Temporal graph

Temporal Graph : Storage,
Traversal, Path discovery
Submitted by
Vinay Sarda
M.Tech(CSE) II Year
13535054
Under the guidance of
Prof. A.K. Sarje

Outline
 Introduction
 Graph Database
 What is a Temporal Graph
 Traversal in Temporal Path
 Types of Path in Temporal Graph
 Algorithms to determine different temporal path
 Conclusion

Introduction
 Graph database are best tool for handling highly connected data
 In Relational or Hierarchical database, some entities have precedence over other
like key, parent etc.
 In Graph No node or edge have a higher precedence over other
 Graph DB are NOSQL, doesn’t have rigid schema
 Graph uses structures like nodes, edges are used to store and represent data
 No need to look-up for neighbours as it provides index-free adjacency
 Graph DB can be used in to determine whether two concept are related or not, or
is it possible to reach from source to destination within specified interval etc.

Example -Graph
Roy is boy
Mary is a girl
Roy and Mary are friends

Accessing Graph DB
 TinkerPop is an open source toolkit that provides
interfaces to implement Blueprint
 Blueprint is a generic graph API, serves as a base layer for
graph DB architectures
 Pipes is a dataflow network using process graphs
 Gremlin is a graph traversal language for accessing and
manipulating graphs
 Frames allows developer to frame a graph element in
terms of java interface
 Furnace is a graph algorithm package
 Rexster is a graph server that exploits blueprint via REST
 Example Titan Graph DB
TinkerPop Stack

Temporal Graph
 Nodes and Edges are active for a specific time instance
 Visiting nodes or edges at any time other than specific time instance is not possible
 Example – SMS Network, flight scheduling, stock exchange network
 Did mail from A to D had any impact on mail send from D to B
Sender Receipent Time
A B 0
A C,E 1
E D 3
B C 5
B D 9
D B 14
A D 20
C A
B D E
1
C A
5 0
20
1
9
B D E
14
3
Non-Temporal Graph Temporal Graph

Temporal Graph Traversal
 Depth First Traversal
 Temporal Constraint – In traversal from node a to e set of edges {(a,b,1),(b,e,6)} is valid,
but edge set {(a,b,9),(b,e,6)} is not
 Multiple edges – When multiple edges occur among the same pair of nodes, only one
edge has to be chosen avoiding combinatorial effect
 Edge with highest timestamp value (most recent connection)
 Edge with lowest timestamp value (first connection)
A
B C
E
D
F
3
2
8
6
7
1
4
5
9

Depth First Traversal
 Create a DFS tree rooted at node ‘a’ with lowest timestamp value
 Initially every node’s visiting time is infinite
 At node ‘a’, Eset = {(a,b,9), (a,b,1), (a,e,8), (a,c,2)}
 Take edge (a,b,1) and visit node ‘b’ and mark time 1
 Now Eset = {(b,d,7), (b,e,6)} and take edge (b,e,6), mark time t[e] = 6, and then (b,d,7), mark
time t[d] = 7, and so on
a
b c
e
d
2
7
1
5
a
c
e
b
d
3 f
2
8
6
7
1
4
9
5
6

Breadth First Traversal
 Challenges
 Temporal Constraint
 Edges (b, d, 3) and (d, e, 3) forms a valid set of consecutive edges, but (c, d, 5), (d, e, 3) doesn’t.
 Smallest path may not be earliest path
 Path from a to d takes only one hop at time t = 7, while the earliest time at which d can be reached from
a is 3 using edges (a, b, 2) and (b, d, 3)
 If we mark node ‘d’ using edge a-d and don’t consider it again then ‘e’ cannot be reached
 Multiple occurrences of a node are possible
 Terminology
 u – node
 dist(u) – distance of node u from source s
 σ(u) – earliest arrival time to reach node u from source node s
 p(u) – parent node of u
 Temporal edge is denoted as (u, v, t)
a
b c
d
e
2
5
4
7
3 5
3

 While Q is not empty for each node v
 If v is not in Q (whether v has been visited or not) traverse e, and if σ(v) > t, then visit v
and push (v, dist(v) = dist(u) + 1, σ (v) = t, p(v) = u) into Q.
 Else
 If there exists (v, dist(v), σ (v), p(v)) in Q such that dist(v) = dist(u) + 1: traverse e, and if σ(v) > t,
then visit v and update σ(v) = t and p(v) = u in Q
 Else (i.e., dist(v) = dist(u)): traverse e, and if σ(v) > t, then visit v and push (v, dist(v) = dist(u) + 1,
σ (v) = t, p(v) = u) into Q.
a
b c
d
e
2
5
4
7
3 5
3
c
a
b
d
e
2
4
3
d
7
3
(a, 0, 1, φ) (b, 1, 2, a) (d, 1, 7, a) (c, 1, 4, a) (d, 2, 3, b) (e, 3, 3, d)
Q (u, dist(u), σ(u), p(v))

Paths in Temporal Graph
 A path ρ = {(s, u1, t1), (u1, u2, t2), (u2, u3, t3),……..,(ui-1, ui, ti), ….. (un-1 ,d, tn)} from
source s to destination d is a temporal path if ti ≤ ti+1
 Terminology
 Pset - set of all paths from source s to destination d
 [ta, tb] – time interval
 t[u] – time at which node u can be reached from source s
 tstart(ρ) - timestamp of a path ρ at which it starts at source s
 tend(ρ) - timestamp of a path ρ at which it ends at destination d
 dist(ρ) - number of nodes in path ρ
 duration(ρ) – time taken by path ρ to reach destination node from source node, tend(ρ) -
tstart(ρ)
 Transition time – time taken to reach from node to its neighbour node (λ), by default
assumed to be 1

 Foremost – In a given interval, path that reaches destination as early
as possible, path ρ ∈ Pset is a foremost path if all path ρ’ ∈ Pset, such
that tend (ρ’) ≤ tend (ρ)
 Example – Pset(A,H) = {{(A,B,1), (B,H,3)}, {(A,B,2), (B,H,3)}, {(A,C,4),
(C,H,6)}}, but only A-B-H is the foremost path
 Latest- departure path - In a given interval, path that reaches
destination as early as possible, path ρ ∈ Pset is a latest-departure
path, if all path ρ’ ∈ Pset, such that tstart (ρ’) ≤ tstart (ρ)
 Example - Pset(A,H) = {{(A,B,1), (B,H,3)}, {(A,B,2), (B,H,3)}, {(A,C,4),
(C,H,6)}}, but only A-C-H is the latest-departure path
 Fastest path – Path that spends minimum time to reach destination d
from source s, path ρ ∈ Pset is a fastest path if for all path ρ ∈ Pset,
duration (ρ) <= duration (ρ’)
 Example – Pset(B,K) = ({(B,H,3),(H,K,7)}, {(B,G,3),(G,K,6)}}, fastest path is
B-G-K as it takes only 3 units of time
A
1
2
4
2
3
B C F
3
3
6 5
G H I
6
2 7 8 9
J K L

 Shortest Path – Path that takes minimum distance to reach the
destination, path ρ ∈ Pset is a shortest temporal path if for all
path ρ’ ∈ Pset, dist (ρ) ≤ dist (ρ′)
 Example – Pset (A,I) = {(A→I), (A→F →J→I)}, then shortest path
from source A to destination I is A→I
 Most recent path – path where source and destination node are
connected using edges with recent timestamp
 Example – Among two paths from A to K { {(A,B,1), (B,G,3),
(G,K,6)}, {(A,C,4), (C,H,6), (H,K,7)} }, path A-C-H-K is most recent
path
 Edge Constrained Path – path from s to d where adjacent edges
have timestamp difference lower than certain threshold value
 Example - Pset(A,L) = {{(A,F,3), (F,I,5), (I,L,8)}, {(A,F,3), (F,I,5), (I,L,9)},
{(A,I,2), (I,L,8)}, {(A,I,2), (I,L,9)}} with threshold value of 3 only path
A-F-I-L satisfies the constraint
A
1
2
4
2
3
B C F
3
3
6 5
G H I
6
2 7 8 9
J K L

Observation
 Optimal path in Temporal Graph v/s Non-Temporal Graph
 Optimal path in Non-Temporal Graph has property that sub-path of optimal path is
always optimal
 But it doesn’t holds in case of Temporal Graph, example shortest path
A
B
C
D
2
3
7
4

Temporal Path Discovery Techniques
 Stream Representation of Temporal Graph
 Transformation Graph

Stream Representation
 Edge sequence in the order of their creation is called stream representation of
edges in a temporal graph
 Example – {(A→B)/2, (B→C)/3, (C→D)/4, (A→C)/7}
A
B
C
D
2
3
7
4

Foremost Path
 Foremost Path for node ‘s’ in time interval [ta,tb]
 For each node u in graph G t[u] = ∞
 Considering each edge in the sequence in the order
 Check for two conditions if true then update t[v] = t+λ
 t+λ ≤ tb and t ≥ t[u]
 t+λ < t[v]
 if t+ λ > tb then exit

Example
 Edge representation is {(a, b, 1, λ), (a, c, 2, λ), (d, f, 3, λ), (b, d, 7, λ)}
 Foremost path for node ‘a’ in [1,4] assuming λ = 1
 For all node t[u] =∞
 Mark t[a] = 0
 For edge (a, b, 1, 1), t + λ = 2, satisfies conditions, update t[b] = 2
 For edge (a, c, 2, 1), t + λ = 3, satisfies conditions, update t[c] = 3
 For edge (d, f, 3, 1), t + λ = 4, doesn’t satisfies condition t ≥ t[d] as t[d] = ∞ and t =
3
t+λ ≤ tb and t ≥ t[u]
t+λ < t[v]
a
c
b
d f
3
2
7
1
∞
∞
∞
∞ ∞
0 a
3
c
b
d f
3
2
7
1
2 ∞ ∞

Fastest Path
 Lv – sorted list of foremost time for each node v from source node ‘s is maintained,
(s[v], a[v]), where s[v] represents start time from ‘s’ and a[v] arrival time at v
 If s’[v] > s[v] and a’[v] ≤ a[v], or s’[v] = s[v] and a’[v] < a[v], then (s’[v], a’[v])
dominates (s[v], a[v])
A
1
2
1,1 2,2
B D
3
5
C E
4
5
F
6
1,3
1,4
2,5
2,6
S.No. s[v] a[v]
a 1 4
b 2 4
c 4 6
d 4 5

Transformation Graph
 Two Step process
 Node Creation - for each node in V create nodes in V’ for each oncoming (V’in) and
outgoing edge (V’out) with distinct label as (name , time)
 Edge Creation –
 For each node (v, tin) in V’in(v), add an directed edge from (v, tin) to (v, tout) ∈ V’out(v), such
that tout is minimum in V’out(v), tout ≥ tin. and no other edge from (v, t’in) to (v, tout) is added
to G’.
 Given V’in(v) = {(v, t1), (v, t2), (v, t3), ….,(v, tk)}, where t1 ≤ t2 ≤ t3 ≤ …..tk, then add an edge for
(v, ti) to (v, ti+1) for 1 ≤ i ≤ k. with weight 0. Similarly add edges for V’out(v).
 For each edge e = <x, y, t, λ> ∈ E, add a directed edge from (x, t) ∈ V’out(x) to (v, t+ λ) ∈
V’out(y).

4
a’,0
b,2
1 0 1
b,3
0 0
c,3
c,5
f,6
f,7
g,8
0
0
0
1
0 0
1
1
0 0 0
1
1
a
1
2 2
b c
f g
5
6
7
Example
Vout
Vin
Edge in original
graph
Within Vin/Vout
From Vin to Vout

Foremost Path
 Process the transformed graph G’ for a simple BFS Traversal
 If the time t of any node is not in interval, then stop the BFS from that node
 The minimum time of all the nodes visited (v, t) in G’ is the foremost time from a to
v in G’
 From node a, in first round we visit (b, 2), (b, 3), (c, 3), and (c, 5)
 Thus, the foremost time from a to b is 2, and from a to c is 3 and so on

Cost Optimal Path with time constraint
 A graph in which cost of an edge changes with time is called time dependent
graph
 Two factors are considered in time dependent graph
 Time
 Cost of path
 Constraints
 Departure time from source and arrival time at destination must be within specified
interval
 Cost of the path must be minimum
 Problem of finding cost-optimal path is
 Optimal path from source to destination
 Optimal waiting time at each node

 Earliest arrival time for each node is calculated so , λ1 = 0, λ2 =
10, λ3 = 15 and λ4 = 25 and time domain of g1(t), g2(t), g3(t) and
g4(t) are [0, 60], [10, 60], [15, 60] and [25, 60] respectively
 Initially ti for all node is infinite except for source s
 Priority Q is maintained containing all nodes with respect to
their ti
 In first iteration, v1 is dequeued as t1= 0 is minimum
 g2(t) and g3(t) are computed and v1 is removed from Q
 t2 = 20 and t3 = 5 are set
 In second iteration, v3, is dequeued from Q
 As T3 = 30, so S3 is updated as [30,60]
 g4(t) is updated and t3 = 20
 In third iteration, v2 is dequeued
 g3(t) and g4(t) are computed and v2 is removed from Q
 Similarly v3 is dequeued again in fourth iteration and v4 is
dequeued in fifth iteration, and thus algorithm terminates

Conclusion
 Graph DB used to store connected data and different technologies used to access
them
 Temporal Graph are used to withhold the temporal information about the data
 Breadth first and depth traversal techniques on Temporal Graph
 Different temporal paths and algorithms to determine those path

Temporal graph

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