Segmenting Sequences of Node-labeled Graphs

Segmenting Sequences of Node-labeled Graphs

Sorour E. Amiri, Liangzhe Chen, B. Aditya PrakashDepartment of Computer Science

Virginia Tech

ICDM, DaMNet, Barcelona, Spain, December 12, 2016

Outline Motivation Background Our Proposed Method: SnapNETS Experiments Conclusion

Amiri, Chen, Prakash 2

Network SequencesEpidemiology: disease spreads over contact networks

Social Media: Information spreads over friendship networks

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Flu

Meme

Amiri, Chen, Prakash

Making sense of network sequences

4

Flu

when do the infection patterns change?

Star Bridge Near Clique

Reason:• Virus mutation• Vaccination• …


Making sense of network sequences

5

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.

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Star Bridge Near CliqueAmiri, Chen, Prakash

Outline Motivation Background Our Proposed Method: SnapNETS Experiments Conclusion

7Amiri, Chen, Prakash

Alternative 1: Feature Ext. &Time-series

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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]


G1 G2 G3 G4-1

0

1

2

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 periodo Threshold


G1 G2 G3 G4-1

0

1

2

Features time series

F1 F2 F3

Alternative 2: Plain-graph-based analysis

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[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 11

Entire graph Active subgraph

Desirable Properties P1. Parameter-free:

• No threshold, No fixed granularity

P2. Comprehensive: • Use the entire graph


Outline Motivation Background Our Proposed Method: SnapNETS

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 edgesDefining 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

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Reminder: P1. Parameter-free P2. Comprehensive






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]

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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

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Given: A graph with labeled nodes and a compression ratio.Find: a coarsened graph such that:






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

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How can we measure the distance between two segments?Amiri, Chen, Prakash

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

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Edge Weights






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?

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S t. . .

S t0.01 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.

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Given a segmentation graphFind 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

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Time complexity:






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

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SnapNETS outperforms the baselines Clear patterns in summary graphs

We found Ground truth segmentation

As-Oregon


Case studies: Memetracker

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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

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New community New segment






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?

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Funding:

Code at: https://github.com/SorourAmiri/SnapNETS

Sorour E. Amiri Liangzhe Chen B. Aditya Prakash

Goal 1 Goal 2 Goal 3Finding the best segmentation

Successively merge nodesKeep leading eigenvalueKeep same set of labels

Graph summarization Segmentation graph Nodes Edges Edge weights

ALP

SnapNETS Result

Data & Analytics

Segmenting Sequences of Node-labeled Graphs