origin {NetOrigin}

Origin Estimation for Propagation Processes on Complex Networks

Description

This is the main function for origin estimation for propagation processes on complex networks. Different methods are available: effective distance median ('edm'), recursive backtracking ('backtracking'), and centrality-based source estimation ('centrality'). For details on the methodological background, we refer to the corresponding publications.

origin_edm for effective distance-median origin estimation (Manitz et al., 2016)

Usage

origin(events, type = c("edm", "backtracking", "centrality", "bayesian"), ...)

origin_edm(events, distance, silent = TRUE)

origin_backtracking(events, graph, start_with_event_node = TRUE, silent = TRUE)

origin_centrality(events, graph, silent = TRUE)

origin_bayesian(
  events,
  thres.vec,
  obs.vec,
  mu.mat,
  lambda.list,
  poss.candidate.vec,
  prior,
  use.prior = TRUE
)

Arguments

events

numeric vector of event counts at a specific time point; if type is 'bayesian', 'events' is a matrix, number of nodes x time points; entries represent number of cases

type

character specifying the method, 'edm', 'backtracking', 'centrality' and 'bayesian' are available.

...

parameters to be passed to origin methods origin_edm, origin_backtracking, origin_centrality or origin_centrality

distance

numeric matrix specifying the distance matrix (for type='edm')

silent

locigal, should the messages be suppressed?

graph

igraph object specifying the underlying network graph (for type='backtracking' and type='centrality')

start_with_event_node

logical specifying whether backtracking only starts from nodes that experienced events (for type='backtracking')

thres.vec

vector, length represents number of cities/nodes, representing thresholds for cities/nodes that they are infected

obs.vec

list of cities ids used as observers

mu.mat

matrix- number of cities/nodes x number of observers, each row represents - if this node is the source, the mean of arrival time vector

lambda.list

a length-number of cities/nodes list, each element is a number of observers x number of observers matrix - if a node is the source, the covariance matrix for arrival time vector

poss.candidate.vec

a boolean vector indicating if a node has the potential to be the source

prior

vector, length - number of cities/nodes, prior for cities

use.prior

boolean, TRUE or FALSE, if use prior, default TRUE

Value

origin_edm returns an object of class origin, list with

• est origin estimate

• aux data.frame with auxiliary variables

  • id as node identifier,

  • events for event magnitude,

  • wmean for weighted mean,

  • wvar for weighted variance, and

  • mdist mean distance from a node to all other nodes.

• type = 'edm' effective distance median origin estimation

origin_backtracking returns an object of class origin, list with

• est origin estimate

• aux data.frame with auxiliary variables

  • id as node identifier,

  • events for event magnitude, and

  • bcount for backtracking counts, how often backtracking identifies this source node.

• type = 'backtracking' backtracking origin estimation

origin_centrality returns an object of class origin, list with

• est origin estimate

• aux data.frame with auxiliary variables

  • id as node identifier,

  • events for event magnitude, and

  • cent for node centrality (betweenness divided degree).

• type = 'centrality' centrality-based origin estimation

a dataframe with columns 'nodes' and 'probab', indicating nodes indices and their posteriors

Author(s)

Juliane Manitz with contributions by Jonas Harbering

Jun Li

References

• Comin, C. H. and da Fontoura Costa, L. (2011). Identifying the starting point of a spreading process in complex networks. Physical Review E, 84. <doi: 10.1103/PhysRevE.84.056105>

• Manitz, J., J. Harbering, M. Schmidt, T. Kneib, and A. Schoebel (2017): Source Estimation for Propagation Processes on Complex Networks with an Application to Delays in Public Transportation Systems. Journal of Royal Statistical Society C (Applied Statistics), 66: 521-536. <doi: 10.1111/rssc.12176>

• Manitz, J. (2014). Statistical Inference for Propagation Processes on Complex Networks. Ph.D. thesis, Georg-August-University Goettingen. Verlag Dr.~Hut, ISBN 978-3-8439-1668-4. Available online: https://ediss.uni-goettingen.de/handle/11858/00-1735-0000-0022-5F38-B.

• Manitz, J., Kneib, T., Schlather, M., Helbing, D. and Brockmann, D. (2014). Origin detection during food-borne disease outbreaks - a case study of the 2011 EHEC/HUS outbreak in Germany. PLoS Currents Outbreaks, 1. <doi: 10.1371/currents.outbreaks.f3fdeb08c5b9de7c09ed9cbcef5f01f2>

• Li, J., Manitz, J., Bertuzzo, E. and Kolaczyk, E.D. (2020). Sensor-based localization of epidemic sources on human mobility networks. arXiv preprint Available online: https://arxiv.org/abs/2011.00138.

See Also

Other origin-est: origin_multiple()

Examples

data(delayGoe)

# compute effective distance
data(ptnGoe)
goenet <- igraph::as_adjacency_matrix(ptnGoe, sparse=FALSE)
p <- goenet/rowSums(goenet)
eff <- eff_dist(p)
# apply effective distance median source estimation
om <- origin(events=delayGoe[10,-c(1:2)], type='edm', distance=eff)
summary(om)
plot(om, 'mdist',start=1)
plot(om, 'wvar',start=1)
performance(om, start=1, graph=ptnGoe)

# backtracking origin estimation (Manitz et al., 2016)
ob <- origin(events=delayGoe[10,-c(1:2)], type='backtracking', graph=ptnGoe)
summary(ob)
plot(ob, start=1)
performance(ob, start=1, graph=ptnGoe)

# centrality-based origin estimation (Comin et al., 2011)
oc <- origin(events=delayGoe[10,-c(1:2)], type='centrality', graph=ptnGoe)
summary(oc)
plot(oc, start=1)
performance(oc, start=1, graph=ptnGoe)

# fake training data, indicating format
nnodes <- 851
max.day <- 1312
nsimu <- 20
max.case.per.day <- 10
train.data.fake <- list()
for (j in 1:nnodes) {
  train.data.fake[[j]] <- matrix(sample.int(max.day, 
    size = nsimu*nnodes, replace = TRUE), nrow = nsimu, ncol = nnodes)
}
obs.vec <- (1:9)
candidate.thres <- 0.3
mu.lambda.list <- compute_mu_lambda(train.data.fake, obs.vec, candidate.thres)
# matrix representing number of cases per node per day
cases.node.day <- matrix(sample.int(max.case.per.day, 
  size = nnodes*max.day, replace = TRUE), nrow = nnodes, ncol = max.day)
nnodes <- dim(cases.node.day)[1] # number of nodes
# fixed threshold for all nodes - 10 infected people
thres.vec <- rep(10, nnodes)
# flat/non-informative prior
prior <- rep(1, nnodes) 
result2.df <- origin(events = cases.node.day, type = "bayesian",
                     thres.vec = thres.vec,
                     obs.vec = obs.vec,
                     mu.mat=mu.lambda.list$mu.mat, lambda.list = mu.lambda.list$lambda.list, 
                     poss.candidate.vec=mu.lambda.list$poss.candidate.vec,
                     prior=prior, use.prior=TRUE)