Mining attributed static graphs

Céline ROBARDET

Mon, 29.09. Resource-aware Machine Learning

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Different kinds of torrents of data .

Resource-aware Machine Learning

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Potential increase of our knowledge .


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Viewed as attributed dynamic graphs .


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Mining network data .

Network data brings several questions: • Working with network data is messy • Not just “wiring diagrams” but also dynamics and data (features, attributes) on nodes and edges • Computational challenges • Large scale network data • Algorithmic models as vocabulary for expressing complex

scientific questions • Social science, physics, biology +

Understanding how network structure and node attribute values relate and affect each other.


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Database versus Inductive queries Database queries: retrieve transactions that match a search criteria

+

. Data mining queries: retrieve sets of data, called patterns, that match some criteria

the criteria is computed on individual data or on sets of data


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Constraint-based pattern mining .

Developed for extracting itemsets in transaction databases


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Can be used to analyze graphs . Graphs that are often • Dynamic: when nodes and edges appear/disappear through time • Attributed: when another relation describes the nodes or the

edges themselves Relational graph +

Nodes are uniquely identified through time


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Some inductive queries on graphs . • What are the vertex

attributes that strongly co-vary with the graph structure?

Co-authors that published at ICDE with a high degree and a low clustering coefficient Resource-aware Machine Learning

• What are the sub-graphs

whose vertex attributes evolve similarly?

Airports whose arrival delays increased over the three weeks following Katrina hurricane

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Outline of the talk . Constraint-based pattern mining framework Topological patterns in static attributed graphs Mining dynamic graphs Trend dynamic sub-graphs Triggering patterns of topology changes Conclusions and perspectives


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.

Constraint-based pattern mining framework



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Pattern mining .

A Pattern φ describes a subgroup of the data D • observed several times • or associated with characteristic properties

The pattern shape is fixed: φ ∈ L +

whose cartinality is exponential or infinite in the size of the data

C evaluates the adequacy of the pattern to the data C(φ, D) → Boolean

Pattern mining task: Find all interesting subgroups Th(L, D, C) = {φ ∈ L | C(φ, D) is true } Resource-aware Machine Learning


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Pattern constraints .

Constraints are needed for: • only retrieving patterns that describe an interesting subgroup of

the data • making the extraction feasible

Constraint properties are used to infer constraint values on (many) patterns without having to evaluate them individually. Ü

They are defined up to the partial order ⪯ used for listing the patterns



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Constraint properties - 1 .

Monotone constraint

Anti-monotone constraint

∀φ1 ⪯ φ2 , C(φ1 , D) ⇒ C(φ2 , D)

∀φ1 ⪯ φ2 , C(φ2 , D) ⇒ C(φ1 , D)

abcde

abcde

abcd abce abde acde bcde


abc

abd

abe

acd

ace

ade

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bde

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C(φ, D) ≡ b ∈ φ ∨ c ∈ φ Resource-aware Machine Learning

a

b

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d

e

C(φ, D) ≡ a ̸∈ φ ∧ c ̸∈ φ


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Constraint properties - 2 .

Loose AM constraints

Convertible constraints

⪯ is extended to the prefix order ≤ so C(φ, D) ⇒ ∃e ∈ φ : C(φ \ {e}, D) that ∀φ1 ≤ φ2 , C(φ2 , D) ⇒ C(φ1 , D) abcde

abcde



abc

abd

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C(φ, w) ≡ avg(w(φ)) > σ w(a) ≥ w(b) ≥ w(c) ≥ w(d) ≥ w(e) Resource-aware Machine Learning

a

e

b

c

d

e

C(φ, w) ≡ var(w(φ)) ≤ σ

Pei and Han - 2000 Constraint-based pattern mining framework

Bonchi and Lucchese - 2007 15 / 78

Enumeration strategy .

Binary partition: the element 'a' is enumerated abcde


abc

abd

abe

acd

ace

ade

bcd

bce

bde

cde

ab

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ad

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a


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Enumeration strategy .

Binary partition: the element 'a' is enumerated R∨

R∨

.

a ∈ R∨ \ R∧

R∧ ∪ {a} R∨ \ {a}

R∧

R∧



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Constraint evaluation .

Monotone constraint R∨ C(R∨ , D) is false . empty

R∧


Anti-monotone constraint R∨

. empty C(R∧ , D) is false R∧


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R∧





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R∧





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R∧





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A new class of constraints .

Piecewise monotone and anti-monotone constraints 1.

C involves p times the pattern φ: C(φ, D) = f(φ1 , · · · φp , D)

2.

fi,φ (x) = (φ1 , · · · φi−1 , x, φi+1 , · · · , φp , D)

3.

∀i = 1 . . . p, fi,φ is either monotone or anti-monotone: { fi,φ (x) ⇒ fi,φ (y) iff fi,φ is monotone ∀ x ⪯ y, fi,φ (y) ⇒ fi,φ (x) iff fi,φ is anti-monotone


L. Cerf, J. Besson, C. Robardet, J-F. Boulicaut - 2009 Constraint-based pattern mining framework 18 / 78

An example . • ∀e, w(e) ≥ 0 • C(φ, w) ≡ avg(w(φ)) > σ ≡

∑

w(e) |φ|

e∈φ

> σ.

C(φ, D) is piecewise monotone and anti-monotone with ∑ e∈φ1 w(e) f(φ1 , φ2 , D) = |φ2 | ∀x ⪯ y, • f1,φ is monotone: f(x, φ2 , D) = • f2,φ is anti-monotone: ∑

f(φ1 , y, D) =

e∈φ1


|y|

w(e)

∑

e∈x w(e) |φ2 |

∑

>σ⇒

e∈φ1

|x|

w(e)

∑

>σ⇒

w(e) |φ2 |

e∈y

>σ

>σ


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Piecewise constraint exploitation . R∨

Evaluation

If f(R∨ , R∧ , D) = then R is empty.

∑

e∈R∨ w(e) |R∧ |

≤σ

. empty

R∧

Propagation • ∃e ∈ R∨ \ R∧ such that f(R∨ \ {e}, R∧ , D) ≤ σ, then e is moved in

R∧

• ∃e ∈ R∨ \ R∧ such that f(R∨ , R∧ ∪ {e}, D) ≤ σ, then e is removed

from R∨



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Algorithmic principles . Function Generic_CBPM_enumeration(R∨ , R∧ ) 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11:

if Check_constraints(R∧ , R∨ ) then (R∧ , R∨ )← Constraint_Propagation(R∧ , R∨ ) if R∧ = R∨ then output R∧ else for all e ∈ R∨ \ R∧ do Generic_CBPM_Enumeration(R∧ ∪ {e}, R∨ ) Generic_CBPM_Enumeration(R∧ , R∨ \ {e}) end for end if end if



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Case studies . Mining of • Formal concepts [IDA journal 05] • Fault-tolerant patterns [KDID 05, ICCS 06] • Closed patterns in n-ary relations [SDM 08] • Parallel episodes [SDM 09] • Subspace clustering [SDM 09] • Topological patterns in static attributed graphs [TKDE 13] • Evolution patterns in dynamic graphs [ICDM 09] • Trend dynamic sub-graphs [DS 12, PKDD 13] • Triggering patterns [ASONAM 14]



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.

Topological patterns in static attributed graphs



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Outline . Constraint-based pattern mining framework Topological patterns in static attributed graphs Topological properties Rank correlation constraint Mining dynamic graphs Trend dynamic sub-graphs Triggering patterns of topology changes Conclusions and perspectives Resource-aware Machine Learning


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Mining attributed static graphs .

• Composed by two relations • Are these two relations

linked? + Do the attribute values depend on the role played by the vertex in the graph?

New pattern domain that identifies sets of node attributes that co-vary with the graph structure +

the graph structure is captured by topological properties


A. Prado, M. Plantevit, C. Robardet, J-F. Boulicaut - 2013 Topological patterns in static attributed graphs 25 / 78

Topological node properties .

Topological properties are derived from the edges of the graph • Direct neighborhood ▶

degree, clustering

• Connection to all other

vertices ▶ Centrality measures Betw. Cent. EigenVector Cent. Size of the community Clust. Coeff. Degree cent. Microscopic View


Macroscopic View Closeness Cent.

PageRank Size of the largest quasi-cliques #quasi-cliques involving v

Vertex attributes


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Topological patterns . h i t 2 5.1 10

A h 9

Topological pattern language

i 3.5

t 11

h i t 25 1.5 18

0 h

i 2 5.2

h 17

ms ≡ (m, s) ∈ M × {+, −}

• L = 2M×{+,−}

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1

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0

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14

h 10

12

Betweenness centrality


i 4

0

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G

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54

h 0

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D

• M × {+, −}

i 4.25

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B

• M: set of all properties +

h 7

0

L

i 3

1

h i 6 4.5

t 4

t 5

Pattern examples .

Co-authorship network • Node attributes: number of publications in some conferences

(KDD, ICDM, VLDB, etc.).

Patterns that can be discovered • The higher the number of publications at VLDB and SIGMOD, the

higher the centrality value. • The higher the number of publications at SAC • The lower the number of publications at KDD and ICDM. • The lower the centrality value.



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How to extract such patterns? . “Propositionalization": the topological properties are associated to each node.

+

Tabular data



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Topological constraint .

Sets of properties that behave in a similar manner for a large number of vertex pairs • Kendall tau rank correlation coefficient + +

based on the ranks of pattern property values counts the number of vertex pairs that are ordered similarly on all pattern properties

Age .

. Income

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Example .

P = {KDD+ , ClusCoef− } • Supported by 5 pairs • Ex: (A2,A1) • suppτ (P) =

 5  

5 2




=

1 2


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Constraint properties .

• Let P ∈ L, Suppτ (P) = {

▷s ≡

< when s is + > when s is −

|{(u,v)∈V2 | ∀ms ∈P:m(u)▷s m(v)}|

(2n)

with

and n is the number of vertices

Ctopo (P, D) ≡ Suppτ (P) ≥ σ

• Ctopo is anti-monotone • Redundant symmetrical patterns

Suppτ (A+ , B− ) = Suppτ (A− , B+ ) • Support computation quadratic on the number of vertices: + +

An upper bound to avoid, in linear time, some support computation ECLAT mining algorithm with range trees to ease the search of supporting vertices



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• Let P ∈ L, Suppτ (P) = {

▷s ≡

< when s is + > when s is −

|{(u,v)∈V2 | ∀ms ∈P:m(u)▷s m(v)}|

(2n)

with

and n is the number of vertices

Ctopo (P, D) ≡ Suppτ (P) ≥ σ

• Ctopo is anti-monotone • Redundant symmetrical patterns

Suppτ (A+ , B− ) = Suppτ (A− , B+ ) • Support computation quadratic on the number of vertices: + +

An upper bound to avoid, in linear time, some support computation ECLAT mining algorithm with range trees to ease the search of supporting vertices



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Two case studies .

Movie Graph

DBLP Graph

• Netflix and IMDb (1998-2005). • Users rate movies (1 to 5 stars). • 5,972 nodes (movies). • 64 338 edges (common actor). • 5 local attributes: Release year, num_users (that rate the movie), avg_rating, stdev_rating, et num_actors • 9 topological properties


• 42 252 nodes. • 210 320 edges (common

publication). • 29 local attributes: Nb of publications (1990-2013) in 29 journals and conference venues • 9 topological properties


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Some results . ä

{avg_rating+ , num_customers+ }

“People rate the films they like." ä

{num_customers+ , Degree+ } “People rate movies with major actors." ( e.g., R de Niro, S. Connery, and T. Hanks )

ä

{stdev_rating+ , PageRank− } “Controversial movies are isolated."



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Some results . ä


“People rate the films they like." ä

{num_customers+ , Degree+ } “People rate movies with major actors." ( e.g., R de Niro, S. Connery, and T. Hanks )

ä

{stdev_rating+ , PageRank− } “Controversial movies are isolated."



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Some results ä


.

“People rate the films they like." ä ä

{num_customers+ , Degree+ } “People rate movies with major actors." ( e.g., R de Niro, S. Connery, and T. Hanks ) {stdev_rating+ , PageRank− } “Controversial movies are isolated."



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Results on DBLP - 1 .

Is publishing at SAC penalizing? • {SAC+ , ECML/PKDD− }, {SAC+ , KDD− }, {SAC+ , VLDB− } • {SAC+ , PageRank− } • Of course not! • Bias in the data (SAC has a much wider spectrum than databases

and data mining).



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Is publishing at SAC penalizing? • {SAC+ , ECML/PKDD− }, {SAC+ , KDD− }, {SAC+ , VLDB− } • {SAC+ , PageRank− } • Of course not! • Bias in the data (SAC has a much wider spectrum than databases

and data mining).



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{Prk+ , Deg+ , Betw+ , ClusCoef− } • Dense part ≡ database • Other parts ≡ NLP, ML

Conclusion: These behaviors are not specific to a community!



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PageRank and conferences . Top 5 publications related to the emergence of {Deg+ } and {Betw+ } for Prk+ (A) and the top 5 authors (B) (A) Rank 1 2 3 4 5

Deg Publication ECML/PKDD+ IEEE TKDE+ PAKDD+ DASFAA+ ICDM+

Factor 2.5 2.28 2.21 2.09 1.95


Between Publication Factor PVLDB+ 5.67 EDBT+ 5.11 VLDB J.+ 4.35 SIGMOD+ 4.25 ICDE+ 3.42

(B) Prk+ Deg+ Prk+ Between+ + + ECML/PKDD PVLDB Christos Faloutsos Gerhard Weikum Jiawei Han Jiawei Han Philip S. Yu David Maier Bing Liu Philip S. Yu C. Lee Giles Hector GarciaMolina


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Mining dynamic graphs



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Outline . Constraint-based pattern mining framework Topological patterns in static attributed graphs Mining dynamic graphs Constrained subgraphs in static graphs Temporal relationships Trend dynamic sub-graphs Triggering patterns of topology changes Conclusions and perspectives Resource-aware Machine Learning


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Objective .

• Study the evolution of a

graph over time • Capture strong interactions

in the graph and their evolution over time + Evolving communities of social dynamic graphs



C. Robardet - ICDM 2009 40 / 78

The data . Time 1

Time 2

Time 3

Focus on the microscopic level and propose a constraint-based mining approach to uncover evolving patterns.



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Mine subgraphs in each static graph . Time 1

Time 2

Time 3

⇓

⇓

⇓

Compute highly connected subgraphs that are also isolated.



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Pseudo-clique constraint . Subgraphs of interest are usually those made of vertices that have a high density of edges. ⇒ Cliques are subgraphs with maximal density ⇒ Pseudo-cliques relax this strong property using a user-defined threshold σ ∈ [0, 1]: S is a pseudo-clique iff

2|ES | ≥σ |S|(|S| − 1)

Properties The pseudo-clique constraint is not anti-monotonic: Expanding a subgraph by adding a vertex could make the density increase or decrease. Resource-aware Machine Learning


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Loose anti-monotone constraint .

Pseudo-cliques can always be grown from a smaller pseudo-clique with one vertex less [F. Zhu et al. (PAKDD 2007)] • Let S be a pseudo-clique • Let v⋆ be a vertex of S having the smallest degree on S • Thus S \ {v⋆ } is also a pseudo-clique σ=

2 3


S \ {v⋆ }

S


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Other useful constraints .

Maximality and isolated constraint Not all the pseudo-cliques of a graph are of importance: • Some are redundant (because non maximal) • Others have many links to external vertices

The isolation constraint imposes a maximum to the average number of external links per vertex: ∑ u∈S (deg(u) − degS (u)) ≤γ |S|



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Other useful constraints . σ = 0.7

Without Isolated constraint

9 patterns


With γ = 1

1 pattern


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Temporal relationships among subgraphs .

Time 1

Time 2

Time 3

⇓

⇓

⇓

⇒

Combine patterns from consecutive time stamps to construct a global model of the dynamic of the graphs.



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Global model of dynamic graph .

Objective

Structured the numerous pseudo-cliques obtained to answer the following questions: • Do the strong interactions growth, diminish or remain stable over

time? • When do the change occur?

Temporal relationships between time consecutive subgraphs • Stability: S remains the same between t − 1 and t • Growth: S enlarges at time t • Diminution: S shrinks at time t • Extinction: S disappears at time t • Emergence: S appears at time t Resource-aware Machine Learning


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Lyon's shared bicycle system Velo'v .

Velo'v system • 340 stations spread in Lyon • 4000 bikes available in those stations • rental at any station, return it at any other one

Velo'v data • More than 13 millions of bicycle trips • Time-stamps of the trip • Rent and return station IDs Resource-aware Machine Learning


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Results for Velo'v graph .

Evolving patterns (σ = 0.8 and γ = 5) of Velo'v stations and their localization in Lyon. Patterns are shadowed on the map. Resource-aware Machine Learning


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.

Trend dynamic sub-graphs



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Outline . Constraint-based pattern mining framework Topological patterns in static attributed graphs Mining dynamic graphs Trend dynamic sub-graphs Definition of patterns Definition of constraints Triggering patterns of topology changes Conclusions and perspectives Resource-aware Machine Learning


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Dynamic attributed graph .

The data {Gt = (V, Et ), t = 1, · · · , tmax } and A a set of ordinal attributes: ai : V × T → Di , with Di the domain of ai



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Trend Dynamic Sub-graphs .

What are the sub-graphs whose vertex attributes evolve similarly? Mining maximal sub-graphs that satisfy some constraints on the graph topology and on the attribute values • The connectivity of the dynamic subgraphs is constrained by a

maximum diameter value. • To be more robust towards intrinsic inter-individual variability, we

do not compare raw numerical values, but their trends.


E. Desmier, M. Plantevit, C. Robardet, J-F. Boulicaut - PKDD 2013 Trend dynamic sub-graphs 54 / 78

Language of patterns . L = {(U, S, Ω) | U ⊂ V, S = ⟨t1 , · · · , ts ⟩, Ω ⊂ A × {+, −}} − Example: ({A, C, D}, ⟨t0 , t1 ⟩, {a+ 1 ,a3 }) ∈ L

a1 0

a2 2

D

a2 1

t0


a3 4

a1 2

a2 0

C

a1 3

a2 2

a3 4

a1 1

a2 3

.

a3 2

F

C E

D

a3 0

a1 4

a2 2

t1


a3 1

a1 3

a2 1

a2 1

a3 0

.

.

. F E

a1 2

a1 2 B

A

.

a3 4

.

a1 a2 1 3

a3 0

.

.

a3 5

.

C

a2 1

a1 a2 6 3

a3 0

B

A

a1 2

a2 1

.

E

a1 2

a1 5

a2 4

D .

a3 1

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.

a3 1

a2 2

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.

. .

F

a1 5

.

a3 5

.

a2 3

a3 5

B

A

a1 0

a2 4

.

a1 2

.

a3 3

.

a1 a2 3 5

a3 3

a1 5 t2

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a2 1

a3 0

a3 3

Definition of constraints .

A pattern (U, S, Ω) satisfies

• Cdiam ((U, S, Ω), D) ≡ diameterGt (U) ≤ k, ∀t ∈ S

...

1

...

...

Diameter

• Ctrend ((U, S, Ω), D) ≡ ∀u ∈ U, ∀t ∈ S, ∀(a, m) ∈ Ω

{

a(u, t) < a(u, t + 1), if m = + a(u, t) > a(u, t + 1), if m = −

• Cmax ((U, S, Ω), D) ≡ U, S and Ω cannot be enlarged without

invalidating one or both of the above constraints.



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• Ctrend ((U, S, Ω), D) is anti-monotone • Cdiam ((U, S, Ω), D) is piecewise monotone:

diameterGt (U) ≤ k ≡ max dGt (U) (v, w) ≤ k v,w∈U

f(U1 , U2 , D) = max dGt (U2 ) (v, w) v,w∈U1

• f1,U is anti-monotone i.e. if it is satisfied on U1 , it is also satisfied for

any of its subsets; • f2,U is monotone i.e. if it is satisfied on Gt (U2 ), then, adding some

vertices and edges to Gt (U2 ) will not increase its value +

We can use the propagation mechanisms

• Cmax ((U, S, Ω), D) is pushed using specific mechanisms Resource-aware Machine Learning


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Katrina hurricane results .

Top pattern w.r.t. time specificity (in red)

Top pattern w.r.t. trend specificity (Yellow)

• 71 airports whose arrival delays increase over 3 weeks.

• 5 airports whose number of flights increased over 3 weeks

• Arrival delays never increased in these airports during another week.

• Substitutions flights were provided from these airports during this period.

• The hurricane strongly influenced the domestic flight punctuality.

• This behavior is rather rare in the rest of the graph



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.

Triggering patterns of topology changes



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Outline . Constraint-based pattern mining framework Topological patterns in static attributed graphs Mining dynamic graphs Trend dynamic sub-graphs Triggering patterns of topology changes Triggering patterns Conclusions and perspectives



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Context & motivation . Networks structurally change over time: How to describe these dynamics? a: 4 b: 1 c: 5 deg: 1

deg +

c-

a+ b + u2

u2

u1

u3

u5

u4

a: 2 b: 0 c: 6 deg: 2

u2

u2

a: 7 b: 4 u1 c: 5 deg: 1

u3

u5

u4

a: 2 b: 1 c: 6 deg: 2

a: 7 b: 5 c: 2 deg: 1

u1

u3

u5

u4

a: 5 b: 4 c: 5 deg: 2

a: 8 b: 5 c: 1 deg: 4

u3

u5

u4

2

a: 5 b: 5 c: 5 deg: 2

3

4

u2

a: 9 b: 6 u1 c: 1 deg: 4

u3

u5

u4

c-

a+ b +

1

u2

u1

a: 6 b: 5 c: 2 deg: 2

a: 9 b: 6 u1 c: 0 deg: 4

u3

u5

u4

a: 7 b: 5 c: 1 deg: 4

deg +

5

6

Intuition Consider attributed graphs evolving through time The variation of some attribute values (nodes attributes) of a node can lead in several cases to a structural change (topological properties).


M. Kaytoue, Y. Pitarch, M. Plantevit, C. Robardet -- ASONAM 2014 Triggering patterns of topology changes 61 / 78

Context & motivation .

a: 4 b: 1 c: 5 deg: 1

deg +

c-

a+ b + u2

u2

u1

u3

u5

u4

a: 2 b: 0 c: 6 deg: 2

u2

u2

a: 7 b: 4 u1 c: 5 deg: 1

u3

u5

u4

a: 2 b: 1 c: 6 deg: 2

a: 7 b: 5 c: 2 deg: 1

u1

u3

u5

u4

a: 5 b: 4 c: 5 deg: 2

a: 8 b: 5 c: 1 deg: 4

u2

u1

u3

u5

u4

a: 5 b: 5 c: 5 deg: 2

2

u3

u5

u4

a: 6 b: 5 c: 2 deg: 2

a: 9 b: 6 u1 c: 0 deg: 4

u3

u5

u4

c-

a+ b +

1

u2

a: 9 b: 6 u1 c: 1 deg: 4

3

4

deg +

6

5

Triggering pattern example: ⟨{a+ , b+ }, {c− }, {deg+ }⟩ a+ b+ c− deg+

Updating his status more often giving positive opinions about others receiving less negative opinions from the others is often followed by an increase of user's popularity



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a: 7 b: 5 c: 1 deg: 4

Dynamic attributed graphs . Let G = {G1 , . . . , Gt } be a sequence of t static attributed graphs • Gi = (V, Ei , F) with T = {1, . . . , t} • F the set of numerical attributes that map each vertex-time pair to

a real value: ∀f ∈ F, f : V × T → R. ä F gathers the node attributes and the node topological properties

a: 4 b: 1 c: 5 deg: 1

deg +

c-

a+ b + u2

u2

u1

u3

u5

u4

a: 2 b: 0 c: 6 deg: 2

u2

u2

a: 7 b: 4 u1 c: 5 deg: 1

u3

u5

u4

a: 2 b: 1 c: 6 deg: 2

a: 7 b: 5 c: 2 deg: 1

u1

u3

u5

u4

a: 5 b: 4 c: 5 deg: 2

a: 8 b: 5 c: 1 deg: 4

u2

u1

u3

u5

u4

a: 5 b: 5 c: 5 deg: 2

2


u3

u5

u4

a: 6 b: 5 c: 2 deg: 2

a: 9 b: 6 u1 c: 0 deg: 4

u3

u5

u4

c-

a+ b +

1

u2

a: 9 b: 6 u1 c: 1 deg: 4

3

4


deg +

6

5

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a: 7 b: 5 c: 1 deg: 4

Characterizing vertex behaviors Vertex descriptive sequence

.

• A discretization function gives a variation symbol to a

vertex/attribute/time triple, e.g.  discr(v, f, i) =

 + if f(v, i) − f(v, i − 1) ≥ 2 and i > 1 − if f(v, i) − f(v, i − 1) ≤ −2 and i > 1  ∅ otherwise

• A vertex v is described by a sequence of itemsets

δ(v) = ⟨{discr(v, f, 1) | f ∈ F}, . . . , {discr(v, f, t) | f ∈ F}⟩ • ∆ = {δ(v) | v ∈ V} is the set of all sequences.

Example δ(u1 ) = ⟨{a+ , b+ }, {c− }, {deg+ }⟩ ∆ = {δ(u1 ), δ(u2 ), δ(u3 ), δ(u4 ), δ(u5 )} Resource-aware Machine Learning


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Triggering patterns . A triggering pattern is a sequence P = ⟨L, R⟩ with • L a sequence of node attribute variations • R a single topological property variation

Support measure supp(P, ∆) = {v ∈ V | P ⪯ δ(v)} where p ⪯ q means that p is a super-sequence of q

Example • L = ⟨{a+ , b+ }, {c− }⟩ • R = ⟨{deg+ }⟩ • supp(⟨L, R⟩, ∆) =

{u1 , u3 } Resource-aware Machine Learning

a: 4 b: 1 c: 5 deg: 1

deg +

c-

a+ b + u2

u2

u1

u3

u5

u4

a: 2 b: 0 c: 6 deg: 2

u2

u2

a: 7 b: 4 u1 c: 5 deg: 1

u3

u5

u4

a: 2 b: 1 c: 6 deg: 2

a: 7 b: 5 c: 2 deg: 1

u1

u3

u5

u4

a: 5 b: 4 c: 5 deg: 2

a: 8 b: 5 c: 1 deg: 4

u3

u5

u4

2

a: 5 b: 5 c: 5 deg: 2

u2

a: 9 b: 6 u1 c: 1 deg: 4

u3

u5

u4

c-

a+ b +

1

u2

u1

3


4

a: 6 b: 5 c: 2 deg: 2

a: 9 b: 6 u1 c: 0 deg: 4

u3

u5

u4

deg +

5

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6

a: 7 b: 5 c: 1 deg: 4

Assessing the strength of a pattern .

Triggering pattern growth rate Let P = ⟨L, R⟩, we denote by ∆R ⊆ ∆ the set of vertex descriptive sequences that contain R. The growth rate of P is given by: GR(P, ∆R ) =

|∆ \ ∆R | |supp(L, ∆R )| × |∆R | |supp(L, ∆ \ ∆R )|

G. Dong and J. Li. Efficient mining of emerging patterns: Discovering trends and differences. In KDD, pages 43--52, 1999.



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Quantitative results .

(i) Runtimes and number of patterns, (ii) Distribution of execution times, (iii) Support distribution, (iv) Growth rate vs support, (v) Scalability. Resource-aware Machine Learning


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The DBLP data Detecting asynchronous events

.

++ , degree++ } → {degree++ } {eigenvector++ 1 }, {VLDB 2 3



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US flights data . Synchronous events • RITA1: daily in September 2001 {#Canceled+ }{Deg− , Close− , NbCliq− , Prk− , Betw− } → Deg+

Airports that absorb the traffic two days after • RITA2: monthly in (sept. 2000 ; Sept. 2002 ) {#Canceled+ }{#Canceled− }, {nbCliq− , Betw+ } → nbCliq+

A "back to normal" around March 2002 • RITA3: Aug./Sept. 2005 (Katrina Hurricane) {#Canceled+ , DelayAtDep+ }, {#Diverted− , #Depar− , #Arrival− } → {close− }

All the airports supporting this pattern are located in the US West coast where Katrina raged.



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.

Conclusions and perspectives



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Conclusion .

• Constraint-based pattern mining framework can be used to

analyze dynamic graphs • Patterns combine information about the topological structure, the

vertex attribute values and their tendencies. + A wide variety of data can be mined + Provide new insights on the techniques that can be used to analyze dynamic graphs



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Challenges and perspectives - 1 .

Enhancing the quality of the extracted information • by capturing changes in the graph structure • while allowing changes in trend direction +

Combining sub-graph and sequence mining



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Avoiding the known drawbacks of pattern mining approaches • How to get rid of redundant patterns? • How to fix the threshold values?

Returning patterns whose constraint values are surprising • with respect to what could be expected from randomized data • with respect to what happened in the past (real-time data mining) +

Trigger alerts when changes in behavior



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Avoiding the known drawbacks of pattern mining approaches • How to get rid of redundant patterns? • How to fix the threshold values?

Returning patterns whose constraint values are surprising • with respect to what could be expected from randomized data • with respect to what happened in the past (real-time data mining) +

Trigger alerts when changes in behavior



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Pattern without threshold values: Skyline patterns Retrieves the patterns that are not dominated by any other pattern Tid t1 t2 t3 t4 t5 t6 Patterns ABCDEF AB AC A ABCD C

Items A B C D E F A B C D E F A B D A C E freq 2 3 3 4 2 3

length 6 2 2 1 4 1


area 12 6 6 4 8 3


A. Soulet, C. Raissi, M. Plantevit, B. Cremilleux - 2011 74 / 78

Ongoing research projects .

Vél'innov - ANR INOV 2012 - labex IMU project • Constraint-based pattern mining of Vélo'V data • Characterizing the usage of the bicycle sharing system • Modeling the system to simulate the impacts of local changes (in the city or in the system) • Pitfall: • Vertex attributes are static (INSEE data) or derived from the edges (e.g., number of users of a considered category) + Searching for relationships between attributes and the graph structure does not make sense • Envisioned solutions: • Considering the user trajectories and looking for paths that characterize a user group • Considering an attributed static graph with temporal usage vectors associated to edges and seeking for sub-graphs homogeneous on vertex and edges attributes Resource-aware Machine Learning


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Ongoing research projects - 1 .

GRAISearch: enhancing a location-based social media search engine • FP7-PEOPLE-2013-IAPP: Industry-Academia Partnerships and

Pathways • Benefits for DM2L team • 40 researcher months • Access to data with high added value • An industrial cooperation with critical potential benefits for the

company



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Ongoing research projects - 2 .

GRAISearch: Research tasks • Geo-localized demographic flow prediction using geo-tagged

social media uploaded data

• Geo-localized event detection algorithm • Integration into a geo-located social media recommender system



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Acknowledgments .



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Mining attributed static graphs

Mining attributed static graphs

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