SnapNETS: Automatic Segmentation of Network Sequences with Node Labels

SnapNETS: Automatic Segmentation of Network Sequences

with Node Labels

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

Virginia Tech

AAAI, San Francisco, USA, February 9, 2017

Outline Motivation Alternative Approaches 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

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G1 G2 G3 G4

G1 G2 G3 G4

Uninfected

Infected

Inactive

Active

Making sense of network sequences

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Flu

when do the infection patterns change?

Star Bridge Near Clique

Reason:• Virus mutation• Vaccination• …

Amiri, Chen, Prakash

G1 G2 G3 G4Uninfected

Infected

Making sense of network sequences

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Meme Reason:• Event• …

Star Clique

when do the activation patterns change?

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G1 G2 G3 G4

Inactive

Active

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

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In this work: Binary labels {0, 1}

Desirable Properties P1. Parameter-free:

• No threshold, No fixed granularity

P2. Comprehensive: • Use the entire graph

P3. Scalable

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Outline Motivation Alternative Approaches Our Proposed Method: SnapNETS Experiments Conclusion

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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]

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G1 G2 G3 G4-1

0

1

2

Features time series

F1 F2 F3

Step 1: Feature Extraction

Step 2: Time-series segmentationG1 G2 G3 G4

…

Alternative 1: Feature Ext. &Time-series

Drawbacks: Laborious feature-engineering

o # Cliqueso # Ladders

“Local” change detection:o One aggregation time periodo Threshold

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G1 G2 G3 G4-1

0

1

2

Features time series

F1 F2 F3

G1 G2 G3 G4

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

Amiri, Chen, Prakash

Step 2: Dynamic graph segmentation

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G1 G2 G3 G4 G1 G2 G3 G4

Alternative 2: Plain-graph-based analysis

Drawbacks: Inactive nodes are important to detect different patterns

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Entire graphDynamic graph segmentation

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G1 G2 G3 G4 G1 G2 G3 G4

Chain Roles are different

Desirable Properties P1. Parameter-free:

• No threshold, No fixed granularity

P2. Comprehensive: • Use the entire graph

P3. Scalable

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Comparison of SnapNETS

Outline Motivation Alternative Approaches Our Proposed Method: SnapNETS

Main Idea and Overview Goal 1: Summarizing Act-snapshots Goal 2: Constructing the segmentation graph Goal 3: Finding the best segmentation

Experiments Conclusion

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Nodes: For each segment there is a node + {Source (‘s’), Target (‘t’)} Source (‘s’) = start time Target (‘t’) = end time

Edges: There is a directed edge between adjacent nodes

Main Idea: Segmentation graph

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Best segmentation problem Path optimization problem

Inpu

t

Segmentation G

raph

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

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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 P3. Scalable

Amiri, Chen, Prakash

Outline Motivation Alternative Approaches Our Proposed Method: SnapNETS

Main Idea and Overview Goal 1: Summarizing Act-snapshots Goal 2: Constructing the segmentation graph Goal 3: Finding the best segmentation

Experiments Conclusion

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Goal 1: Summarizing graph snapshots

We want to preserve Structural properties Nodes labels

Role of Eigenvalue:

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Epidemic threshold in most diffusion models [Prakash et al. ICDM 2011]

Same Same diffusive properties

Leading eigenvalue of Adjacency matrix

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Our summarization approach We want to get a smaller graph with similar eigenvalues:

Successively merge nodes

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

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Keep leading eigenvalue Matrix perturbation approach

Based on CoarsNet [Purohit et al. KDD 2014] Successively merge nodes Do not merge nodes with different labels

Our Approach

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

Amiri, Chen, Prakash

0.10.1 0.1

0.2

0.2

…

…

Outline Motivation Alternative Approaches Our Proposed Method: SnapNETS

Main Idea and Overview Goal 1: Summarizing Act-snapshots Goal 2: Constructing the segmentation graph Goal 3: Finding the best segmentation

Experiments Conclusion

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Nodes: For each segment there is a node + {Source (‘s’), Target (‘t’)} Source (‘s’) = start time Target (‘t’) = end time

Edges: There is a directed edge between adjacent nodes

Goal 2: Segmentation graph

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Edge Weights

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

w ?

Our Approach Step 1: Extract features from summary graphs:

Easier and more efficient than on original graphs. No complex features

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F = [3.9, 13,..., 2.2]

Step 2: Distance of adjacent segments

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Edge Weights

Amiri, Chen, Prakash

w

Outline Motivation Alternative Approaches Our Proposed Method: SnapNETS

Main Idea and Overview Goal 1: Summarizing Act-snapshots Goal 2: Constructing the segmentation graph Goal 3: Finding the best segmentation

Experiments Conclusion

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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

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Possible approach Longest path? Given a segmentation graph Find the longest path from ‘s’ to ‘t’

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Over segmentation problem

s t. . .

s t0.01 0.01 0.01 0.01

0.9 0.9 0.9

Sum = 3

Sum = 2.7Amiri, Chen, Prakash

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’

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Solving ALP Finding the ALP in general graphs is NP-hard. The segmentation graph is a DAG ALP can be solved in

polynomial time State-of-the-art algorithm [Waggoner et al. WACV 2013]

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Time complexity:

Cubic: Not scalable!

Our Solution: LAYERED-ALP

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Dynamic Programming Optimal solution

lp1 = Longest path with 1 segment

lp2 = Longest path with 2 segments

lp4 = Longest path with 4 segments

Our Solution: LAYERED-ALP

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Time Complexity:

Linear!

Build Layers

Find LP in each layer

Find ALP

Complete algorithm

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Time complexity:

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Sub-quadratic

Complete algorithm: Parallel

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Time complexity:

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Outline Motivation Alternative Approaches Our Proposed Method: SnapNETS

Main Idea and Overview Goal 1: Summarizing Act-snapshots Goal 2: Constructing the segmentation graph Goal 3: Finding the best segmentation

Experiments Conclusion

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Experiments: datasets Different Domains with range of sizes:

BA-degree: Random Barabasi Albert graph AS-Oregon: Autonomous Systems peering information Higgs: Tweets dataset (with the follower-followee network) Portland: Contact network between people of Portland Memetracker: Who-copies-from-whom blog and website network IranElect: Follower-followee network of Twitter related to the Iran

election. DBLP: Co-authorship network related to ‘network’ topic.

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Experiments: baselines DYNAMMO [Li et al. KDD 2009]:

Change point detection ( Reconstruction errors) # segments = # segments of SnapNETS .

K-means [Likas et al. Pattern Recognition 2003]: segment when a new cluster is detected

VOG [Koutra et al. SDM 2014]: 10 most important sub-structures Cut when the set of sub-structures changes significantly

o (threshold = the one gives the best result)

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Feature Extraction & time series

Dynamic graph

Experiments: baselines-variations SN-ORIG: Original graphs instead of summary graphs SN-LP: Longest Path instead of ALP SN-GREEDY: Greedy Approach instead of ALP

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Experiments: Quantitative analysis

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SnapNETS outperforms the baselines Clear patterns in summary graphs

Infection moves to new community

As-Oregon

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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)

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Can I call you joe?

Case studies: AS-Oregon

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New community New segment

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Scalability

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Scalability of SNAP NETS Speedup by parallelizing construction of segmentation graph

Near-linear

Outline Motivation Alternative Approaches Our Proposed Method: SnapNETS

Main Idea and Overview Goal 1: Summarizing Act-snapshots Goal 2: Constructing the segmentation graph Goal 3: Finding the best segmentation

Experiments Conclusion

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Discussion: SnapNets 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.

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Graph summarization and features

Average Longest Path

Properties: P1. Parameter-freeP2. ComprehensiveP3. Scalable

Future Work Handle dynamic graphs with varying

nodes and edges More node labels and real valued features Work with partially observed graphs

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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