Temporal Graph : Storage, Traversal, Path discovery

Submitted by

Vinay Sarda

M.Tech(CSE) II Year

13535054

Under the guidance ofProf. 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

1C A

B D E

5 0

201

9

143

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)

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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), marktime t[d] = 7, and so on

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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)

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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.

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(a, 0, 1, φ) (b, 1, 2, a) (d, 1, 7, a) (c, 1, 4, a) (d, 2, 3, b) (e, 3, 3, d)

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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

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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

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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

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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}

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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]

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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])

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S.No. s[v] a[v]

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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).

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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

Thank YouQ & A