This document discusses representing complex systems as higher-order networks (HON) to more accurately model dependencies. Conventionally, networks represent single entities at nodes, but HON breaks nodes into higher-order components carrying different relationship types. This captures dependencies beyond first order in a scalable way. The document presents applications of HON, including more accurately clustering global shipping patterns and ranking web pages based on clickstreams. HON provides a general framework for network analysis tasks like ranking, clustering and link prediction across domains involving complex trajectories, information flow, and disease spread.
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From complex Systems to Networks: Discovering and Modeling the Correct Network"
1. From complex systems to networks:
discovering and modelign the higher order network
Nitesh Chawla
Frank M. Freimann Professor of Computer Science and Engineering
Director, iCeNSA
3/2/17 11:17 AM
2. 2
Our world is complex
Ship 1 Shanghai Ă Singapore Ă Los Angeles Ă âŚ
Ship 2 Tokyo Ă Singapore Ă Seattle Ă âŚ
Ship 3 Shanghai Ă Singapore Ă Hong Kong Ă âŚ
Ship 4 Hong Kong Ă Singapore Ă Seattle Ă âŚ
⌠⌠⌠âŚ
Ship trajectories
3. 3
Our world is complex
Ship 1 Shanghai Ă Singapore Ă Los Angeles Ă âŚ
Ship 2 Tokyo Ă Singapore Ă Seattle Ă âŚ
Ship 3 Shanghai Ă Singapore Ă Hong Kong Ă âŚ
Ship 4 Hong Kong Ă Singapore Ă Seattle Ă âŚ
⌠⌠⌠âŚ
Ship trajectories Global shipping network
4. 4
Our world is complex
User 1 Company ranking Ă Job listing Ă ApplyĂ âŚ
User 2 Weather Ă Homepage Ă News Ă âŚ
User 3 News Ă Sports Ă Scores Ă âŚ
User 4 Events Ă Homepage Ă Weather Ă âŚ
⌠⌠⌠âŚ
Web page clickstreams
5. 5
Our world is complex
User 1 Company ranking Ă Job listing Ă ApplyĂ âŚ
User 2 Weather Ă Homepage Ă News Ă âŚ
User 3 News Ă Sports Ă Scores Ă âŚ
User 4 Events Ă Homepage Ă Weather Ă âŚ
⌠⌠⌠âŚ
Web page clickstreams Web traffic network
13. 13
Enriching the network
Conventionally: every node
represents a single entity
(location, state, etc.)
( )
( )
1
1 1( | ) t t
t t t t
tj
W i i
P X i X i
W i j
+
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14. 14
Enriching the network
Conventionally: every node
represents a single entity
(location, state, etc.)
Now: break down nodes into
higher-order nodes that carry
different dependency relationships
( )
( )
1
1 1( | ) t t
t t t t
tj
W i i
P X i X i
W i j
+
+ +
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= = =
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15. 15
Enriching the network
Conventionally: every node
represents a single entity
(location, state, etc.)
Now: break down nodes into
higher-order nodes that carry
different dependency relationships
( )
( )
1
1 1( | ) t t
t t t t
tj
W i i
P X i X i
W i j
+
+ +
ÂŽ
= = =
ÂŽĂĽ
( )1
( | )
| ( | )
( | )
t t
k
W i h j
P X j X i h
W i h k
+
ÂŽ
= = =
ÂŽĂĽ
16. 16
Enriching the network
Conventionally: every node
represents a single entity
(location, state, etc.)
Now: break down nodes into
higher-order nodes that carry
different dependency relationships
( )
( )
1
1 1( | ) t t
t t t t
tj
W i i
P X i X i
W i j
+
+ +
ÂŽ
= = =
ÂŽĂĽ
( )1
( | )
| ( | )
( | )
t t
k
W i h j
P X j X i h
W i h k
+
ÂŽ
= = =
ÂŽĂĽ
Compatible with existing tools!
17. 17
Fixed-order Variable-order
Assuming a fixed order
beyond the second order
becomes impractical
because âhigher-order
Markov models are more
complexâ due to
combinatorial explosion
--- Rosvall et al. (Nature
Comm. 2014)
20. 20
How to construct HON?
Raw data
⢠Sequential data
Rule
extraction
⢠Which nodes need to
be split into higher-
order nodes, and how
high the orders are
Network
wiring
⢠Connecting nodes
representing different
orders of dependency
HON
⢠Use HON like the
conventional network
for analyses
22. 22
Network wiring
A
⢠Convert all first-order rules into edges
B
⢠Convert higher-order rules
⢠Add higher-order nodes when necessary
C
⢠Rewire edges
⢠The edge weights are preserved
D
⢠Rewire remaining edges
24. 24
Higher-order dependencies revealed by HON
Data # Records
Dependencies
revealed
Similar observations
Ship movement 3,415,577 Up to 5th order N/A
Clickstream 3,047,697 Up to 3rd order
â⌠appear to saturate at k = 3 for
Yahoo⌠browsing behavior
across websites is definitely not
Markovian but can be captured
reasonably well by a not-too-high
order Markov chain.â
--- Chierichetti et al. (2012)
Retweet 23,755,810 N/A
26. 26
Invasive species
Zebra mussels @ Great Lakes
Clogging water pipes, attach to boats
Photos: Great Lakes Environmental Research Lab; TIME & LIFE Images, Getty Images
$120 billion / year
damage & control costs
35. 35
Ranking on clickstream network
⢠26% pages show more
than 10% changes in
ranking
⢠More than 90% pages lose
PageRank scores, while a
few pages gain significant
scores
No changes
to the ranking algorithm
40. 40
Summary
Data
⢠Ship movements
⢠Web clickstreams
⢠Phone call cascades
⢠⌠âŚ
Network
representation
⢠Global shipping
network
⢠Web traffic network
⢠Social network
⢠⌠âŚ
Network
analysis
⢠Clustering
⢠Ranking
⢠Link prediction
⢠Anomaly detection
⢠⌠âŚ
41. 41
Full paper
⢠Jian Xu, Thanuka L. Wickramarathne, and Nitesh V. Chawla.
"Representing Higher-order Dependencies in Networks."
Science Advances 2, e1600028 (2016)
⢠Jun Tao, Jian Xu, Chaoli Wang, and Nitesh V. Chawla. âHonVis:
Visualizing and Exploring Higher Order Networks." IEEE PacificViz,
2017.
