ER graphs (above ) and to the relatively high value of . That is, the loss of a few edges does not impact the overall ability to control the network.

Link loss. In the second experiment, we analyze the impact of for a fixed graph average degree. From Figure 3, we note that as increases, i.e., fewer links are lost, the MSE reduces leading to an easier to control network for both models. This is because a higher yields realizations with fewer edge losses, thus more similar to the underlying (mean) graph.

Control time. In the third and last experiment, we analyze the impact of the control time horizon . From Figure 4, we observe that the proposed strategies are not significantly affected by changes in (they only improve slightly).

From this set of experiments, we make three key observations. First, the proposed strategies offer the best performance. Second, the biased controller outperforms the unbiased one. This is an intuitively expected result since it levers the bias-variance tradeoff to minimize the overall MSE, while the unbiased controller has the additional constraint of zero bias and minimum energy (since we are solving problem (20)). By combining these two remarks, we also note that the biased controller strategy achieves the lowest MSE. Third, not accounting for the graph randomness seriously affects the performance, even for . In fact, the competing alternatives of Percolation and Min. Energy that do not account for the graph randomness have a performance that is worse by orders of magnitude than the proposed techniques. This contrast is particularly evident when comparing with the simulations for a time-invariant network in Figure 1. This could be explained by the fact that losing a link has a huge impact in the topology of the graph and severely affects the eigenbasis, thereby changing the subspace of signals that are bandlimited on a given graph.

(a) Zachary’s Karate Club
(b) Facebook
Figure 5: Impact of the bandwidth and the control signal shape. \subrefbandwidthZachary Zachary’s Karate Club social network. \subrefbandwidthFB Subnet of Facebook social network. An increased bandwidth () leads to a higher MSE, since these signals are harder to control for a fixed and . Higher graph frequency content signals are also harder to control.
(a) Zachary’s Karate Club
(b) Facebook
Figure 6: Impact of the number of selected nodes and sampling strategy. More control nodes lead to a smaller MSE satisfying our intuition since the degrees of freedom increase. The proposed greedy heuristic is close to the optimal combinatorial solution and represents a substantial improvement compared the random node selection.

Vi-B Real world graphs

We here consider two social networks, namely Zachary’s Karate Club [38] composed of nodes and a Facebook subnetwork [39] having nodes. The situation is akin to the spread of a rumor, and thus the control actions aim at curbing this information spread towards the desired state from a few selected persons.

We set the control time horizon to , the number of selected nodes to , and . For simplicity of presentation, we focus here only on the biased controller strategy which has consistently yielded the best performance. We again averaged the performance over different realizations.

Bandwidth and spectrum of the desired state. In this experiment, we analyze the impact that the desired state bandwidth and its GFT have on the controllability performance. We consider four different GFTs for the desired state, namely: (i) a step low-pass for , (ii) a step high-pass, where the active frequencies correspond to the eigenvectors with highest total variation, (iii) a linear decay response given by for , and (iv) an exponential decay response with for . We normalize all desired states to have unit energy for fair comparison and analyze different values of , which are set to be proportional to (a fraction between and ). The results are depicted in Figure 5. First, we observe that controlling the system to a state with a higher bandwidth is harder since the set of graph frequencies that we need to guarantee controllability increases. Second, we observe that the high-pass response is harder to achieve and that responses that decay to zero (like the linear decay and the exponential decay) yield lower MSE. This is because high-pass responses are translated in the vertex domain as states that have dissimilar values in adjacent nodes, therefore, rendering the control of network dynamics to such a state more challenging.

Sampling heuristics. In the last experiment, we focus on the impact of the control nodes. In particular, we compare the proposed constrained greedy heuristic for choosing the nodes (Algorithm 1) with the optimal combinatorial solution and a uniformly random sampling scheme. We fix and consider the linear decay desired state . The obtained results are shown in Figure 6. First, we observe that the MSE decreases as more control nodes are selected. Second, we note that the greedy heuristic yields a performance similar to the optimal solution and represents a considerable improvement over the random selection.

Vii Conclusions

In this paper, we studied the problem of sparse controllability of graph signals. We considered a random time-varying network to be controlled to a desired graph-bandlimited state. To address the randomness of the underlying support, we introduced the concept of controllability in the mean, where we postulate to control the system as if it were running on the expected graph. We then carried out a detailed mean squared analysis to quantify the deviation from the desired signal, when the control is designed for the expected graph but ran on any given realization of the random network. We used this analysis to propose two different control design strategies and evaluated their performance on both synthetic graph models and real-world social networks. We concluded that it is of paramount importance to take into account the random nature of the underlying topology. We leave as future work the analysis of more complex random network models, as well as other control strategies involving spectral or energetic constraints. Another direction worth investigating is the proposal of other heuristic solutions to the respective optimization problems.

Appendix A Proof of Lemma 1.

Proof.

For model , and the system transition matrix is for . First, we prove that . Note that for every and, therefore, from the Laplacian interlacing property [40] this condition always hold. The proof of Assumption 2 is straightforward, i.e., from , which means that and share the same eigenvectors. For the last condition, note that since . Therefore, is upper bounded by some finite .

For model , and the system transition matrix is . To prove that , recall that for connected graphs, the largest eigenvalue is positive and real [41]. Then, since is considered to be normal and Assumption 1 holds, . Likewise, since , then and therefore for all . The proofs of the last two conditions are straightforward since and . This completes the proof. ∎

Appendix B Proof of Proposition 1 and Corollary 1

Proof of Proposition 1.

Recall that is the set of the selected nodes and that , where with if and , otherwise. System (10) is equivalent to

(26)

where denotes the zero-extended control signal such that if and , otherwise. Then, system (26) is controllable iff the matrix

(27)

is full rank. Observe that

(28)

holds from . Therefore, to ensure the full rank of , must hold, for some . This concludes the proof. ∎

Proof of Corollary 1.

Recall that, for two matrices and [34]

(29)

The mean system (14) is controllable, iff

(30)

has rank with .

The first term in (30) has rank

(31)

while the second term has rank