Estimating Infection Sources in Networks Using Partial Timestamps

We study the problem of identifying infection sources in a network based on the network topology, and a subset of infection timestamps. In the case of a single infection source in a tree network, we derive the maximum likelihood estimator of the source and the unknown diffusion parameters. We then introduce a new heuristic involving an optimization over a parametrized family of Gromov matrices to develop a single source estimation algorithm for general graphs. Compared with the breadth-first search tree heuristic commonly adopted in the literature, simulations demonstrate that our approach achieves better estimation accuracy than several other benchmark algorithms, even though these require more information like the diffusion parameters. We next develop a multiple sources estimation algorithm for general graphs, which first partitions the graph into source candidate clusters, and then applies our single source estimation algorithm to each cluster. We show that if the graph is a tree, then each source candidate cluster contains at least one source. Simulations using synthetic and real networks, and experiments using real-world data suggest that our proposed algorithms are able to estimate the true infection source(s) to within a small number of hops with a small portion of the infection timestamps being observed.Comment: 15 pages, 15 figures, accepted by IEEE Transactions on Information Forensics and Securit

A General Framework for Sensor Placement in Source Localization

When an epidemic spreads in a given network of individuals or communities, can we detect its source using only the information provided by a small set of nodes? We propose a general framework that incorporates two dimensions. First, we can either rely exclusively on a set of selected nodes (i.e., sensors) which always reveal their state independently of any particular epidemic (these are called static), or we can add some sensors (called dynamic) as an epidemic spreads, depending on which additional information is required. Second, the method can either localizes the source after an epidemic has spread through the entire network (offline), or while the epidemic is ongoing (online). We empirically study the performance of offline and online localization both with and without dynamic sensors. Our analysis shows that, by using dynamic sensors, the number of sensors necessary to localize the source is reduced by up to a factor of 10 and that, even with high-variance transmission delays, the source can be localized by using fewer than 5% of the nodes as sensors

Localizing the Source of an Epidemic Using Few Observations

Localizing the source of an epidemic is a crucial task in many contexts, including the detection of malicious users in social networks and the identification of patient zeros of disease outbreaks. The difficulty of this task lies in the strict limitations on the data available: In most cases, when an epidemic spreads, only few individuals, who we will call sensors, provide information about their state. Furthermore, as the spread of an epidemic usually depends on a large number of variables, accounting for all the possible spreading patterns that could explain the available data can easily result in prohibitive computational costs. Therefore, in the field of source localization, there are two central research directions: The design of practical and reliable algorithms for localizing the source despite the limited data, and the optimization of data collection, i.e., the identification of the most informative sensors. In this dissertation we contribute to both these directions. We consider network epidemics starting from an unknown source. The only information available is provided by a set of sensor nodes that reveal if and when they become infected. We study how many sensors are needed to guarantee the identification of the source. A set of sensors that guarantees the identification of the source is called a double resolving set (DRS); the minimum size of a DRS is called the double metric dimension (DMD). Computing the DMD is, in general, hard, hence estimating it with bounds is desirable. We focus on G(N,p) random networks for which we derive tight bounds for the DMD. We show that the DMD is a non-monotonic function of the parameter p, hence there are critical parameter ranges in which source localization is particularly difficult. Again building on the relationship between source localization and DRSs, we move to optimizing the choice of a fixed number K of sensors. First, we look at the case of trees where the uniqueness of paths makes the problem simpler. For this case, we design polynomial time algorithms for selecting K sensors that optimize certain metrics of interest. Next, turning to general networks, we show that the optimal sensor set depends on the distribution of the time it takes for an infected node u to infect a non-infected neighbor v, which we call the transmission delay from u to v. We consider both a low- and a high-variance regime for the transmission delays. We design algorithms for sensor placement in both cases, and we show that they yield an improvement of up to 50% over state-of-the-art methods. Finally, we propose a framework for source localization where some sensors (called dynamic sensors) can be added while the epidemic spreads and the localization progresses. We design an algorithm for joint source localization and dynamic sensor placement; This algorithm can handle two regimes: offline localization, where we localize the source after the epidemic spread, and online localization, where we localize the source while the epidemic is ongoing. We conduct an empirical study of offline and online localization and show that, by using dynamic sensors, the number of sensors we need to localize the source is up to 10 times less with respect to a strategy where all sensors are deployed a priori. We also study the resistance of our methods to high-variance transmission delays and show that, even in this setting, using dynamic sensors, the source can be localized with less than 5% of the nodes being sensors