Segmenting Sequences of Node-labeled Graphs

Segmenting Sequences of
Node-labeled Graphs
Sorour E. Amiri, Liangzhe Chen, B. Aditya Prakash
Department of Computer Science
Virginia Tech
ICDM, DaMNet, Barcelona, Spain, December 12, 2016

Outline
 Motivation
 Background
 Our Proposed Method: SnapNETS
 Experiments
 Conclusion
Amiri, Chen, Prakash 1

Network Sequences
 Epidemiology: disease spreads over contact networks
 Social Media: Information spreads over friendship networks
2
Flu
Meme
Amiri, Chen, Prakash

Making sense of network sequences
3
Flu
when do the infection patterns change?
Star Bridge Near Clique
Reason:
• Virus mutation
• Vaccination
• …

Making sense of network sequences
4
Meme Reason:
• Event
• …
Star Clique
when do the infection patterns change?

Problem 1: Network sequence segmentation
 Given a sequence of networks with labeled nodes,
 Find the best segmentation which captures:
 Different distribution of node labels.
5
Star Bridge Near Clique

Outline
 Motivation
 Background
 Experiments
 Conclusion
6Amiri, Chen, Prakash

Alternative 1: Feature Ext. &Time-series
7
0 0 0 … 2F1: #cliques (of active subgraph)
F2: #ladders (of inactive subgraph)
F3: #ladders (of active subgraph)
1 1 0 … 0
0 0 0 … 1
[Henderson et al. 2010]
[Likas, Vlassis, and Verbeek 2003]
[Li et al. 2009]
-1
0
1
2
G1 G2 G3 G4
Features time series
F1 F2 F3
Step 1: Feature Extraction
Step 2: Time-series
segmentation

Alternative 1: Feature Ext. &Time-series
 Drawbacks:
 Laborious feature-engineering
 “Local” change detection:
o One aggregation time period
o Threshold
-1
0
1
2
G1 G2 G3 G4
Features time series
F1 F2 F3

Alternative 2: Plain-graph-based analysis
9
[Shah et al. 2015]
[Sun et al. 2007]
[Lin et al. 2009]
[Qu et al. 2014]
Step 1: Extract active subgraphs
Step 2: Dynamic graph segmentation

Alternative 2: Plain-graph-based analysis
 Drawbacks:
Inactive nodes are important to detect different patterns
Amiri, Chen, Prakash 10
Entire graph Active subgraph

Desirable Properties
 P1. Parameter-free:
• No threshold, No fixed granularity
 P2. Comprehensive:
• Use the entire graph

Outline
 Motivation
 Background
Overview
 Goal 1: Summarizing Act-snapshots
 Goal 2: Constructing the segmentation graph
 Goal 3: Finding the best segmentation
 Experiments
 Conclusion

Overview of SnapNETS
 Goal 1. Summarize each graph:
Keep structural and label dependent properties
 Goal 2. Construct Segmentation graph:
Define nodes and edges
Defining edges weights
o extract the features of summarized graphs
 Goal 3. Find the best segmentation:
Define the best segmentation (path)
Compute the best segmentation

Technical Challenges
 Using the entire graph snapshots:
 Summarize graph while satisfying P2
 Finding the number of segments:
 Compute segmentation while satisfying P1
14
Reminder:
 P1. Parameter-free
 P2. Comprehensive

Outline
 Motivation
 Background
Overview
 Experiments
 Conclusion

Goal 1: Summarizing graph snapshots
 We want to preserve
 Structural properties
 Nodes labels
 Role of Eigenvalue:
Same leading eigenvalue ( ) of Adjacency matrix Same diffusive
properties
Leading eigenvalue Epidemic threshold [Prakash et al. 2012]

Our Approach
 We want to get a smaller graph with similar eigenvalues:
Successively merge nodes

Problem 2: Graph summarization
 Given: A graph with labeled nodes and a compression ratio.
 Find: a coarsened graph such that:

 CoarseNet algorithm [Purohit et al.2014]
 Matrix perturbation approach
 Successively merge nodes
 Keep leading eigenvalue
 Our tweak
 Do not merge nodes with different labels
Problem 2: Graph
summarization
19
Given: A graph with labeled nodes and a compression ratio.
Find: a coarsened graph such that:

Outline
 Motivation
 Background
 Overview
 Experiments
 Conclusion

 Nodes:
 For each segment there is a node + {Source (‘s’), Target (‘t’)}
 Edges:
 There is a directed edge between adjacent nodes
Goal 2: Segmentation graph

Edge Weights
22
How can we measure the distance between two segments?

Our Approach
 Step 1: Extract features from summary graphs:
Easier and more efficient than on original graphs.
No complex features

Step 2: Distance of adjacent segments
24
Edge Weights

Outline
 Motivation
 Background
 Overview
 Experiments
 Conclusion

Goal 3: Finding the best segmentation
 Observation:
For each segmentation there is a path from ‘s’ to ‘t’
For each path from ‘s’ to ‘t’ there is a segmentation
 Therefore,
• Best segmentation problem ≡ Path optimization problem

Possible approach
 Longest path?
27
S t. . .
S t
0.01 0.01 0.01 0.01
0.9 0.9 0.9
Sum = 3
Sum = 2.7
Over segmentation problem

Problem 3: Finding the best segmentation
 Our idea: Average longest path
 Advantages:
 Parameter free
 Naturally balances weight of the path with the number of segments.
28
Given a segmentation graph
Find the average longest path from ‘s’ to ‘t’

Solving ALP
 Finding the ALP in general graphs is NP-hard.
 The segmentation graph is a DAG ALP can be solved in
polynomial time
 Negative cycle detection [Waggoner et al. 2013]

Complete algorithm
30
Time complexity:

Outline
 Motivation
 Background
 Overview
 Experiments
 Conclusion

Experiments: datasets
 Different Domains with range of sizes:
 BA-degree: Random Barabasi Albert graph
 Higgs: Tweets dataset (with the follower-followee network)
 Memetracker: Who-copies-from-whom blog and website network
 DBLP: Co-authorship network related to ‘network’ topic.

Experiments: baselines
 DYNAMMO [Li et al. 2009]:
 Feature Etraction & time series
 Change point detection ( Reconstruction errors)
 # segments = # segments of SnapNETS .
 VOG [Koutra et al. 2014]:
 Get active sub-graph
 10 most important sub-structures
 Cut when the set of sub-structures changes significantly
o (threshold = the one gives the best result)
 SN-LP:
 Longest Path instead of ALP

Experiments: Quantitative analysis
34
 SnapNETS outperforms the baselines
 Clear patterns in summary graphs
We found Ground truth segmentation
As-Oregon

Case studies: Memetracker
35
Televised vice-presidential debates
 Summary graphs are close to
the case when all nodes have
the same label (f5)
 Random nodes are active (f8)
 Summary graphs are
substantially sparser (f2).
 Many active nodes got merged
into important nodes such as
CNN and BBC to form hubs (f6)

Case studies: AS-Oregon
36
 New community  New segment

Outline
 Motivation
 Background
 Overview
 Experiments
 Conclusion

Conclusion: SnapNets
 Properties:
 P1. Parameter-free
 P2. Comprehensive
 Patterns:
 the ‘placement’ and ‘connection’ of active/inactive nodes:
• structural (e.g. community/role/centrality)
• rate changes.
 Global method:
 SnapNETS is a ‘global’ method and not simply a change-point detection
method.

Future Work
 Faster ALP:  Linear?
 Handle dynamic graphs with varying
nodes and edges
 More node labels and real value features
 Work with partially observed graphs

Any questions?
40
Funding:
Code at: https://github.com/SorourAmiri/SnapNETS
Sorour E. Amiri Liangzhe Chen B. Aditya Prakash
Goal 1 Goal 2 Goal 3
Finding the best segmentation
Successively merge nodes
Keep leading eigenvalue
Keep same set of labels
Graph summarization Segmentation graph
 Nodes
 Edges
 Edge weights
ALP
SnapNETS Result

Segmenting Sequences of Node-labeled Graphs

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