(laboratoriodi) sistemi)informaci )avanza...
TRANSCRIPT
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(Laboratorio di ) Sistemi Informa3ci Avanza3
Giuseppe Manco
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SEARCH
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Approcci alle re3 di grandi dimensioni
Heavy-‐tails e power laws (su scale di grandi imensioni): • forte eterogeneità locale, mancanza di struEura • base per i modelli preferen3al aEachment
Local clustering/structure (su scale di piccole dimensioni): • situazioni locali hanno una struEura “geometrica”
• punto di partenza per modelli small world che partono con una “geometria” globale e aggiungono link random per oEenere un diametro piccolo e preservare la geometria a livello locale
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Le problema3che di interesse
• Quali sono le sta3s3che di base (degree distribu3ons, clustering coefficients, diametro, etc.)?
• Ci sono raggruppamen3/par3zioni naturali?
• Come evolvono/rispondono alle perturbazioni le re3?
• Come avvengono I processi dinmaici -‐ search, diffusion, etc. – nelle re3?
• Come fare classificazione, regressione, ranking, etc.?
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Osservazioni sulle re3 reali
• Diametro – Costante
• Coefficiente di clustering – Costante
• Degree distribu3on – Power-‐law
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Applicazioni: Search
• Small world – È possibile navigare la rete
• Preferen3al aEachment – Ci sono alcuni grossi hubs
• Come sfruEare tali informazioni?
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Singular Value Decomposi3on
• Tecnica di decomposizione matriciale • Basata sull’analisi speErale • Tante applicazioni
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La Singular Value Decomposi3on (SVD)
• Data una matrice A, m x n, essa può essere decomposta come prodoEo di tre matrici:
• p: rango di A • U, V: matrici ortogonali (UTU=Im, VTV=In) contenen3
rispe_vamente i veEori singolari destri e sinistri di A
• ∑: matrice diagonale contenente i valori singolari di A, in ordine non-‐crescente σ1≥σ2≥... ≥σp ≥0
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Interpretazione a Layer della SVD
m x n
A mxn
= u1vT1 + u1vT1 +... σ1 σ2
Importanza decrescente
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VeEori Singolari, Intuizione
I cerchi blu rappresentano m pun3 nello spazio euclideo. La SVD della matrice mx2 sarà cos3tuita da: -‐ Primo veEore singolare (destro): direzione della varianza max -‐ Secondo veEore singolare (destro): direzione della max varianza dopo aver rimosso la proiezione dei da3 lungo il primo veEore singolare
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VeEori Singolari, Intuizione
• σ1: misura quanta varianza dei da3 è “caEurata/spiegata” dal primo veEore singolare
• σ2: misura quanta varianza dei da3 è “caEurata/sp iegata” da l secondo veEore singolare
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Low Rank Approxima3on
• Si tronca la SVD ai primi k termini:
• k= rango della decomposizione
• Uk, Vk: matrici ortogonali contenen3 rispe_vamente i primi k veEori singolari destri e sinistri di A
• ∑k: matrice diagonale contenente i primi valori k singolari di A
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Proprietà
• Anche per matrici con da3 posi3vi, la SVD è mista in segno
+ +/-‐ +/-‐
+
• U e V sono dense • Unicità: nonostante ci siano diversi algoritmi, ques3
producono la stessa SVD (A troncata)
• Proprietà: mantenere i primi k valori singolari di A fornisce la migliore rank-‐k approxima3on di A rispeEo alla Frobenius norm
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Low Rank Approxima3on • Usa Ak al posto di A
A mxn
Umm Ukm ∑ mxn
∑kk VT nxn
VT kxn nxn
≈
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Sommario della Truncated SVD • Pro:
– Usare Ak al posto di A implica un aumento delle performance generale degli algoritmi di mining
– la riduzione del rumore isola le componen3 essenziali della matrice da3
– Best rank-‐k approxima3on – Ak è unica e o_ma secondo la Frobenious norm
• Contro: – Storage (Uk e Vk sono dense) – L’interpretazione di U e V è difficile perchè hanno segno misto
– Un buon punto di troncamento k è difficile da determinare
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Applicazioni della SVD all’analisi dei da3
• Dimensionality reduc3on: la truncated SVD fornisce una rappresentazione compressa di da3 ad alta dimensionalità (con mol3 aEribu3).
• La compressione SVD minimizza la perdita di informazione, misurata secondo la Frobenious norm
• Se i da3 originali contengono rumore, la riduzione di dimensionalità può essere considerata come una tecnica di aEenuazione del rumore
• Se fissiamo k=2 o k=3, allora è possibile ploEare le righe di U. La rappresentazione grafica rende possibile un’interpretazione visuale della struEura del dataset
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SVD e Latent Seman3c Indexing
KDD'09 Faloutsos, Miller, Tsourakakis P1-23
CMU SCS
SVD - Example
• A = U ΛΛΛΛ VT - example:
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 3
0 0 0 1 1
datainf.
retrieval
brainlung
0.18 0
0.36 0
0.18 0
0.90 0
0 0.53
0 0.80
0 0.27
=CS
MD
9.64 0
0 5.29x
0.58 0.58 0.58 0 0
0 0 0 0.71 0.71
x
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SVD e Latent Seman3c Indexing
KDD'09 Faloutsos, Miller, Tsourakakis P1-24
CMU SCS
SVD - Example
• A = U ΛΛΛΛ VT - example:
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 3
0 0 0 1 1
datainf.
retrieval
brainlung
0.18 0
0.36 0
0.18 0
0.90 0
0 0.53
0 0.80
0 0.27
=CS
MD
9.64 0
0 5.29x
0.58 0.58 0.58 0 0
0 0 0 0.71 0.71
x
CS-conceptMD-concept
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SVD e Latent Seman3c Indexing
KDD'09 Faloutsos, Miller, Tsourakakis P1-24
CMU SCS
SVD - Example
• A = U ΛΛΛΛ VT - example:
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 3
0 0 0 1 1
datainf.
retrieval
brainlung
0.18 0
0.36 0
0.18 0
0.90 0
0 0.53
0 0.80
0 0.27
=CS
MD
9.64 0
0 5.29x
0.58 0.58 0.58 0 0
0 0 0 0.71 0.71
x
CS-conceptMD-concept
Affinità documento-‐conceEo
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SVD e Latent Seman3c Indexing
Importanza del conceEo
KDD'09 Faloutsos, Miller, Tsourakakis P1-24
CMU SCS
SVD - Example
• A = U ΛΛΛΛ VT - example:
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 3
0 0 0 1 1
datainf.
retrieval
brainlung
0.18 0
0.36 0
0.18 0
0.90 0
0 0.53
0 0.80
0 0.27
=CS
MD
9.64 0
0 5.29x
0.58 0.58 0.58 0 0
0 0 0 0.71 0.71
x
CS-conceptMD-concept
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SVD e Latent Seman3c Indexing
KDD'09 Faloutsos, Miller, Tsourakakis P1-24
CMU SCS
SVD - Example
• A = U ΛΛΛΛ VT - example:
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 3
0 0 0 1 1
datainf.
retrieval
brainlung
0.18 0
0.36 0
0.18 0
0.90 0
0 0.53
0 0.80
0 0.27
=CS
MD
9.64 0
0 5.29x
0.58 0.58 0.58 0 0
0 0 0 0.71 0.71
x
CS-conceptMD-concept
Affinità termine-‐conceEo
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Riduzione di dimensionalità
KDD'09 Faloutsos, Miller, Tsourakakis P1-38
CMU SCS
SVD - Interpretation #2
• A = U ΛΛΛΛ VT - example:
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 3
0 0 0 1 1
0.18 0
0.36 0
0.18 0
0.90 0
0 0.53
0 0.80
0 0.27
=9.64 0
0 5.29x
0.58 0.58 0.58 0 0
0 0 0 0.71 0.71
x
v1
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Riduzione di dimensionalità
KDD'09 Faloutsos, Miller, Tsourakakis P1-39
CMU SCS
SVD - Interpretation #2
• A = U ΛΛΛΛ VT - example:
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 3
0 0 0 1 1
0.18 0
0.36 0
0.18 0
0.90 0
0 0.53
0 0.80
0 0.27
=9.64 0
0 5.29x
0.58 0.58 0.58 0 0
0 0 0 0.71 0.71
x
variance (‘spread’) on the v1 axisVarianza lungo l’asse v1
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Riduzione di dimensionalità
• Eliminiamo elemen3 a bassa varianza
KDD'09 Faloutsos, Miller, Tsourakakis P1-42
CMU SCS
SVD - Interpretation #2
• More details
• Q: how exactly is dim. reduction done?
• A: set the smallest singular values to zero:1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 3
0 0 0 1 1
0.18 0
0.36 0
0.18 0
0.90 0
0 0.53
0 0.80
0 0.27
=9.64 0
0 5.29x
0.58 0.58 0.58 0 0
0 0 0 0.71 0.71
x
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Riduzione di dimensionalità
• Eliminiamo gli elemen3 a bassa varianza
KDD'09 Faloutsos, Miller, Tsourakakis P1-44
CMU SCS
SVD - Interpretation #2
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 3
0 0 0 1 1
0.18 0
0.36 0
0.18 0
0.90 0
0 0.53
0 0.80
0 0.27
~9.64 0
0 0x
0.58 0.58 0.58 0 0
0 0 0 0.71 0.71
x
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Riduzione di dimensionalità
• Eliminiamo gli elemen3 a bassa varianza
KDD'09 Faloutsos, Miller, Tsourakakis P1-45
CMU SCS
SVD - Interpretation #2
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 3
0 0 0 1 1
0.18
0.36
0.18
0.90
0
0
0
~9.64
x
0.58 0.58 0.58 0 0
x
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Riduzione di dimensionalità
• Eliminiamo gli elemen3 a bassa varianza
KDD'09 Faloutsos, Miller, Tsourakakis P1-46
CMU SCS
SVD - Interpretation #2
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 3
0 0 0 1 1
~
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
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Applicazioni della SVD all’analisi dei da3
• Clustering: nello spazio della trasformazione SVD troncata, la relazioni tra i pun3 sono più eviden3 e il processo di clustering ne trae direEo vantaggio
• Applicazioni al clustering: • Clustering sul nuovo spazio • U3lizzo direEo delle proprietà dell’SVD • Spectral clustering: i pun3 che giacciono nel cono intorno al
primo asse (prodoEo con il primo asse <1/2) sono raggruppa3 in un cluster
• Quelli con la stessa proprietà rispeEo al secondo asse vengono raggruppa3 nel secondo cluster e così via
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Raggruppamen3, blocchi
KDD'09 Faloutsos, Miller, Tsourakakis P1-54
CMU SCS
SVD - Interpretation #3
• finds non-zero ‘blobs’ in a data matrix
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 3
0 0 0 1 1
0.18 0
0.36 0
0.18 0
0.90 0
0 0.53
0 0.80
0 0.27
=9.64 0
0 5.29x
0.58 0.58 0.58 0 0
0 0 0 0.71 0.71
x
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Raggruppamen3, blocchi
KDD'09 Faloutsos, Miller, Tsourakakis P1-55
CMU SCS
SVD - Interpretation #3
• finds non-zero ‘blobs’ in a data matrix
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 3
0 0 0 1 1
0.18 0
0.36 0
0.18 0
0.90 0
0 0.53
0 0.80
0 0.27
=9.64 0
0 5.29x
0.58 0.58 0.58 0 0
0 0 0 0.71 0.71
x
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Raggruppamen3, blocchi
KDD'09 Faloutsos, Miller, Tsourakakis P1-56
CMU SCS
SVD - Interpretation #3
• finds non-zero ‘blobs’ in a data matrix =
• ‘communities’ (bi-partite cores, here)
1 1 1 0 0
2 2 2 0 0
1 1 1 0 0
5 5 5 0 0
0 0 0 2 2
0 0 0 3 3
0 0 0 1 1
Row 1
Row 4
Col 1
Col 3
Col 4Row 5
Row 7
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Applicazioni della SVD all’analisi dei da3 • Ranking: • Ogni riga di U può essere rappresentata come un punto nello spazio k-‐
dimensionale. Supponiamo di tracciare una freccia dall’origine verso ciascuno dei pun3
• L’angolo (coseno) tra i due veEori denota la correlazione tra i pun3 • Ogge_ altamente correla3 o altamente non correla3 con altri pun3
tendono a piazzarsi intorno all’origine
• Pun3 colloca3 lontano dall’origine corrispondono ad ogge_ che esibiscono una correlazione inusuale con altri ogge_
• Pun3 colloca3 vicino all’origine sono meno “interessan3”
• Il rank degli ogge_ può essere effeEuato tenendo conto della distanza dall’origine
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Proprietà (A)
UTU = ImVTV = InΛk = diag σ1
k,σ 2k,...,σ r
k( )AT =VΛUT
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Proprietà (B)
• Similarità documento-‐documento
• Similarità termine-‐termine
AAT =UΛ2UT
ATA =VΛ2VT
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Proprietà (B)
• Inoltre:
• v1 autoveEore rela3vo a σ1 (l’autovalore più grande)
ATA( )k =VΛ2kV T
ATA( )k ≈ v1σ12kv1
T
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Proprietà (C)
• Per qualsiasi veEore v – Conseguenza: procedura itera3va per il calcolo degli autoveEori
ATA( )k v ≈ λv1T
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Proprietà (C)
• AmmeEe soluzione
Ax = b
x =VΛ−1UTb
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Proprietà (C)
• conseguentemente
Av1 =σ1u1u1T A =σ1v1
ATAv1 =σ 1
2v1
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PCA e MDS Principal Components Analysis (PCA) • Da3{Xi}i=1,…,n con Xi veEori reali,
trova il soEospazio k-‐dimensionale P e il mapping Yi=PXi
t.c.. Variance(Y) è massima (o Error(Y) è minimo)
• SVD sulla matrice di covarianza C =XXT
Mul3dimensional Scaling (MDS) • Da3 {Xi}i=1,…,n con Xi veEori reali,
trova il soEospazio k-‐dimensionale P e il mapping Yi=PXi
t.c. Dist(Yi-‐Yj) = Dist(Xi-‐Xj) (ovvero distanze preservate)
• SVD sulla matrice matrix G = XT X
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LSI/SVD e power laws
• Gli autovalori più grandi della matrice di adiacenza di un grafo scale-‐free sono distribui3 con una power-‐law.
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Caso di Studio: Social Network Analysis
• Obie_vo: iden3ficare proprietà e relazioni tra i membri di al Qaeda
• Il dataset fornito da Marc Sageman con3ene informazioni su 366 membri dell’associazione terorris3ca all’inizio del 2004
• AEribu3:
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al Qaeda Dataset
• Grafo delle relazioni: 366 nodi e 2171 archi. • Il grado massimo del grafo è 44, mentre
quello medio è 6.44. • Il diametro è 11 • Bavelas-‐LeaviE Centrality: rapporto tra la
somma dei cammini geodesici aven3 come sorgente/des3nazione il nodo considerato e la somma dei cammini geodesici dell’intero dataset
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al Qaeda Dataset:
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al Qaeda Dataset: Link Analysis
• Analisi della matrice di adiacenza 366 x 366 – Conta_ e relazioni tra i membri
Plot of the low rank (3) SVD of al Qaeda members using only rela3onship aEribute
• 4 cluster
• Hambali ha un ruolo di connessione
• bin Laden non è l’elemento estremo del cluster che iden3fica la leadership
Algerians"South East Asian"
Leaders and core Arabs"
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The University of Arizona group have analyzed this dataset and used multidimensional scalingto produce a picture of the group’s connectivity (Jie Xu, personal communication, 2004). Thisshows that the dataset is naturally clustered into 13 almost-cliques, with about 60 members notallocated to a single clique.
A graph of the links within al Qaeda is maintained by Intelcenter and can be viewed on theirweb site (www.intelcenter.com/linkanalysis.html). While the graph is compendious, it is hard toextract actionable information from it.
4 Analysis using matrix decompositions
4.1 Using the links between individuals
In this section we consider only the results of enhanced link analysis, that is we consider the graphof relationships among al Qaeda members. The base dataset is a 366 £ 366 adjacency matrix forthe graph that includes: acquaintances, family, friends, relations, and contacts after joining.
−0.35−0.3
−0.25−0.2
−0.15−0.1
−0.050
−0.1
0
0.1
0.2
0.3
0
0.1
0.2
0.3
0.4
U1
U2
U3
Figure 3: SVD plot of al Qaeda members using only relationship attributes.
Figure 3 shows a 3-dimensional (truncated) view of the relationships among al Qaeda membersextracted from their links. The most obvious fact is that there is a clear division into three (perhapsfour) clusters. This radial pattern is typical: those points at the extremities represent individualswith the most interesting connections to the rest of the group. Many members are either connected
8
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SVD e centralità
Misura l’importanza di un nodo • degree centrality – numero di link di un nodo
• betweenness centrality –numero di cammini che lo contengono
• closeness centrality -‐ potenziale di comunicazione indipendente
• eigenvector centrality – connessioni a nodi con high-‐degree, itera3vamente
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Eigenvector centrality
• Riformulato, risulta essere
x j ≈ aij xii=1
N
∑
Ax =σ x
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IL WEB
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StruEura del Web 13.6. EXERCISES 395
7
6
8
11
2
12 13 14
9
3
4
5
10
15
16
17
1
18
Figure 13.8: A directed graph of Web pages.
(b) Name an edge you could add or delete from the graph in Figure 13.8 so as to
increase the size of the set IN.
(c) Name an edge you could add or delete from the graph in Figure 13.8 so as to
increase the size of the set OUT.
3. In Exercise 2, we considered how the consistuent parts of the bow-tie structure change
as edges are added to or removed from the graph. It’s also interesting to ask about
the magnitude of these changes.
(a) Describe an example of a graph where removing a single edge can reduce the size
of the largest strongly connected component by at least 1000 nodes. (Clearly you
shouldn’t attempt to draw the full graph; rather, you can describe it in words,
and also draw a schematic picture if it’s useful.)
(b) Describe an example of a graph where adding a single edge can reduce the size
of the set OUT by at least 1000 nodes. (Again, you should describe the graph
rather than actually drawing it.)
Source: David Easley, Jon Kleinberg Networks, Crowds, and Markets, Cambridge University Press (2010)
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StruEura del web 13.3. THE WEB AS A DIRECTED GRAPH 387
I'm a student at Univ. of X
Company Z's home page
Our Founders
Press Releases
Contact Us
Univ. of X
Classes
Networks
Networks class blog
Blog post about college rankings
I teach at Univ. of X
USNews: College
Rankings
USNews: Featured Colleges
Blog post about
Company Z
I'm a applying to college
My song lyrics
Figure 13.6: A directed graph with its strongly connected components identified.Source: David Easley, Jon Kleinberg Networks, Crowds, and Markets, Cambridge University Press (2010)
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Bow-‐Tie
SCCIN OUT
tendrils
tubes
disconnected components
Source:A. Broder, et at.. Graph structure in the Web. In Proc. WWW, pages 309–320, 2000.
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Il problema della Ricerca
• Inserisci un termine nella pagina di google – Analizza i risulta3
• Il primo elemento è quello che 3 aspeEavi? • Come ha faEo google a calcolare il risultato?
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Search
• Un problema difficile – Informa3on retrieval: ricerca in grosse repositories, sulla base di keywords
– Keywords limitate e inespressive, e: • sinonimia (modi mul3pli per dire la stessa cosa: casa, abitazione) • Polisemia (significa3 mul3pli per lo stesso termine: Jaguar, Apple)
– Differen3 modalità di authoring • Esper3, novizi, etc.
– Estrema dinamicità del web – Shi�
• Scarcity -‐> abundance
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Hubs, Authori3es
• Un problema di links – Perché wikipedia è in cima agli elemen3 suggeri3?
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Hubs, authori3es
• Molte pagine contengono il termine “re3 sociali” – Perché wikipedia è più rilevante?
• Indicheres3 wikipedia come riferimento?
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Hubs, authori3es
• Votazione 400 CHAPTER 14. LINK ANALYSIS AND WEB SEARCH
Wall St.
Journal
New York
Times
USA Today
Yahoo!
Amazon
2 votes
4 votes
3 votes
1 vote
3 votes
3 votes
SJ Merc
News 2 votes
Figure 14.1: Counting in-links to pages for the query “newspapers.”
A List-Finding Technique. It’s possible to make deeper use of the network structure
than just counting in-links, and this brings us to the second part of the argument that links
are essential. Consider, as a typical example, the one-word query “newspapers.” Unlike
the query “Cornell,” there is not necessarily a single, intuitively “best” answer here; there
are a number of prominent newspapers on the Web, and an ideal answer would consist of a
list of the most prominent among them. With the query “Cornell,” we discussed collecting
a sample of pages relevant to the query and then let them vote using their links. What
happens if we try this for the query “newspapers”?
What you will typically observe, if you try this experiment, is that you get high scores for a
mix of prominent newspapers (i.e. the results you’d want) along with pages that are going to
receive a lot of in-links no matter what the query is — pages like Yahoo!, Facebook, Amazon,
and others. In other words, to make up a very simple hyperlink structure for purposes of
Source: David Easley, Jon Kleinberg Networks, Crowds, and Markets, Cambridge University Press (2010)
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Hubs, authori3es 402 CHAPTER 14. LINK ANALYSIS AND WEB SEARCH
Wall St.
Journal
New York
Times
USA Today
Yahoo!
Amazon
11
7
3
6
3
3
5
new score: 19
new score: 31
new score: 24
new score: 5
new score: 15
8
SJ Merc
News
6
new score: 19
new score: 12
Figure 14.3: Re-weighting votes for the query “newspapers”: each of the labeled page’s newscore is equal to the sum of the values of all lists that point to it.
of links to on-line newspapers; for “Cornell,” one can find many alumni who maintain pages
with links to the University, its hockey team, its Medical School, its Art Museum, and so
forth. If we could find good list pages for newspapers, we would have another approach to
the problem of finding the newspapers themselves.
In fact, the example in Figure 14.1 suggests a useful technique for finding good lists. We
notice that among the pages casting votes, a few of them in fact voted for many of the pages
that received a lot of votes. It would be natural, therefore, to suspect that these pages have
some sense where the good answers are, and to score them highly as lists. Concretely, we
could say that a page’s value as a list is equal to the sum of the votes received by all pages
that it voted for. Figure 14.2 shows the result of applying this rule to the pages casting votes
in our example.
• Compilazione di liste – Ogni pagina “rappresenta” quelle che la puntano
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Hubs, authori3es
• Miglioramento itera3vo • Normalizzazione • Authori3es
– Le pagine che rappresentano gli end-‐points • Hubs
– Le pagine che rappresentano molte altre pagine (e il cui voto conseguentemente conta tanto)
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Hubs, authori3es
• Hubs
• Authori3es
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Authority value
• Dato un nodo i:
• Generalizzando su ogni nodo:
ai = hk + hl + hm
a =ATh
KDD'09 Faloutsos, Miller, Tsourakakis P1-82
CMU SCS
Kleinberg’s algorithm
Then:
ai = hk + hl + hm
that is
ai = Sum (hj) over all j that (j,i) edge exists
or
a = AT h
k
l
m
i
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Hub value
• Dato un nodo i:
• Generalizzando su ogni nodo:
hi = an + ap + aq
h =Aa
KDD'09 Faloutsos, Miller, Tsourakakis P1-83
CMU SCS
Kleinberg’s algorithm
symmetrically, for the ‘hubness’:
hi = an + ap + aq
that is
hi = Sum (qj) over all j that (i,j) edge exists
or
h = A a
p
n
q
i
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Algoritmo HITS
• In conclusione, s3amo cercando due veEori h e a tali che
a =AThh = Aa