Rooting out the Rumor Culprit from Suspects (1301.6312v4)

Abstract: Suppose that a rumor originating from a single source among a set of suspects spreads in a network, how to root out this rumor source? With the a priori knowledge of suspect nodes and an observation of infected nodes, we construct a maximum a posteriori (MAP) estimator to identify the rumor source using the susceptible-infected (SI) model. The a priori suspect set and its associated connectivity bring about new ingredients to the problem, and thus we propose to use local rumor center, a generalized concept based on rumor centrality, to identify the source from suspects. For regular tree-type networks of node degree {\delta}, we characterize Pc(n), the correct detection probability of the estimator upon observing n infected nodes, in both the finite and asymptotic regimes. First, when every infected node is a suspect, Pc(n) asymptotically grows from 0.25 to 0.307 with {\delta} from 3 to infinity, a result first established in Shah and Zaman (2011, 2012) via a different approach; and it monotonically decreases with n and increases with {\delta}. Second, when the suspects form a connected subgraph of the network, Pc(n) asymptotically significantly exceeds the a priori probability if {\delta}>2, and reliable detection is achieved as {\delta} becomes large; furthermore, it monotonically decreases with n and increases with {\delta}. Third, when there are only two suspects, Pc(n) is asymptotically at least 0.75 if {\delta}>2; and it increases with the distance between the two suspects. Fourth, when there are multiple suspects, among all possible connection patterns, that they form a connected subgraph of the network achieves the smallest detection probability. Our analysis leverages ideas from the Polya's urn model in probability theory and sheds insight into the behavior of the rumor spreading process not only in the asymptotic regime but also for the general finite-n regime.

• The paper develops a Maximum a Posteriori estimator to identify rumor sources in SI model networks with suspect nodes, utilizing the concept of the local rumor center.
• Key findings show detection probability is significantly influenced by network degree, the number of observed infected nodes, and the configuration of the suspect set, particularly whether suspects are connected or clustered.
• The research has practical implications for improving source detection strategies in network security, epidemic tracking, and information spread analysis by considering network topology and suspect node characteristics.

Analysis of Rumor Source Detection in Networks

This paper by Wenxiang Dong, Wenyi Zhang, and Chee Wei Tan provides a thorough investigation into the problem of accurately identifying the source of a rumor within networked structures, specifically under the assumptions of a susceptible-infected (SI) model. The authors address situations where the rumor originates from a known subset of suspect nodes, using the Maximum a Posteriori (MAP) estimator for source detection.

Model and Methodology

The paper revolves around a network modeled as a regular tree with node degree $\delta$. The SI model prescribes that once infected, nodes remain permanently infected and continue spreading the rumor. A MAP estimator is developed based on observing infected nodes at a snapshot in time, leveraging prior knowledge about suspect nodes. The notion of the local rumor center, an advanced form of rumor centrality, is utilized to maximum effect in this estimation process.

Key Findings

1. Detection Probability for Full Suspect Set:
   • When all nodes in the network are suspects ($\delta \geq 3$), the detection probability $\mathbf{P_c}(n)$ approaches a constant value in the asymptotic limit; specifically, the probability grows from 0.25 to 0.307 as $\delta$ increases, which aligns with prior findings.
   • The detection probability decreases with an increasing number of observed infected nodes ($n$) and increases with the node degree ($\delta$).
2. Detection with Connected Suspects:
   • When the suspect nodes form a connected subgraph, the MAP estimator shows enhanced reliability, particularly as node degree becomes large ($\delta \geq 3$). $\mathbf{P_c}(n)$ exceeds 0.5, defying prior probability assumptions.
   • Reliable detection converging to probability 1 is achievable with a sufficiently large node degree.
3. Two Suspect Nodes:
   • With just two suspect nodes, the distance between them (measured in hops) critically affects detection probability. With $\delta \geq 3$, $\mathbf{P_c}(n) \geq 0.75$, and grows as the separation increases.
4. Scenario with Multiple Suspects:
   • When suspects form a general subgraph, the configuration where suspects are clustered yields the lowest detection probability, highlighting the complexity of network structure on estimator performance.

Theoretical and Practical Implications

This research influences both theoretical models of epidemic-like spread in networks and practical applications in social media, epidemic control, and cybersecurity. By casting the problem in the probabilistic framework of the Pólya's urn model, the paper bridges epidemiological models with advanced network theory. Practically, the findings suggest enhanced detection protocols in network security measures and strategic intervention in epidemic control based on suspect node prioritization and network topology configuration.

Future Directions

Further research could explore adapting these models to dynamically changing networks, which are more indicative of real-world social structures. Additionally, extending this theoretical framework to include nodes with varying transmission rates or mixed network topologies could significantly enrich practical detection strategies.

This paper is a methodological advancement in identifying influencers or sources in network structures, paving the way for more robust network analysis and control strategies in fields ranging from public health to information technology.