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Temporal Graph Algebra
VERA ZAYCHIK MOFFITT JOINT WORK WITH JULIA STOYANOVICH SEPTEMBER 1, 2017
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https://www.kenedict.com/apples-internal-innovation-network-unraveled-part-1-evolving-networks/
Temporal Graph Algebra VERA ZAYCHIK MOFFITT JOINT WORK WITH JULIA - - PowerPoint PPT Presentation
Temporal Graph Algebra VERA ZAYCHIK MOFFITT JOINT WORK WITH JULIA - - PowerPoint PPT Presentation
Temporal Graph Algebra VERA ZAYCHIK MOFFITT JOINT WORK WITH JULIA STOYANOVICH SEPTEMBER 1, 2017 Graph Evolution https://www.kenedict.com/apples-internal-innovation-network-unraveled-part-1-evolving-networks/ 2 Interesting and Important
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Interesting and Important Questions
§ What is the likelihood of an individual to join a community? § Which roads exhibit abrupt congestion and at what time? § Which websites have the highest increase in popularity/rank over the past year? § What is the rate of densification of the graph? § Have any changes in network connectivity been observed? § At what time scale can interesting trends be observed?
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Goal
Principled and systematic support for querying and analytics of evolving graphs
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Existing Models – Time as Data
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Approach – Add time property
- Need a new node for each change of property or period of
validity
! Time needs special treatment
Are Alice and Bill connected?
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Snapshot Reducibility
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Existing Models – Snapshot Sequence
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! No explicit references to time
Which pairs of people are connected by a journey?
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Contributions
§ Conceptual representation of an evolving graph
§ Captures evolution of both topology and properties
§ Temporal Graph Algebra (TGA)
§ Concisely express wide range of common analysis tasks
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Temporal Graph Model
Definition 3.1.1 (TGraph). A TGraph is a 7- tuple (V, E, Π, ρ, ξ, λv, λe), where:
- V – set of nodes,
- E – set of edges,
- P – set of available properties,
- ρ : E → (V x V ) total function,
- ξ𝑈 : (V ∪ E) x T → B total function,
- λ𝑈: (V ∪ E) x P x T → Val partial function
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type=co-author cnt=3
e1
type=person name=Alice school=Drexel
v1
type=person name=Bob school=CMU
v2
type=person name=Bob
v2
type=person name=Cathy school=Drexel
v3
[2015/1,2015/7) [2015/2,2015/5) [2015/2,2015/5) type=co-author cnt=3
e1
[2015/5,2015/6) [2015/1,2015/10) [2015/7,2015/10) [2015/5,2015/10) type=co-author cnt=4
e1
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TGA Operators
§ Provide temporal versions of common graph operations:
§ subgraph § aggregation § vertex- and edge-map § union, intersection, difference
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Node Creation
§attribute-based node creation
§add new nodes representing a matching input pattern
§ window-based node creation
§Change temporal resolution of
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Attribute-based node creation
school=x2
x1
type=studentAt
fe(x1,studentAt,x2)
type=school students = count(x1)
fv(x2)
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§ Add new nodes to represent matching pattern adds nodes Drexel and CMU and edges to them
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Attribute-based node creation
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type=co-author cnt=3
e1
type=person name=Alice school=Drexel
v1
type=person name=Bob school=CMU
v2
type=person name=Bob
v2
type=person name=Cathy school=Drexel
v3
[2015/1,2015/7) [2015/2,2015/5) [2015/2,2015/5) type=co-author cnt=3
e1
[2015/5,2015/6) [2015/1,2015/10) [2015/7,2015/10) [2015/5,2015/10) type=co-author cnt=4
e1
type=school students=2
Drexel
[2015/1,2015/7) type=school students=1
Drexel
[2015/7,2015/10) type=school students=1
CMU
[2015/5,2015/10) type=studentAt
e3
[2015/1,2015/7) type=studentAt
e4
[2015/1,2015/7) type=studentAt
e4
[2015/7,2015/10) type=studentAt
e5
[2015/5,2015/10)
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Window-based node creation
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type=person name=Alice school=Drexel
v1
type=person name=Bob school=CMU
v2
type=person name=Cathy school=Drexel
v3
[2015/1,2015/7) type=co-author cnt=3
e1
[2015/4,2015/7) [2015/1,2015/10) [2015/7,2015/10) [2015/4,2015/10) type=co-author cnt=4
e1
3 months
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Example: NYC Cabs
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Node Influence over Time
Are there high influence nodes and is that behavior persistent over time?
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Node Influence, with TGA
1. Select a subset of the data representing the 5 years of interest, using trim: 2. Compute in-degree (prominence) of each node during each time point using aggregation and pattern p1
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x3 x2
deg=count(x2)
x1
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Node Influence, with TGA
3. Aggregate degree information per node across the timespan of G2 using the window-based node creation operator: 4. Transform the attributes of each node using the vertex-map operator:
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Summary
§ TGraph model represents evolution of graph topology and properties § TGA provides a concise set of operations over TGraphs
§ Precise semantics § More expressive than current state of the art § Desirable temporal properties
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Thank You! Questions?
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