Rumor Source Detection in Unicyclic Graphs Pei-Duo Yu † , Chee Wei Tan † and Hung-Lin Fu * City University of Hong Kong † , National Chiao Tung University ∗ {peiduoyu,cheewtan}@cityu.edu.hk, hlfu@math.nctu.edu.tw Abstract—Detecting information source in viral spreading has important applications such as to root out the culprit of a rumor spreading in online social networks. In particular, given a snapshot observation of the network topology of nodes having the rumor, how to accurately identify the initial source of the spreading? In the seminal work by Shah and Zaman in 2011, this problem was formulated as a maximum likelihood estimation problem and solved using a rumour centrality approach for graphs that are cycle-free. This however is optimal only for degree-regular trees, and even the special case of a single cycle is an open problem. In this paper, we address the maximum likelihood estimation problem by an generalized rumor central- ity for spreading in graphs with cycles. We derive analytical characterization of the optimal solution and a polynomial-time algorithm to solve the problem. I. I NTRODUCTION Networks represent a fundamental medium for the spreading and diffusion of information. Network nodes are said to be infected when they possess this information, and network topologies govern how the spreading processes increase the susceptibility of other nodes to be infected leading to the successive spread of information from a few initial nodes to a much larger set. An example of such a viral spreading phenomenon is rumor spreading in online social networks. From a cybersecurity enforcement viewpoint, this begs the question of detecting and rooting out malicious information sources in a reliable and efficient manner [1]–[5]. In particular, given a snapshot observation of the infected nodes, who is the culprit source of the rumor spreading? In a recent seminal work in [6], Shah and Zaman formulated this as a maximum likelihood estimation problem, and pro- posed rumor centrality, a form of network centrality, to solve this problem for degree-regular tree graphs assuming that the underlying graph has no cycle. This means that the resultant observation graph is cycle free. The infected node with the most number of ways to spread to other nodes is the rumor center that coincides with the maximum likelihood estimate. This rumor centrality approach was subsequently extended to various problem settings, e.g., extension in [4], [5], [7] to random trees, extension in [5], [8] to constrained observations, extension in [9] to multiple source detection, extension in [10] to detection with multiple snapshot observations, and extension in [11] to prove that when the number of infected vertices is large enough, the probability of the source vertex not in a confidence set is less than a given error ǫ. In [12], [13], the authors established its equivalence to the graph theoretic centroid. There is however a key limitation in the rumor centrality ap- proach. A main modeling assumption in all the aforementioned work is that the underlying graph (i.e., number of susceptible nodes) is cycle-free. This is never true in general of practical real-world networks where inter-connections are diverse and the presence of cycles cannot be ignored, and makes this constrained maximum likelihood estimation problem a much harder combinatorial problem. In essence, the cycle effects allow the dynamical spreading process to spread thruogh mul- tiple alternate path thus increasing the likelihood that nodes near the cycle to be infected. Hence, the number of cycles and their location can significantly shape spreading and therefore the estimation performance. To be exact, existing algorithms in the literature, e.g., [6]–[10], are no longer optimal even with the presence of a single cycle in degree-regular pseudo-tree graph. Rumor source detection over graphs with cycles is clearly more challenging, but is more realistic and also significantly generalizes all previous work [6]–[10] that mostly use the breadth-first search heuristic for graphs with cycles. Thus, another key issue is the design of optimal inference algorithms that work for a variety of network topologies. Finding the max- imum likelihood estimate as opposed to a suboptimal heuristic to detect the source is important. A focus in this paper is thus to extend the rumor centrality and propose optimal algorithm design for pseudo-tree graph, leveraging the theorem derived in this paper, that have practical computational complexity. A. Our Contributions The main contributions are summarized as follows: • This paper considers a more general underlying network, i.e., network with cycles. We analyze the graph with a single cycle, and characterize the optimal solution of the maximum likelihood estimation problem. • For a degree-regular graph with cycles say G n , we prove that under certain condition, the rumor center of G n is equal to the rumor center any spanning tree of G n . • We propose a polynomial-time algorithm to find the rumor center on the graph with a single cycle. II. PRELIMINARIES OF RUMOR CENTRALITY We model an online social network of nodes by an undi- rected graph G = (V,E), where the set of vertices V represents the nodes in the underlying network, and the set of edges E represents the links between the nodes. Following [6], we use the Susceptible-Infectious (SI) model in [14] to