Using Graph Algorithms for Advanced Analytics - Part 2 Centrality

Graph Gurus 27
Using Graph Algorithms for Advanced Analytics
Part 2 - Centrality
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Today's Presenter
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Victor Lee
Head of Product Strategy
● BS in Electrical Engineering and Computer
Science from UC Berkeley, MS in Electrical
Engineering from Stanford University
● PhD in Computer Science from Kent State
University focused on graph data mining
● 20+ years in tech industry

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Today’s Outline
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1
3
2
Recap of Part 1: The What and Why of
Graph Algorithms
Demo
Running and modifying GSQL
Centrality Algorithms
Centrality Algorithms
How to select and use the right one

Graph algorithms are functions for measuring characteristics
of graphs, vertices, or relationships.
A graph algorithm library serves as a toolkit and as building
blocks for analyzing your data.
Specialized functions Combine to make
something greater
Graph Algorithms
6

Example Questions/Analyses for Graph Algorithms
Which entity is most centrally
located?
● For delivery logistics or greatest visibility
● Closeness Centrality, Betweenness
Centrality algorithms
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How much influence does this
entity exert over the others?
● For market penetration & buyer
influence
● PageRank algorithm
Which entity has similar relationships
to this entity?
● For grouping customers, products, etc.
● Cosine Similarity, SimRank, RoleSim
algorithms
What are the natural community
groupings in the graph?
● For partitioning risk groups, workgroups,
product offerings, etc.
● Community Detection, MinCut
algorithms

GSQL Graph Algorithm Library
● Written in GSQL - high-level, parallelized
● Open-source, user-extensible
● Well-documented
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docs.tigergraph.com/graph-algorithm-library

TigerGraph GSQL Graph Algorithm Library
✓ Call each algorithm as a GSQL query
or as a RESTful endpoint
✓ Run the algorithms in-database (don't
export the data)
✓ Option to update the graph with the
algorithm results
✓ Able to modify/customize the
algorithms. Turing-complete
language.
✓ Massively parallel processing to
handle big graphs
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Review: Analytics with Graph Algorithms
● Graph Algorithms answer fundamental questions about
connected data
● Each algorithm in a library is tool in an analytics toolkit.
● Building Blocks for more complex business questions
● Some questions/algorithms are faster or slower than others
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Picking the Right Algorithm for the Job
To become a master graph craftsperson:
● Learn what each tool can do
● Learn to combine building blocks to solve complex issues
This webinar series is designed to show you what each tool
can do. We'll also show you some application examples.
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Some Types of Graph Algorithms
● Search
● Path Finding & Analytics
● Centrality / Ranking
● Clustering / Community Detection
● Similarity
● Classification
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Centrally Located
A
B
C
D
E
F
H
G
In a network, which node is the most centrally
located?
● Location might not be physical
● Edges = road segment, cables
who-knows-whom
Abstract concept of Travel
● Something (goods, info) is moving across the
network
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Example Questions for Centrality
A
B
C
D
E
F
H
G
Logistics:
● Where to locate a Distribution Center?
● Where to locate a Starbucks or gas station?
Social Dynamics:
● Which entity in a social group is the most
influential? What do we mean by influence?
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Centrality for a Distribution Center
● Truck deliveries
● Deliveries might be to
any destination in the
service area.
● Complications:
○ Some destinations
are more popular
than others.
○ Multi-delivery route
with several stops
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https://www.mdpi.com/1999-4893/12/2/45/htm

Centrality for Airline Hubs
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© 2018 TigerGraph. All Rights
Reserved
Influence as Centrality
● Who are the most influential medical providers,
for a given region, for a given specialty?
○ If you change the behavior of this provider,
it will have great impact.
○ Influence through referral from one provider
to another?
● Which seller in a interrelated market has
the most pricing power?
○ Or player in a game → game theory

Some Centrality Measures
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Degree
# of neighbors (in, out, or both)
CD
(v) = degree(v)
Closeness
inverse of sum of shortest path distances
CC
(v) = (n-1) / ∑z
d(v,z)
Harmonic
sum of inverse of shortest path distances
CH
(v) = 1/(n-1) * ∑z
1/d(v,z)
Betweenness
On the most shortest paths
CB
(v) = ∑A ≠ B
{v on path(A,B)}
Eigenvector A*c = ƛ* c, A = Adjacency matrix
PageRank CP
(v) = (1-d)/N + d∑(x,v)
CP
(x)/deg(x)
Percolation https://en.wikipedia.org/wiki/Centrality#Percolation
_centrality

Closeness Centrality
Compute average distance from a vertex to every other vertex.
Closeness = inverse of distance.
19
A
B
C
D
E
F
H
G
1
2
1
1
22
2
● Weighted paths: need to run SSSP, e.g. Dijkstra's
● Directed: What if there no path between 2 vertices?
○ distance is infinite, so closeness = 0?
● Unweighted paths: count hops
○ For F, avg distance =
(2+2+1+2+1+2+1)/7 = 11/7
■ Closeness = inverse(11/7) = 0.636
○ For A, avg distance =
(1+1+1+2+2+3+3)/7 = 13/7
■ Closeness = inverse(13/7) = 0.538

Betweenness Centrality
Count how often a vertex is on a shortest path,
for APSP (All Pairs Shortest Path)
● n(n-1)/2 ~ n2
/2 pairs, if undirected
● n(n-1) ~ n2
pairs, if directed
○ But some pairs might not have a directed path
● Even for this small graph, it's too hard to
compute all the paths by inspection.
● Betweenness is harder (slower) to compute
than Closeness.
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A
B
C
D
E
F
H
G
?

PageRank Centrality
Original purpose: find the most "authoritative" page in the Web.
Search engine Ranking problem:
Engine finds many (millions) of pages which match your search terms.
Too many! Which pages to list at the top, in what order?
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PageRank's Principles
A page's authority increases if
more pages point to it.
A page is more authoritative if
the pages that point to it are
more authoritative. (Recursive!)
A B
Rank(B) > Rank(A)
C D
C1 C2 D1 D2
If Rank(D1),Rank(D2) > Rank(C1),Rank(C2), then
Rank(D) > Rank(C)
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Formulating PageRank
PageRank
○ "A node is authoritative if several authoritative nodes point to it."
○ Authority by Referral:
■ PRV
= Authority of V = sum(Referrals).
■ Referral by X = (Authority of X)/outdegree(X)
○ Random network surfer: Start anywhere. Pick a random
out-edge to go to a neighboring node. Repeat "forever."
■ PRV
= Fraction of time that the random surfer spends on node V.
Does it fit your need?
○ For directed graphs with one type of node (e.g., social network)
○ # Computational steps ~ (# connections)*(#iterations)
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PageRank Computation by Iteration
Initialization:
At time t0, set Rank(V,t0) = 1,
for every node V.
V0
V1
V5
V3
V2
V4
?
Rank(V0, t0)/3
Rank(V0,t0)
Rank(V0, t0)/3
Rank(V0, t0)/3
Iteration - 1. Consider your paths.
At time t, Surfer at V0
"gives" to out-neighbor V1
Rank(V0,t)/out-degree(V0)
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PageRank Computation by Iteration
Iteration - 2. Arrive at a neighbor:
Rank at time t+1 = sum of all
the "gifts" from in-neighbors:
Rank(V,t+1)
= Rank(V0,t)/out-degree(V0)
+ Rank(V1,t)/out-degree(V1)
+ …
+ Rank(Vn,t)/out-degree(Vn)
Repeat the 2-phase iterations until
Rank scores converge to stable
values.
V0
V1
V5
V3
V2
V4
?
Rank(V0, t0)/3
Rank(V0,t0)
Rank(V0, t0)/3
Rank(V0, t0)/3
Rank(V4, t0)/2
Rank(V5, t0)/1
25

Small Example
Time V0 V1 V2
0 1 1 1
1 0.333 1.333 1.333
2?
V0
V1
V2
0 1 0
0 0 1
0.33 0.33 0.33
Transition Matrix
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0.443 0.778 1.778
3 0.593 1.036 1.371

Same Example, Different Initial Condition
Time V0 V1 V2
0 2 1 0
1 0 2 1
2
V0
V1
V2
0 1 0
0 0 1
0.33 0.33 0.33
Transition Matrix
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0.333 0.333 2.333
3 0.778 1.111 1.111
4 0.370 1.148 1.481

Teleportation - Solves 2 Problems
Add a new way to travel:
Teleport to ANY node in the graph.
15%: pick a destination at random.
85%: follow the edges
0.85 = damping factor
Solves 2 problems:
Math: Nodes with no in-edges or
no out-edges.
Real-world: Surfers don't always
follow referrals.
Without Teleportation, PageRank
= Eigenvector Centrality
V0
V1
V5
V3
V2
V4
~15%
Teleport!
85% - Traverse
the graph as
usual
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PageRank in GSQL
CREATE QUERY pageRank (FLOAT maxChange, INT maxIter, FLOAT damping) FOR GRAPH generic
{
SumAccum<float> @score = 1; # PageRank of each vertex. Initial value = 1.
SumAccum<float> @received_score = 0; # sum of scores which each vertex receives from neighbors
MaxAccum<float> @@maxDiff = 9999; # global max score change in an iteration
Start = {Person.*};
WHILE @@maxDiff > maxChange LIMIT maxIter DO
@@maxDiff = 0;
V = SELECT s FROM Start:s -(Refer_to:e)-> :t
ACCUM t.@received_score += s.@score/(s.outdegree("Refers_to")) # traverse
POST-ACCUM s.@score = (1.0-damping) + damping * s.@received_score, # teleport
s.@received_score = 0,
@@maxDiff += abs(s.@score - s.@score'); # score' means previous score
END;
PRINT Start.@scores;
}
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Personalized PageRank
● A.k.a. Random Walk with Restart
● Make the following changes to PageRank:
○ Select one or more vertices as the Home vertices
○ Initial scores: 1 if a Home vertex. 0 otherwise
○ Teleportation: Teleport only to Home vertices
● Random Walk interpretation: It's rare to wander far from
home, so scores tend to be higher closer to Home.
30

DEMO
GSQL Graph Algorithms in TigerGraph Cloud
31

Real-World Example: Airline Routes
● Global commercial airline routes from www.openflights.org
○ 7,698 airports
○ 67,664 direct flight routes
● Closeness Centrality
○ Unweighted edges: Which airport on average requires the fewest
connections to get to ANY other airport?
○ Weighted edges: Which airport on average requires the fewest
miles traveled to get to ANY other airport?
○ Note: We can limit the search to only one region, e.g. USA
○ We can also limit the maximum number of connections
32

Real-World Example: Airline Routes
● Betweenness Centrality
○ Unweighted edges: When travelling from ANY city to ANY other
city, which airport is most often/likely to be a connection city?
● PageRank
○ Random flyer model: If you are equally likely to fly ANY route from a
given airport, which airport will be your destination most often?
● How does real-world air travel differ from these calculations?
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Edits made to Library Algorithms
● closeness_cent
○ Split into main (closeness_cent) and subquery (cc_subquery)
○ Change graph name to MyGraph
○ Change vertex type to Airport
○ Change edge type to flight_to
○ Change maxHops to be a parameter
○ Add country name filter to Start set
○ Add Airport name to output tuples for readability
● betweenness_cent
○ Split into main (betweenness_cent) and subquery (bc_subquery)
○ Change graph name, vertex type and edge type
○ Add country name filter to Start set
○ Put results in local accum so we can ORDER BY and LIMIT results
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Summary
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1
4
3
Centrality Algorithms - abstract
concepts of location and travel
Customizing GSQL Library algorithms is
easy and familiar, like procedural SQL
PageRank - uses directed referral edges
to find the most influential nodes.
Personalized PageRank is localized.
2 Closeness and Betweenness use shortest
paths. Betweenness is more complex.

Q&A
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Using Graph Algorithms for Advanced Analytics - Part 2 Centrality

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