The shortest path is not always a straight line

THE SHORTEST PATH IS NOT ALWAYS A STRAIGHT LINE

leveraging semi-metricity in large-scale graph analysis

Vasiliki Kalavri (kalavri@kth.se) KTH Royal Institute of TechnologyTiago Simas (tiago.simas@telefonica.com) Telefonica Research Dionysios Logothetis (dionysios@fb.com) Facebook

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Alice42 likes

Weighted graphs capture relationship strength

distance

similarity social proximity

rating preference

influential nodes

optimal propagation paths

communities

recommendations

BobMax

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Sparsification techniques reduce the graph size and still give exact or good

approximate results

G G’f(G) ~ f(G’)

THE METRIC BACKBONE

Reduces the graph size while maintaining relevant structure

The minimum subgraph of a weighted graph, that preserves the shortest paths of the original graph

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WHAT CAN WE USE IT FOR?• Exact computations

• any algorithm that depends on the shortest paths• reachability, connectivity• betweenness centrality, closeness centrality

• Approximation• PageRank, random walks• eigenvector centrality• community detection, clustering

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WHAT CAN WE USE IT FOR?• Exact computations

• any algorithm that depends on the shortest paths• reachability, connectivity• betweenness centrality, closeness centrality

• Approximation• PageRank, random walks• eigenvector centrality• community detection, clustering

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Improves community detection modularity and recommender

systems accuracy

IMPACT ON LARGE-SCALE SYSTEMS• Graph Databases

• fewer edges => smaller path search space

• Batch Graph Processing• CPU and memory requirements depend on #messages

• #messages proportional to #edges

• fewer edges => improved analysis performance

• Graph Compression• fewer edges => storage reduction

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BACKGROUND

SEMI-METRICITYIn a weighted graph, an edge is semi-metric, if there exists a shorter indirect path between its endpoints

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SEMI-METRICITYIn a weighted graph, an edge is semi-metric, if there exists a shorter indirect path between its endpoints

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CE is 1st-order semi-metric:

C-D-E is a shorter2-hop path

SEMI-METRICITYIn a weighted graph, an edge is semi-metric, if there exists a shorter indirect path between its endpoints

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AD is 2nd-order semi-metric:

A-B-C-D is a shorter 3-hop path

CE is 1st-order semi-metric:

C-D-E is a shorter2-hop path

SEMI-METRICITYIn a weighted graph, an edge is semi-metric, if there exists a shorter indirect path between its endpoints

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CE is 1st-order semi-metric:

C-D-E is a shorter2-hop path

AD is 2nd-order semi-metric:

A-B-C-D is a shorter 3-hop path

AB, BC, CD, DE are metric

BACKBONE ALGORITHM

BACKBONE CALCULATION• Calculating the backbone:

• find all semi-metric edges: 1 BFS per edge?• compute APSP and store O(N2) paths

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BACKBONE CALCULATION• Calculating the backbone:

• find all semi-metric edges: 1 BFS per edge?• compute APSP and store O(N2) paths

Can we calculate or approximate the backbone

without solving APSP?

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ORDER OF SEMI-METRICITY

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ORDER OF SEMI-METRICITY

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Most semi-metric edges are1st-order semi-metric

A 3-PHASE BACKBONE ALGORITHM

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Find 1st-order semi-metric edges: only look at triangles

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A 3-PHASE BACKBONE ALGORITHM

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Find 1st-order semi-metric edges: only look at triangles

1. Scalable & practicalfor large graphs

EXAMPLE

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EXAMPLE

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Phase 1

EXAMPLE

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Phase 1

A 3-PHASE BACKBONE ALGORITHM

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Find 1st-order semi-metric edges: only look at triangles

1. Scalable & practicalfor large graphs

A 3-PHASE BACKBONE ALGORITHM

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Find 1st-order semi-metric edges: only look at triangles

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Identify metric edges in 2-hop paths

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Scalable & practicalfor large graphs

A 3-PHASE BACKBONE ALGORITHM

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Find 1st-order semi-metric edges: only look at triangles

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Identify metric edges in 2-hop paths

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Scalable & practicalfor large graphs

Most semi-metric edgeshave been removed

EXAMPLE

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Phase 2

EXAMPLE

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Phase 2

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The lowest-weight edge of every vertex is metric

EXAMPLE

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Phase 2

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The lowest-weight edge of every vertex is metric

uv2

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any indirect pathfrom u to vwould have

larger weight

EXAMPLE

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Phase 2

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The lowest-weight edge of every vertex is metric

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any indirect pathfrom u to vwould have

larger weight

A 3-PHASE BACKBONE ALGORITHM

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Find 1st-order semi-metric edges: only look at triangles!

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Identify metric edges in 2-hop paths

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Scalable & practicalfor large graphs!

Most semi-metric edgeshave been removed

A 3-PHASE BACKBONE ALGORITHM

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Find 1st-order semi-metric edges: only look at triangles!

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Identify metric edges in 2-hop paths

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Run a BFS for remaining unlabeled edges.

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Scalable & practicalfor large graphs!

Most semi-metric edgeshave been removed

A 3-PHASE BACKBONE ALGORITHM

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Find 1st-order semi-metric edges: only look at triangles!

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Identify metric edges in 2-hop paths

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Run a BFS for remaining unlabeled edges.

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Scalable & practicalfor large graphs!

1%-9% edges

Most semi-metric edgeshave been removed

EXAMPLE

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Phase 3

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BFS

EXAMPLE

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Phase 3

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BFS

Explore paths with shorter

distances only

EXAMPLE

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Phase 3

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BFS

Explore paths with shorter

distances only

If the BFS arrives at the target, the edge

is semi-metric

EXAMPLE

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Metric Backbone

DISTRIBUTED IMPLEMENTATION

code available: http://grafos.ml/okapi.html#analytics

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Implementation in the vertex-centric model

EVALUATION

EVALUATION GOALS

• How does our algorithm compare to APSP?

• Are large, real-world graphs semi-metric?

• Can we improve graph analysis performance?

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COMPARISON TO APSPComputing APSP in Giraph• multiple SSSPs• multiple MSSPs, i.e. SSSPs from

several sources in parallel

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COMPARISON TO APSPComputing APSP in Giraph• multiple SSSPs• multiple MSSPs, i.e. SSSPs from

several sources in parallel

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In the order of months for million-edge graphs

COMPARISON TO APSPComputing APSP in Giraph• multiple SSSPs• multiple MSSPs, i.e. SSSPs from

several sources in parallel

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In the order of months for million-edge graphs

In the order of days for million-edge graphs

COMPARISON TO APSPComputing APSP in Giraph• multiple SSSPs• multiple MSSPs, i.e. SSSPs from

several sources in parallel

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In the order of months for million-edge graphs

In the order of days for million-edge graphs

Our algorithm is 120-180x faster than SSSPand 11-14x faster than MSSP: order of hours for million-edge graphs

ALGORITHM PHASES

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Phase 1 Phase 2 Phase 3

ALGORITHM PHASES

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Phase 1 Phase 2 Phase 3

Very fastand scalable

ALGORITHM PHASES

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Phase 1 Phase 2 Phase 3

Very fastand scalable

Removes up to 90%of semi-metric edges

ALGORITHM PHASES

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Phase 1 Phase 2 Phase 3

Very fastand scalable

Removes up to 90%of semi-metric edges

Moderately fast

ALGORITHM PHASES

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Phase 1 Phase 2 Phase 3

Very fastand scalable

Removes up to 90%of semi-metric edges

Moderately fast

Labels up to 60%of the unlabeled edges

ALGORITHM PHASES

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Phase 1 Phase 2 Phase 3

Very fastand scalable

Removes up to 90%of semi-metric edges

Moderately fast

Labels up to 60%of the unlabeled edges

Slow

ALGORITHM PHASES

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Phase 1 Phase 2 Phase 3

Very fastand scalable

Removes up to 90%of semi-metric edges

Moderately fast

Labels up to 60%of the unlabeled edges

Slow

Labels up to 1-9%of the total edges

ALGORITHM PHASES

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Phase 1 Phase 2 Phase 3

Very fastand scalable

Removes up to 90%of semi-metric edges

Moderately fast

Labels up to 60%of the unlabeled edges

Slow

Labels up to 1-9%of the total edges

Phase 1 is the fastest and most useful phase

PHASE 1 SCALABILITY

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PHASE 1 SCALABILITY

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<200s on a billion-edge graph

PHASE 1 SCALABILITY

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almost linear scalability

<200s on a billion-edge graph

SEMI-METRICITY IN REAL GRAPHS

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Graph |V| |E| metric semi-metricity

Facebook 190M 49.9B custom 26.5%Twitter 40M 1.5B jaccard 39%Tuenti 12M 685M jaccard 59%

Livejournal 4.8M 34M jaccard 40%NotreDame 0.3M 1.5M jaccard, adamic 45%-29%

DBLP 318K 1M jaccard, adamic 23%-9%Twitter-ego 81K 1.7M jaccard, adamic 57%-39%Movielens 1.6K 1.9M jaccard 88%

Facebook 1K 143K #messages, message size 78%-77%

US-Airports 0.5K 6K #passengers 72%C-Elegans 0.3K 2.3K #connections 17%

SEMI-METRICITY IN REAL GRAPHS

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Graph |V| |E| metric semi-metricity

Facebook 190M 49.9B custom 26.5%Twitter 40M 1.5B jaccard 39%Tuenti 12M 685M jaccard 59%

Livejournal 4.8M 34M jaccard 40%NotreDame 0.3M 1.5M jaccard, adamic 45%-29%

DBLP 318K 1M jaccard, adamic 23%-9%Twitter-ego 81K 1.7M jaccard, adamic 57%-39%Movielens 1.6K 1.9M jaccard 88%

Facebook 1K 143K #messages, message size 78%-77%

US-Airports 0.5K 6K #passengers 72%C-Elegans 0.3K 2.3K #connections 17%

% 1st-order semi-metric edges =>

reduction in memory and communication

QUERY SPEEDUP ON NEO4J

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6.7x speedup

APACHE GIRAPH SPEEDUP

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Including the time to calculate the backbone

4x speedup

APACHE GIRAPH SPEEDUP

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6x speedup

COMMUNICATION REDUCTION

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Up to 70% for highly semi-metric graphs

BEST PRACTICESWhen to use the backbone?

• semi-metric weighting schemes, e.g. neighborhood similarity• we can amortize the overhead: e.g. many algorithms on the same graph,

multiple distance queries• lossy compression is ok

When not to use the backbone?

• for metric weighting schemes• we need to run one-off analysis• we need lossless compression

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RECAP: MAIN CONTRIBUTIONS

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• An algorithm for computing the metric backbone without solving APSP

• An open-source distributed implementation• Graph query and graph analytics speedup on

Neo4j and Apache Giraph

THE SHORTEST PATH IS NOT ALWAYS A STRAIGHT LINE

leveraging semi-metricity in large-scale graph analysis

Vasiliki Kalavri (kalavri@kth.se) KTH Royal Institute of TechnologyTiago Simas (tiago.simas@telefonica.com) Telefonica Research Dionysios Logothetis (dionysios@fb.com) Facebook