tsna (version 0.1.3)

paths: Temporally Reachable Paths in a networkDynamic Object

Description

Functions to search out the sequence and distances of vertices in a networkDynamic object reachable from an initial vertex by following paths constrained by edge timing.

Usage

tPath(nd, v, direction=c('fwd','bkwd'), 
                 type=c('earliest.arrive', 'latest.depart'),
                 start, end, active.default = TRUE, graph.step.time = 0)
is.tPath(x)

Arguments

nd

networkDynamic object to be searched for temporal paths

v

integer id of the vertex to be used as the starting point of the search

direction

option indicating the temporal direction in which the network should be searched: 'fwd' means search forwards in time and forward along edge directions, 'bkwd' means search backwards in time and backwards along edge directions.

type

option indicating the type of path (temporal constraint of the path) be searched for:

• 'earliest.arrive'will find the paths that arrive first at the target vertices,
• 'latest.depart'will find the paths that leave

start

time at which to begin searching. Edges that terminate before this time will not be considered. If not specified, defaults to earliest time observed on the network according to get.change.times.

end

time to end the path search. Edges that onset on or after this time will not be considered in the path search.

active.default

Boolean, default TRUE. Should edges with no timing information be considered active by default?

graph.step.time

numeric. How much time should be added for each edge traversal (graph hop)? Default is 0, meaning that path distances returned will be purely temporal and will not incorporate graph path distances and 'transmission' can cross multiple edges in a single i

x

an object to be tested for inheriting the class 'tPath'

Value

• Currently an object of class tPath which is essentially list with several elements providing information on the path found.
• tdist
  A numeric vector with length equal to network size in which each element contains the earliest/latest temporal distance at which the corresponding vertex could reach / be reached from the seed vertex. Values are elapsed time, as measured from the start parameter. Unreachable vertices are marked with Inf
• previous
  A numeric vector with length equal to network size in which each element indicates the previous vertex along (a possible) reachable path. Can be used to reconstruct the path tree. The initial vertex and unreachable vertices are marked with 0
• gsteps
  A numeric vector (of length equal to network size) in which each element indicates the number of steps in the path (number of graph hops) to the vertex along the temporal path found starting at the seed vertex.
• start
  the numeric start value that was used as the earliest bound for the path calculation (may not have been explicitly set)
• end
  the numerid end value that was used as the latest bound for the path calculation (may not have been explicitly set)
• direction
  The direction 'fwd' or 'bkwd' of the path
• type
  The type of temporal constraint for the path

Details

A temporal path in a dynamic network is a sequence of vertices and edges such that the onset times of successive elements are greater than or equal than those of the previous. In other words, the path is a directed traversal of the network that respects the constraints of edge activity spells and permits 'waiting' at intermediate vertices for 'future' edges to form.

When set to use direction='fwd' , type='earliest.arrive' tPath performs a time-minimizing Dijkstra's style Depth First Search to find the set of vertices reachable on a forward temporal path from the initial seed vertex v while respecting the constraints of edge timing.The path found is a earliest arriving (in contrast to the earliest leaving or quickest or latest arriving path). When there are multiple equivalent paths only a single one will be arbitrarily returned. NOTE THAT THE PATH-FINDING ALGORITHM WILL NOT GIVE CORRECT RESULTS IF ANY SPELLS CONTAIN VALUES LESS THAN 0.

When set to direction='bkwd' and type='latest.depart' the path will be found by searching backwards in time from the end point. In other words, it returns the set of vertices that can reach v, along with latest possible departure times from those vertices. Note that in this case the elapsed time values returned for tdist will be negative, indicating time measured backwards from the end bound.

When set to type='fewest.steps' the path returned will be a 'shortest' (fewest steps/graph hops) time-respecting path. This would not be necessiairly the quickest or earliest route, but would pass across the fewest possible number of edges (requires the fewest number of transmission steps).

The graph.step.time parmeter allows specifying an explicit duration for edge traversals. In this case the algorithm considers both the onset and terminus times of activity spells to ensure that suffecient time remains for an edge traversal to be made. If graph.step.time > the remaining duration of an edge's activity spell, the edge is considered non-traverseable. The primary use case for this parameter is to align the paths discovered with those that might be found by a discrete time transmission simulation in which a path can only spread a single graph hop per model timestep.

Vertex activity is currently ignored, and it is assumed that once a path reaches a vertex, all future edges from the vertex are accessible. The path search can be constrained in time using the start and end parameters to bound the time span to be explored by the path search.

'bwkd' 'latest.depart' is essentially the inverse of fwd earliest arrive. It finds the latest time paths backwards from the initial seed vertex. This is the latest-leaving time. Note that the distance returned are positive, but represent the latest distance back in time from the end parameter time at which a vertex can reach v.

The is.tPath function checks if an object has the class tPath.

References

Unpublished discussions with James Moody and Martina Morris and the statnet team.

Useful background information (for a slightly different algorithm) can be found in: B. Bui Xuan, Afonso Ferreira, Aubin Jarry. "Computing shortest, fastest, and foremost journeys in dynamic networks." RR-4589, 2002. https://hal.inria.fr/inria-00071996/document

B. Bui Xuan, Afonso Ferreira, Aubin Jarry. Evolving graphs and least cost journeys in dynamic networks. WiOpt'03: Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks, Mar 2003, Sophia Antipolis, France. 10 p., 2003 https://hal.inria.fr/inria-00466676/document

Examples

require(networkDynamicData)
data(hospital_contact)
hosPath<-tPath(hospital,v=1)