Observer Placement for Source Localization: The Effect of Budgets and Transmission Variance

When an epidemic spreads in a network, a key question is where was its source, i.e., the node that started the epidemic. If we know the time at which various nodes were infected, we can attempt to use this information in order to identify the source. However, maintaining observer nodes that can provide their infection time may be costly, and we may have a budget $k$ on the number of observer nodes we can maintain. Moreover, some nodes are more informative than others due to their location in the network. Hence, a pertinent question arises: Which nodes should we select as observers in order to maximize the probability that we can accurately identify the source? Inspired by the simple setting in which the node-to-node delays in the transmission of the epidemic are deterministic, we develop a principled approach for addressing the problem even when transmission delays are random. We show that the optimal observer-placement differs depending on the variance of the transmission delays and propose approaches in both low- and high-variance settings. We validate our methods by comparing them against state-of-the-art observer-placements and show that, in both settings, our approach identifies the source with higher accuracy.Comment: Accepted for presentation at the 54th Annual Allerton Conference on Communication, Control, and Computin

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

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

The effect of transmission variance on observer placement for source-localization

Abstract Detecting where an epidemic started, i.e., which node in a network was the source, is of crucial importance in many contexts. However, finding the source of an epidemic can be challenging, especially because the information available is often sparse and noisy. We consider a setting in which we want to localize the source based exclusively on the information provided by a small number of observers – i.e., nodes that can reveal if and when they are infected – and we study where such observers should be placed. We show that the optimal observer placement depends not only on the topology of the network, but also on the variance of the node-to-node transmission delays. We consider both low-variance and high-variance regimes for the transmission delays and propose algorithms for observer placement in both cases. In the low-variance regime, it suffices to only consider the network-topology and to choose observers that, based on their distances to all other nodes in the network, can distinguish among possible sources. However, the high-variance regime requires a new approach in order to guarantee that the observed infection times are sufficiently informative about the location of the source and do not get masked by the noise in the transmission delays; this is accomplished by additionally ensuring that the observers are not placed too far apart. We validate our approaches with simulations on three real-world networks. Compared to state-of-the-art strategies for observer placement, our methods have a better performance in terms of source-localization accuracy for both the low- and the high-variance regimes

Budgeted sensor placement for source localization on trees

We address the problem of choosing a fixed number of sensor vertices in a graph in order to detect the source of a partially-observed diffusion process on the graph itself. Building on the definition of double resolvability we introduce a notion of vertex resolvability. For the case of tree graphs we give polynomial time algorithms for both finding the sensors that maximize the probability of correct detection of the source and for identifying the sensor set that minimizes the expected distance between the real source and the estimated one

Ambulatory management of heavy menstrual bleeding.

Heavy menstrual bleeding (HMB) has significant adverse effects on the quality of life of many women, placing an economic burden on both health services and society at large. Thus, it is essential that all women with HMB have easy access to the proper diagnostic and therapeutic work-up in an outpatient fashion, avoiding the more time-consuming inpatient management. This new outpatient approach for HMB is one of the latest development of gynecological practice and can offer both diagnostic and therapeutic procedures. This manuscript aims to show the current possibilities of the modern management of HMB, which can be safely and effectively accomplished in the outpatient setting: global and directed endometrial biopsy, levonorgestrel intrauterine system insertion as well as minimally invasive surgical procedures (encompassing a variety of operative hysteroscopic procedures and second-generation endometrial ablation) are described below

Hysteroscopic treatment of Müllerian duct anomalies

Müllerian duct anomalies are a broad and complex spectrum of abnormalities that are often associated with infertility, obstetric complications as well as gynecological disorders among women of reproductive age. Operative hysteroscopy is the gold standard in the treatment of most of those anomalies amenable to surgical correction. The evidence to date shows an ongoing increase in the release of recommendations in favour of operative hysteroscopic treatment, in concert with the progressive refinement of hysteroscopic technologies and techniques. The aim of this paper was to describe and critically evaluate the role of the currently available hysteroscopic techniques for treating Müllerian duct anomalies, taking into account their indications, feasibility and efficacy as well as their impact on the reproductive outcome. Special attention will be paid to the most recently developed minimally invasive treatments for uterine and vaginal anomalies