Network observability and localization of the source of diffusion based on a subset of nodes

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1 Network observability and localization of the source of diffusion based on a subset of nodes Sabina Zejnilović, João Gomes, Bruno Sinopoli Carnegie Mellon University, Department of Electrical and Computer Engineering, Pittsburgh, PA sabinaz@cmu.edu, brunos@ece.cmu.edu Institute for Systems and Robotics Instituto Superior Técnico (ISR/IST), Technical University of Lisbon (UTL), Portugal jpg@isr.ist.utl.pt Abstract Identifying the patient-zero of an epidemic outbreak, locating the person who started a rumor in a social network, finding the computer that initiated the spreading of a computer virus in a network- these are all applications of localizing the source of diffusion in a network. Since most of the networks of interest are very large, we are usually able to observe only a part of the network. In this paper, we first present a model for the dynamics of network diffusion similar to state update of a linear time-varying system. Based on this model, we provide a sufficient condition for observability of the network, i.e., we establish when is the partial information available to us sufficient to uniquely localize the source. Also, we connect the problem of finding the minimum number of observable nodes to the problem of metric dimension of the graph. We then present different methods to perform source localization depending on network observability. I. INTRODUCTION In today s world, we are a part of many different networks in which diffusion of different phenomena takes place. Infectious diseases are spread over contact networks, information and trends are propagated over social networks, and viruses are disseminated over computer networks. Whether it is for the purpose of identifying the culprit by the authorities, for controlling and preventing further infection, or identifying trendsetters, the task of localizing the source of diffusion is an important one. Recently, there has been a surge of research addressing this challenge for different diffusion scenarios. In [1], the goal is to identify the source of a rumor knowing which nodes have been infected by certain time. A source estimator that depends on a metric denoted as rumor centrality is proposed and it corresponds to the maximum likelihood estimator for regular trees. In [2], the observations not only include the state of the nodes (susceptible/infected), but also the times of infection. However, only a subset of nodes can be observed, as in most real world networks it is unfeasible to have access to all the nodes. An optimal estimator for tree networks, and suboptimal for general networks, is presented. Several strategies for the best choice of observable nodes are experimentally compared. An algorithm for identifying multiple sources of epidemics is Support for this research was provided by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the Carnegie Mellon Portugal Program. proposed in [3], and it is based on the Minimum Description Length principle. The single best source is calculated using the smallest eigenvector of the Laplacian sub-matrix. Once the best source is found, a new source is calculated by removing the previously chosen source and solving again for smaller infected graph. Multiple sources are also localized in [4], by identifying which nodes reduce the most the largest eigenvalue of the adjacency matrix after its removal. With experimental results, it is shown that the proposed technique is able to identify the source nodes if the graph approximates a tree sufficiently well. A greedy algorithm is presented in [5] to identify the sources of rumor involving the observations of preselected number of nodes. Several strategies for choosing the most informative nodes to observe, based on different centrality measures, are experimentally evaluated. In this paper we also assume a propagation model where once infected/informed, the nodes remain in that state as in [1] [5]. This assumption corresponds to trend and rumor propagation, or spreading of certain diseases and viruses where recovery happens on a larger time scale than infection spreading. We address the problem of localizing the source of diffusion based only on the observations of a part of the network, as having access to the states of all the nodes in large scale networks is not a practical assumption. The choice of nodes that are observed influences the performance of source estimators, as shown in [2], [5], where different selection strategies are proposed and evaluated through simulation. We present a new model for dynamics of network diffusion which allows us not only to perform source localization, but also to analyze which choice of observers can lead to correct source identification. Diffusion through the network is modeled similarly to state evolution in linear time-varying systems. The ability to correctly identify the source of diffusion, based only on partial knowledge of the network, corresponds to the concept of system observability. Equivalently to the analysis of system observability, where system noise is not considered, we also only consider the model of deterministic, noiseless propagation. This contrasts with [1], [2], where propagation times between nodes are modeled as random variables. We derive a sufficient condition which, when satisfied, guarantees that the choice of observers is such that a source will be localized,

2 regardless of its placement within a network. Since the resources allocated for the observation of the nodes are usually limited, we are interested in the problem of finding the minimum number of nodes that will allow us to localize the source. This corresponds to finding the minimum number of communities to monitor for the disease outbreak or which individuals to choose to observe their behavior on the social network. Under our propagation assumptions, this problem actually represents a known problem of finding a metric dimension of a graph [6] or determining minimum cardinality of a resolving set [7]. For an arbitrary graph, this problem is NP-hard [6], but explicit results exist for some families of graphs [8]. Here we review some of the results in the context of the minimum number of needed observers and illustrate the performance of the available approximation algorithm. We also address the problem of identifying the source suspects, when the sufficient observability condition is not satisfied. Our formulation of source localization can be viewed as l 0 norm optimization problem and using the results from compressive sensing [9], we resort to l 1 norm relaxation. However, with this relaxation the original optimal solution is still found, due to the structure of our problem. Additionally, when the observations are such that multiple nodes are equally likely suspects for the source, the solution of the relaxed optimization problem correctly identifies all of them. The proposed model for network diffusion is presented in the next section, while network observability and minimum number of observable nodes is discussed in Section III. Source localization approaches for different cases of network observability are described in Section IV and we conclude in Section V. II. MODELING DIFFUSION IN A NETWORK A network of N nodes is represented using a graph G = {V, E}, where V = {1,2,...,N} is the set of vertices representing the nodes and E V V is the set of edges. The nodes in a network can correspond, for example, to people in a social network, or computers in a communication network. There is an edge between nodes i and j, (i,j) E, if nodes i and j can communicate directly. If (i, j) E implies that (j, i) E, then the associated graph is called undirected. We will assume G to be undirected, as infections and rumors spread through contact and ties which are typically bidirectional. Additionally, we will consider a connected graph, meaning that there is a path between any two vertices in a network, otherwise some parts of the network would be isolated and would not be relevant to the diffusion process. We consider the network topology to be known. The distance between two vertices in a graph represents the number of edges in the shortest path connecting them. A walk represents a sequence of vertices (possibly repeated) where each vertex is adjacent to the preceding vertex in the sequence. The length of a walk is the number of edges that it uses. The adjacency matrix A of the graph G is a N N symmetric matrix, with elements a ij = 1 if (i,j) E and a ij = 0 otherwise. The propagation model we will analyze is referred to in the literature as the Susceptible-Infected (SI) model, where once a node is infected or informed it remains in that state. Initially, only one infected node is present in the network, and it is denoted as the source node s. It becomes active at time 0. The source node corresponds, for example, to the patient-zero in an epidemic outbreak or a trendsetter in a social network. We assume deterministic propagation, meaning that once a node is infected at t 1, in the following time instant t, where t is a discrete time index, it will infect all of its neighbors, with probability 1. The time it takes for a certain node to become infected is equal to its distance from the source node s. The nodes whose state can be observed, and whose infection times are known, are denoted as observer nodeso 1,o 2,...,o K. Typically due to limited resources, the number of observer nodesk is much smaller than the total number of nodesn, as not all the individuals report hearing a rumor, nor the infection times of all patients are available. The state of node i at time t is denoted as a binary variable x i (t) and is equal to 1 only if node i has been infected by time t. The vector x(t) R N describes the state of all the nodes at time t. The initial state x(0) is equal to e s, which is a column vector with all entries equal to 0 except for the s-th entry, which is equal to 1 and corresponds to the index of the source node s. The infection time of node i is denoted as t i and it represents the time of change, such that x i (t i 1) = 0 and x i (t i ) = 1. With y(t) R K we denote the states of K observed nodes at time t. The states of observed nodes are obtained from the full state vector through multiplication by K N matrix C = [e o1,e o2,...,e ok ] T, where M T denotes the transpose of matrix M. Below, we denote by M, a binary matrix that has the same sparsity structure as matrix M, but with all nonzero elements replaced with 1, as follows M ij = { 0, if Mij = 0 1, if M ij 0 }. Finally, we can state a theorem that characterizes the evolution of network diffusion under the above stated assumptions. Theorem 1: The dynamics of diffusion in a network, under the deterministic SI propagation model, can be characterized as follows x(t) = Φ(t,0) x(0) y(t) = Cx(t), (1) where Φ(t,0) = A t +A t 1. Proof: The state equation of (1) for node i can be rewritten as x i (t) = j Φ ij (t,0)x j (0) = Φ is (t,0). The last equality holds since x(0) has a single nonzero entry for the source node. Now we refer to the specific properties of the powers of adjacency matrices [10], where the ij-th entry of A t equals to the number of walks of lengthtbetween nodes i and j. Based on this property, we have that if the distance of node i to the source is t i, then Φ is (t,0) = 0 for all t < t i,

3 which consequently gives x i (t) = 0 for all t < t i. At t = t i, both Φ is (t,0) and x i (t) assume the value 1. If there exists a walk of length t i, then there also exists at least one walk of lengtht i +2l, forl = 0,1,..., as any edge included in the walk can be repeated (once in the forward and once in the backward direction) to add a cycle to the walk, increasing its length. Hence A ti+2l > 0, and subsequently Φ is (t i +2l,0) = 1. At times t = t i +(2l+1), Φ is (t,0) is equal to 1, at least due to the term A t 1. Therefore, for all t t i, Φ is (t,0) = 1 and the state of the node is 1, reflecting the fact the node i became infected at time t i. The second equation of (1) models that at each time t, only the state of the observer nodes can be seen. Thus, the equations (1) model the state evolution and available observations for the diffusion process in a general network. III. NETWORK OBSERVABILITY In the previous section, we have presented a model for the dynamics of network diffusion, similar to the space state representation of a linear time-varying system with a constant observation matrix. Our goal is the identification of the source node, based on the infection times only of observer nodes. We now tackle the question of when the choice of observer nodes guarantees correct source identification, and we treat this as a network observability problem. Stacking equations (1) for times 0,...,N 1, we get the following matrix equation or equivalently y(0) y(1). y(n 1) = C CΦ(1,0). CΦ(N 1,0) x(0), Y N 1 = O N 1 x(0). (2) We refer to NK N matrix O N 1 as a network observability matrix. The following theorem states the necessary and sufficient conditions for correct identification of the source based on the infection times of observer nodes. Theorem 2: If the rank of the observability matrix O N 1 is equal to N, then the infection times of the particular choice of observers are sufficient to correctly identify the source, regardless of its position in the network. Then, the initial state can be recovered as x(0) = ( O T N 1 O N 1) 1O T N 1 Y N 1. (3) The necessary condition for correct source identification, for any possible source, is that the observability matrix O N 1 has N unique columns. Proof: In a network of N nodes, the largest distance between any two nodes is at most N 1, meaning that the states of all the observer nodes will be 1, at most by time N 1 and will remain the same for all t N 1. Product CΦ(t,0) has an interesting structure; its ij-th entry is equal to 1, only if there is a path of length smaller or equal to t between observer o i and node j, otherwise it is 0. Hence, all entries of CΦ(t,0) are equal to 1, for t N 1. Therefore, Node 4 Node 5 Node 6 Fig. 1. Node 2 Node 3 Node 1 Example network stacking CΦ(t,0), for t > N 1 will not increase the rank of the observability matrix. This is parallel to the property of linear time-varying systems where, where if the initial state can be recovered, then it can be recovered from observations y(0),...,y(n 1). From (2), if the observability matrix has full column rank, we obtain (3), as a standard result from linear algebra. In order to uniquely identify the source node based on the distances of observer nodes to it, it is necessary that the observer nodes have different distances to all the remaining nodes. If the observability matrix has N unique columns, looking at the structure of the product CΦ(t, 0), this is equivalent to the condition that there are no two nodes with the same distances to the observers, which is exactly needed for correct localization. If the observability matrix O N 1 of a network, with adjacency matrix A and a choice of observers characterized by C, has N unique columns, then we refer to this network as an observable network. Verifying the observability of a network using Theorem 2 does not require knowledge of infection times of the observers. Thus, it is a task that can be efficiently performed offline, before the actual source localization takes place. This would allow timely selection of observer nodes that would guarantee correct source localization, irrespective of the source position in the network. The observability of a network does not depend on the source node, meaning that in such a network regardless of which node is the source, the information from the observers is sufficient to identify it. However, this condition is not necessary for a particular choice of source node, as illustrated by a following example. Example 1 A simple tree network of N = 6 nodes is shown on Figure 1. Assuming that node 3 is the only observer, the observability matrix has only 4 unique columns and rank 4, and therefore the network is not observable with this choice of observers. An example of the observers inability to identify the source would be the case if the infection time of node3was t 3 = 3. This information would be insufficient to distinguish whether the source was either node 4 or 5. However, if t 3 = 2, then we would be able to correctly identify the source as node 2. This illustrates that a network might be generally unobservable given a particular choice of observers, but this

4 does not imply that the information provided by the observers is insufficient to identify the source node in all the cases. In the following section, we present a method to recover the initial state in these special cases, when either the sufficient or necessary condition does not hold, but we are still able to perform correct localization, as well as for the case when there are multiple source suspects and we are interested in identifying all of them. The necessary condition for network observability additionally provides insight into the problem of selecting how many and which observer nodes are needed to achieve network observability, as the next subsection shows. A. Minimum number and location of observers needed for network observability The necessary condition for correct source localization implicitly states that the choice of observers is such that all the nodes in a network have different distances to them. Let us denote with S the set of observer nodes{o 1,o 2,...,o K }, with d(i,o k ) the distance between nodes i and o k and with d(i,s) the k-vector of distances from node i to the set of observer nodes [d(i,o 1 ),...,d(i,o k )]. Then having the set of observer nodes that will satisfy the necessary condition for the correct node localization can be stated as d(i,s) d(j,s) for all i,j pairs of nodes. Stated as such, finding the set of observers with this property corresponds to the problem of finding a resolving set of nodes S of the graph [7]. Determining the resolving set of minimum cardinality is a well-known problem in graph theory called finding the metric basis of the graph and the cardinality of this basis is called the metric dimension [6]. The motivation for this problem came from the placement of detecting devices in a network, such that every vertex can be described in terms of distances to them, and also independently from describing the structure of chemical compounds in pharmaceutical chemistry [11]. Although for an arbitrary graph, computing its metric dimension is NP-hard problem [6], for some families of graphs exact values can be easily determined [8]. Applying these results, we have for example that in path networks the minimal number of observers is one, if the observer is the end (leaf) node, while n 1 observers are needed in complete networks. Explicit results, among others, also exist for tree networks, d-dimensional grids [6] and random networks [12]. For general networks, O(log n) factor approximation algorithm can be used to approximate the metric dimension of a graph in polynomial time [6]. The problem of choosing the minimum number of observers can be cast as the set cover problem where the elements correspond to all pairs of nodes. The approximation algorithm selects one by one the node that distinguishes the highest number of node pairs and the algorithms s performance is illustrated with the following example. Example 2 The Erdős-Rényi is a random graph model, where each pair of nodes is connected with equal probability, independently of other pairs. We generated 50 Erdős-Rényi graphs with 20 nodes, each with different probability of an Minimum number of observers optimal approximation Edge probability Fig. 2. Performance of the approximation algorithm for choosing the minimum number of observers. edge. For each graph, we found the minimal number of observers, by checking all the possible combinations. We compared this to the performance of the approximation algorithm and the results are shown on Figure 2. The example illustrates that as the edge probability increases and the graph becomes more dense, more observers are needed to correctly distinguish between the nodes. For very sparse graphs, the number of observers is small, but as the graph more resembles a complete one, the number of observers tends to n 1. The approximation algorithm chooses at most one observer more than the minimum needed, and in around 70% cases its performance coincides with the optimal. IV. SOURCE LOCALIZATION In the previous section a sufficient condition for network observability was given, as well as the method for recovering the initial state of the network, when the condition is satisfied. As this condition is only sufficient, there are cases where source could be correctly identified, even if the observability matrix is not invertible. This may correspond to the case when the network is actually observable, or as in shown in the Example 1, when the network is unobservable, but it is still possible to identify certain sources. We now present a method for source localization in these cases. Given that the initial state is a binary vector with only one nonzero entry, corresponding to the index of the source node, we can cast source localization as a l 0 norm optimization problem as follows min x(0) x(0) 0 subject to O N 1 x(0) = Y N 1. (4) The optimization problem (4) seeks to find the sparsest vector x(0), i.e., with the fewest nonzero entries, that satisfies the observation model (2), which gives us exactly the desired initial state. However, problem (4) is non-convex and hard to solve. Typically, l 0 optimization problems are relaxed to l 1, which are easier to handle through the use of linear programming [9]. The l 1 relaxed version of problem (4) can

5 be stated as min x(0) x(0) 1 subject to O N 1 x(0) = Y N 1. (5) Generally, the solutions of the original l 0 and relaxed l 1 optimization problems differ. However, given the structure of our problem, a solution of the relaxed problem coincides with the optimal solution of the original problem. Given the described structure of the observability matrix, the constraint O N 1 x(0) = Y N 1 contains N constraints of the form CΦ(t,0)x(0) = y(t). Each row i of these constraints, after simplifying, corresponds to the equation j N l o i x j (0) = I s N l oi, (6) where I is the indicator function and No l i is the l-hop neighborhood of the observer i, for l = 0,...,N 1. Hence for each observer,i = 1,...,k, for l = d(s,o i ) neighborhood that includes the source node, the equation (6) takes the form x j (0) = 1. (7) j N d(s,o i ) o i For all the other neighborhoods l d(s,o i ), equations (6) are equal to 0. Under the above assumptions that the source node can be resolved by the observers, the source node is the only node that is present only in the equations (7). If there is some other node r also at the distance of d(s,o i ) from the observer o i, then there exist at least one other observer o k from which node r is at the different distance than d(s,o k ), otherwise source node and node r could not be distinguished. This means that such node also appears in the equations that are equal to 0. Since the problem (5) minimizes the l 1 norm of x(0), the states of all the nodes, except the source node are set to 0, while the state of the source node is set to 1. Should the state of any other node be different than 0, for example node r, since that node also appears in the equation that is equal to zero, then there should also be another node whose state is nonzero, to satisfy the constraint. This in turn would increase the l 1 norm of the vectorx(0). Therefore, in this case, l 1 minimization yields a solution with the cardinality exactly 1, the sparsest solution possible and a solution to the original l 0 minimization. This allows us to recover the initial state, i.e. identify the source node correctly. In other scenarios, given the choice of observers that make the network unobservable, even if we cannot uniquely identify the source, narrowing down the list of suspect source nodes still can be very useful. This is a likely scenario when there are not enough resources to allocate for the required number of observers to attain observability, and yet we would like to obtain as much information as possible with the existing resources. Hence, we would like to recover all the possible x(0) vectors, with a single nonzero entry, that satisfy the observation model (2). We denote with x i (0) for i = 1,...,p all the possible p solutions of the original problem (4). Again, instead of solving the combinatorial problem (4), and searching for multiple solutions, we resort to solving the relaxed l 1 optimization problem (5). The constraint in the problem still consists of equations of the form (6). However, now there are p nodes that appear only in the equations (7). One of these is the source node itself, but it cannot be distinguished from all the other suspect nodes based on the infection times of the available observers. Again, l 1 minimization sets the state of all the non-suspect nodes to 0, for the same reason as before. Setting the state of one of the suspect node to 1 and all the others to 0 represent each of the p possible solutions with the same value of the cost( function. The combination of these solution, of the form 1 p x 1 (0)+x 2 (0)+...+x p (0) ) is also a solution, from which we can easily recover individual solutions. Hence, l 1 minimization allows us to correctly obtain the list of all the possible suspect source nodes. Note Let us denote with D R k N the distance matrix, whose elements D ij represent the distance between observer o i and node j. Let t R k be the vector of infection times of observers. Then the source localization problem can be stated as finding the column s of the matrix D which is equal to the vector t. In case there are multiple source suspects, then there are multiple columns of D equal to t. Although this is a much simpler way to treat source identification problem compared to l 1 optimization, here we present the first approach. Together with the diffusion model (1), it provides a way to deal with the source localization problem in the case of more realistic assumptions: when the observations are noisy, the activation time of the source is not known and when there is uncertainty in the network topology, which will be our future work. V. CONCLUSION We presented a new model for dynamics of network diffusion, in order to identify the source of diffusion when the infection times are available from only a subset of nodes. We introduced the concept of network observability which reflects when the choice of observable nodes is such that correct source localization is possible. Based on the presented model, we gave necessary and sufficient condition for network observability. We provided a method for source localization when the sufficient condition holds. Also, we showed that under our assumptions, the problem of selecting the minimum number of nodes that makes a network observable is equivalent to the problem of finding the metric dimension of a graph and we reviewed some of the available results in this area. Finally, for the case when the sufficient condition does not hold, we formulated source localization problem as l 1 minimization problem. 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