A Probabilistic Model for Road Selection in Mobile Maps Thomas
C. van Dijk Jan-Henrik Haunert W2GIS, 5 April 2013
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Overview: a little bit of modelling, interpretation,
algorithms, evaluation, outlook,
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Road selection focus -and- context map route map Depends on
user task
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Road selection destination map Kopf et al. focus maps Yamamoto
et al.
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Motivating example Pedestrian with a smartphone Current
location available Wayfinding to a destination Interest in
surroundings Freedom of movement Contrast: in-car sat nav
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From scoring to selection Maximise score using given number of
road segments Minimise number of road segments selecting at least a
given amount of score Additional constraints?
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Betweenness centrality Consider all shortest paths between all
pairs of nodes; count how often each edge is used Folklore
interpretation: People are likely to travel shortest paths Heavily
traveled edges are important
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Betweenness centrality Consider all shortest paths between all
pairs of nodes; count how often each edge is used Probabilistic
interpretation: Start at a uniformly random node, uniformly random
destination, take a uniformly random shortest path Probability that
you visit a certain edge
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PageRank primer Origin in web page ranking (Page, Brinn,
Motwani, Winograd Google) Previous application in predicting human
movement (Jiang) Usually stated as eigenvector calculation Also:
interpretation in terms of random walks
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Random Surfer model Hypothetical user is at a web page
Transitions along a uniformly random link All transitions take
equal time Sometimes jumps to a uniformly random page on road
segment turn Traveler road segment
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Non-uniform transition probability Walking along a road segment
Coming up to a crossing Where to go? Shortest path? Straight on?
Anywhere? Whats the network? Web pages, hyperlinks?
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Whats the network? Web pages? Links? Modelling user behaviour:
routing decisions Road network -> bidirectional line graph
linear dual graph (Winter)
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Bidirected line graph
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Non-uniform transition probability At a node of the line graph
Follow an arc Where to go? Straight on? Constant given road network
Shortest path? Constant given road network and destination
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Non-uniform damping PageRank jumps to uniformly random node
Instead, jump to the current user location! Probabilistic
interpretation: Consider random walk starting at user location
Exponential distribution of walk length Score of node = probability
of going to node
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Data set OpenStreetMap City of Wrzburg and surroundings Lower
Franconia, Germany Approx 10 5 nodes Examples here: crop to the
city itself 3786 nodes, 4987 road segments
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Demo
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Shortest-path bonus
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Non-turning bonus
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Runtime of simple implementation Calculate PageRank Loop over
nodes Distribute score to neighbours Repeat Select nodes with high
score Runtime: ~100 ms
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Direct selection Select all nodes with high rank Select no
nodes with tiny rank O( log n ) time Correctness holds almost
surely Almost immediately from Borgs et als algorithm for
Significant PageRanks.