Confidence Sets for the Source of a Diffusion in Regular Trees

Khim, Justin; Loh, Po-Ling

doi:10.1109/tnse.2016.2627502

Cited by 48 publications

(68 citation statements)

References 24 publications

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“…It would be interesting to study what additional constraints could ensure the agreement of both persistent nodes. Our results show that the top K central nodes obtained according to the centrality measure ψ stabilizes after a finite number of steps. The results in and showed that confidence sets for the root node may be constructed by selecting the most central nodes according to ψ. However, the set so obtained may be suboptimal in terms of the size required as a function of the error probability ϵ .…”

Section: Discussionmentioning

confidence: 99%

“…The results in and showed that confidence sets for the root node may be constructed by selecting the most central nodes according to ψ. However, the set so obtained may be suboptimal in terms of the size required as a function of the error probability ϵ . It would be interesting to see whether other centrality measures such as those corresponding to the maximum likelihood estimator are also “robust” in the sense that they produce a stable output after some finite time. The problem of establishing persistence of centrality in non‐trees (eg, in preferential or uniform attachment models where more than one node is added at each step) appears to be very challenging.…”

Section: Discussionmentioning

confidence: 99%

“…If the required probability is

1 - ϵ

, for some

ϵ > 0

, then the set produced is called a

1 - ϵ

confidence set. It has been shown that one may obtain such confidence sets for the root node in uniform and preferential attachment models and d ‐regular diffusion trees by selecting the nodes that minimize the maximum subtree estimator ψ. Furthermore, the size of the confidence set may be taken as a fixed function

K (ϵ)

of the error probability ϵ, and does not need to grow with n .…”

Section: Persistence Of the Top K Central Nodesmentioning

confidence: 99%

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Persistence of centrality in random growing trees

Jog

2017

Random Struct Algorithms

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We investigate properties of node centrality in random growing tree models. We focus on a measure of centrality that computes the maximum subtree size of the tree rooted at each node, with the most central node being the tree centroid. For random trees grown according to a preferential attachment model, a uniform attachment model, or a diffusion processes over a regular tree, we prove that a single node persists as the tree centroid after a finite number of steps, with probability 1. Furthermore, this persistence property generalizes to the top K ≥ 1 nodes with respect to the same centrality measure. We also establish necessary and sufficient conditions for the size of an initial seed graph required to ensure persistence of a particular node with probability 1 − , as a function of : In the case of preferential and uniform attachment models, we derive bounds for the size of an initial hub constructed around the special node. In the case of a diffusion process over a regular tree, we derive bounds for the radius of an initial ball centered around the special node. Our necessary and sufficient conditions match up to constant factors for preferential attachment and diffusion tree models.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

“…If the required probability is

1 - ϵ

, for some

ϵ > 0

, then the set produced is called a

1 - ϵ

K (ϵ)

of the error probability ϵ, and does not need to grow with n .…”

Section: Persistence Of the Top K Central Nodesmentioning

confidence: 99%

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Persistence of centrality in random growing trees

Jog

2017

Random Struct Algorithms

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“…infection paths, inside the infected subgraph. Their linear time algorithm is an optimal estimator in regular trees and enjoys strong theoretical properties in such idealized settings [34]. Zhou and Ying [26] consider SIR dynamics on a tree and show that the most likely infection path is rooted at a Jordan center (JC) of the infected set O, that is, a node with minimum eccentricity (i.e., maximum distance to others).…”

Section: Introductionmentioning

confidence: 99%

“…Zhou and Ying [26] consider SIR dynamics on a tree and show that the most likely infection path is rooted at a Jordan center (JC) of the infected set O, that is, a node with minimum eccentricity (i.e., maximum distance to others). It has been shown [26,34] that in regular trees, eccentricity ranking generates, with high probability, a confidence set containing the true source, whose size does not grow with the infection size.…”

Section: Introductionmentioning

confidence: 99%

Approximate Identification of the Optimal Epidemic Source in Complex Networks

Kazemitabar

Amini

2020

Proceedings of NetSci-X 2020: Sixth International Winter School and Conference on Network Science

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We consider the problem of identifying the source of an epidemic, spreading through a network, from a complete observation of the infected nodes in a snapshot of the network. Previous work on the problem has often employed geometric, spectral or heuristic approaches to identify the source, with the trees being the most studied network topology. We take a fully statistical approach and derive novel recursions to compute the Bayes optimal solution, under a susceptible-infected (SI) epidemic model. Our analysis is time and rate independent, and holds for general network topologies. We then provide two tractable algorithms for solving these recursions, a mean-field approximation and a greedy approach, and evaluate their performance on real and synthetic networks. Real networks are far from tree-like and an emphasis will be given to networks with high transitivity, such as social networks and those with communities. We show that on such networks, our approaches significantly outperform geometric and spectral centrality measures, most of which perform no better than random guessing. Both the greedy and mean-field approximation are scalable to large sparse networks.Preprint. Under review. and consistent recovery are restricted to regular infinite trees [23,26], and as we show in this paper, the popular and well-cited methods are quite unreliable in a wide range of real networks.Source identification has remained largely unsolved and poorly understood for real complex networks. As we will show through experiments in Section 5, in real networks, even the optimal Bayes estimator applied to small infected sets has difficulty narrowing down to the true source. It is thus important to recover as much information from the likelihood of the model as possible. We develop techniques for computing the full likelihood of the infection, as opposed to identifying the most likely samplepath [26]. Moreover, we fully exploit the information from the boundary of the infection set, in addition to the structure inside the infected subgraph. This idea has been pointed out before [32], but has been mostly neglected by subsequent work; cf. [29,24]. We develop all these ideas without restricting the structure of the network to trees. Our framework also easily extends to the case where there are multiple infecting sources (Appendix A).In this paper, we develop statistical algorithms that outperform the state-of-the-art in a wide range of network topologies. Our contributions are distinct in several ways:1. Our methods are parameter-free, meaning that they do not require knowing the duration of the epidemic or how fast it grows.

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