Spatio-temporal Networks: Modeling, Storing, and Querying Temporally-Detailed Roadmaps - Space-Time Integration in Geography and GIScience - page 83

Finally, since time expanded graphs model the time-dependence of edge param-
eters through the cross edges that connect the copies of the nodes, they lack the
means to separate data (for example, an edge attribute series) from its physical
representation, which, again, can adversely affect physical data independence.

6.8 Time-Aggregated Graphs

Many of the limitations of TEG can be eliminated by a time aggregated graph (TAG)
approach. A Time Aggregated Graph (TAG) is defined as

N; E; TF;

TAG D

f 1 :::f k ;g 1 :::g l ;

.n w 1 ;ne 1 /:::.n w k ;ne p /;
.e w 1 ;ee 1 /:::.e w p ;ee p /; ˇ ˇ

! R TF ! R TF W N I W E I

f i g i

! R TF ;ne i

n w i W N W N ! PD;

! PD ; ! R TF ;ee i

e w i W E W E

where N is the set of nodes, E is the set of edges, TF is the length of the entire
time interval, f 1 ::: f k are the mappings from nodes to the time-series associated
with the nodes (for example, the time instants at which the node is present),
g 1 ::: g l are mappings from edges to the time series associated with the edges,
and ( ew 1 , ee 1 ) ::: ( ew p , ee p ) indicate the time dependent attribute on the edges. PD
indicates a probabilistic error. These attributes are the quantitative descriptors of the
physical relationship between the nodes. To represent the stochastic nature of the
measured values of physical phenomena, each attribute is a pair that represent the
measured value and the associated error.
Time aggregated graphs (TAG), at their simplest, show a network with a time
series of attributes on nodes, edges or both. These time series represent the state of
an object at each time step. If we were to compare it to a more familiar temporal
geography idea, such as space-time paths, we can easily represent the travel of one
person over time in a TAG. Figure 6.5 contains a TAG representation of the sample
network shown earlier as a space-time trajectory (Fig. 6.4 ), snapshots (Fig. 6.4 )and
a TEG (Fig. 6.6 ). Each node, A, B, C and D has a time-series to represent sets of peo-
ple associated at different time instants. For example, node A has person P at t D 1,
and no other people at t D 2ort D 3. Similarly, node B has person P at t D 2, etc.