1 applications of relative importance why is relative importance interesting? web social networks...
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Applications of Relative Importance
Why is relative importance interesting? Web Social Networks Citation Graphs Biological Data
Graphs become too complex for manual analysis
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Existing Techniques Web
PageRank (Google) Social Networks
‘Centrality’
All focus on global measures of node importance – we’re interested in importance relative to a set of root nodes R
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Use Existing Techniques?
Use global algorithm on the subgraph surrounding root nodes?
No preferential treatment of root nodes – just ranking surrounding nodes.
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Organization: Relative importance Algorithms
Notation Problem Formulation General Framework Algorithms
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Notation Digraph
G = (V, E) Edges
Ordered pair of nodes (u, v) Graphs are directed, unweighted, simple Walks from u to v
a.k.a. A walk is a path with no repeated nodes
1 2 ... ku u u u v 1 1 2( , ),( , ),...,( , )ku u u u u v
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Notation k-short paths P(u,v) – set of paths between u and v – set of distinct out-going edges from
u Similarly, we have
( )outS u( ) ( )out outd u S u
( ) ( )in ind u S u
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Problem Formulation
1. Given G and r and t, where , compute the “importance” of t w.r.t. root node r:
{r,t} G
|I t r
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Problem Formulation
2. Given G and node , rank all vertices in T(G), T V, w.r.t. r.
r G
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Problem Formulation
3. Given G, a set of nodes T(G) to rank, and a set of root nodes R(G) where R V, rank all vertices in T w.r.t. R.
This is similar to the last case, except that we compute rather than
Average importance:
|I t r |I t R
1| |
r R
I t R I t rR
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Problem Formulation (3 cont’d.) Rather than average each node’s
importance score, we could define
This requires ‘important’ nodes to have a high importance score among all nodes in R
| min | :I t R I t r r R
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Problem Formulation
4. Given G, rank all nodes where R=T=V.
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General Framework:Weighted Paths
Nodes are related according to the paths that connect them
The longer the path, the less importance:
is a scalar coefficient,
P(r,t) is a set of paths from r to t, pi is the ith path in P.
Importance decays exponentially
,
1
|
i
P r tp
i
I t r 1
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How to choose P(r,t)?
Path examples
A
R
D
E
F
T
C
B
A
R
D
E
F
T
C
B
a. b.
Shortest pathsfrom R to T:{R-C-T. R-D-T}which fail to capture much ofConnectivity fromR to T.
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Shortest Path
e.g.: Transport cargo from r to t
Shortest path doesn’t always give a good approximation of importance. E.g: the web (graph b)
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k-Short Paths Paths of length K Idea: there might often be longer paths than the shortest ones that are
important to take into account Fixes problem of longer, important
paths in Shortest Paths e.g.: graph b., 3-short
Problem: capacity constraints e.g.: network topology
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k-Short Node-Disjoint Paths
No nodes and no edges are repeated Implicitly enforces capacity constraints Motivated by ‘mass flow’ where
importance can ‘flow’ along paths e.g.: graph b.
Breadth-first with some heuristic, with some K and some
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Markov Chains & Relative Importance
Graph viewed as a stochastic process Explanation of Markov Chains Token traversing Chain… Obviously good for modeling the web
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Markov Chains & Relative Importance
Markov Centrality Mean First Passage Time
: expected number of steps until first arrival at node t starting at node r : probability that the chain first returns to
state t in exactly n steps
1
( )rt rtn
m nf n
rtm
( )rtf n
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Markov Chains & Relative Importance
Bias toward ‘central nodes’ COMPLEX!!
Time: O(|V|3) (inversion of |V|x|V| transition matrix)
Space: O(|V2|)
1( | )
1rt
r R
I t Rm
R
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Markov Chains & Relative Importance
PageRank Uses backlinks to assign importance to
web pages
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Markov Chains & Relative Importance
PageRank Less complex
Converges logarithmically 322 million links
processed in 52 iterations
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Markov Chains & Relative Importance
Retrofit PageRank such that all nodes in R have a uniform bias at the start
‘Surfer’ begins at a root node, traverses graph, returning to root set R with probability at each time-step
I(t|R) = probability that surfer visits t during a walk
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Experiments (Simulated Data)
D F
E
J
C HA
B
G
I
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Experiments (Simulated Data)
D F
E
J
C HA
B
G
I
More complex in and out degrees
changed Shortest path
lengths between nodes changed (e.g.: A-B)
Analysis which follows, R={A,F}
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Experiments (Simulated Data)
D F
E
J
C HA
B
G
I
HITSPaA .252F .241G .128C .110E .099H .052D .032J .025I .032B .024
HITSPhF .225A .186D .162B .119E .090I .067H .061J .050G .028C .008
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Experiments (Simulated Data)
D F
E
J
C HA
B
G
I
MarkovCJ .180C .133G .130H .129E .111I .101F .069D .051A .047B .044
KSMarkovH .146G .142E .142J .140C .120I .098F .087D .061A .034B .024
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Experiments (9/11 Terrorist Network)
63 nodes (terrorists) 308 edges (interactions)
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Rank PRankP HITSP WKPaths MarkovC KSMarkov
1 Khemais Khemais Beghal Atta Khemais
2 Beghal Beghal Khemais Al-Shehhi Beghal
3 Moussaoui Atta Moussaoui Al-Shibh Moussaoui
4 Maaroufi Moussaoui Maaroufi Moussaoui Maaroufi
5 Qatada Maaroufi Bensakhria Jarrah Qatada
6 Daoudi Qatada Daoudi Hanjour Daoudi
7 Courtaillier Bensakhria Qatada Al-Omari Bensakhria
8 Bensakhria Daoudi Walid Khemais Courtaillier
9 Walid Courtaillier Courtaillier Qatada Walid
10 Khammoun Khammoun Khammoun Bahaji Khammoun
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Conclusion
Provides a first-step to addressing ‘relative-importance’
Scaling for algorithms such as Markov Chaining can be an issue
Using different algorithms and comparing results can reveal interesting information
…Paper Analysis…
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References White, Smyth. Algorithms for Estimating Relative
Importance in Networks. SIGKDD ’03. Page, Brin, Motwani, Winograd. The PageRank Citation
Ranking: Bringing Order to the Web. Stanford University, Computer Science Department Technical Report.
Wikipedia on Markov Chains http://en.wikipedia.org/wiki/Markov_chain http://en.wikipedia.org/wiki/Examples_of_Markov_chains