Upcoming SlideShare
×

# Social Network Analysis

2,570 views

Published on

Introductive presentation on static social network models.

Published in: Technology
3 Likes
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

Views
Total views
2,570
On SlideShare
0
From Embeds
0
Number of Embeds
876
Actions
Shares
0
62
0
Likes
3
Embeds 0
No embeds

No notes for slide

### Social Network Analysis

1. 1. AOT LAB DII, UNIPR SOCIAL NETWORK ANALYSIS Enrico Franchi (efranchi@ce.unipr.it)1
2. 2. Outline SNA = Complex Network Analysis on Social Networks Notation & Metrics Degree Distribution Path Lengths Transitivity Models Random Graphs Small-Worlds Preferential Attachment Models Discussion Conclusion 2
3. 3. Network Directed NetworkG = (V, E) E ⊂ V 2 k out = ∑ A ij k = ∑ A ji in{(x, x) x ∈V } ∩ E = ∅ i i j j ki = kiin + kiout Undirected NetworkAdjacency Matrix A symmetric ⎧1 if (i,j) ∈EA ij = ⎨ ⎩0 otherwise ki = ∑ A ji = ∑ A ij j j px = # {i ki = x } 1Degree Distribution nAverage Degree k =n −1 ∑k x x∈V 3
4. 4. Measure of Transitivity () −1 kiLocal Clustering Coefficient Ci = 2 T (i) T(i): # distinct triangles with i as vertex 1Clustering Coefficient C = ∑ Ci n i∈VC= ( number of closed paths of length 2 ) = ( number of triangles ) × 3 ( number of paths of length 2 ) ( number of connected triples ) 4
5. 5. Shortest Path Length and Diameter scalar operations AB = A + .⋅ B The matrix product depends from ( A,+,⋅) [ AB]ij = ∑ A ik ⋅ Bkj the operations of the semi-ring kSet of Adjacency Matrices min Other matrix products make sense: e.g., ( A,+,^ ) or ( A,^,+ ) We consider: ( Sk (M) = M + .^ M k ^ .+ M k ) Shortest path lengths matrix: L = ( Sn … S1 ) ( M ) Diameter: d = max L Average shortest path:  = Lij ij 5
6. 6. Computational Complexity of ASPL: All pairs shortest path matrix based (parallelizable): ( ) α ≈ 3/ 4 O n 3+α All pairs shortest path Bellman-Ford: O (n )3 All pairs shortest path Dijkstra w. Fibonacci Heaps: O ( n log n + nm ) 2Computing the CPL x = M q (S) q#S elements are ≤ than x and (1-q)#S are > than x x = Lqδ (S) q#S(1-δ) elements are ≤ than x and (1-q)#S(1-δ) are > than xHuber Algorithm 2 2 (1 − δ ) 2 Let R a random sample of S such that #R=s, then s = 2 ln q  δ 2 Lqδ(S) = Mq(R) with probability p = 1-ε. 6
7. 7. 2 2 (1 − δ ) 2s = 2 ln q  δ 2 7
8. 8. Facebook Hugs Degree Distribution10000000 Nodes: 1322631 Edges: 1555597 m/n: 1.17 CPL: 11.74 1000000 Clustering Coefficient: 0.0527 Number of Components: 18987 100000 Isles: 0 10000 Largest Component Size: 1169456 1000 For large k we have 100 statistical fluctuations 10 1 1 10 100 1000 For small k power-laws do not hold 8
9. 9. Many networks havepower-law degree distribution. pk ∝ k −γ γ >1• Citation networks k r =?• Biological networks• WWW graph• Internet graph• Social Networks Power-Law: ! gamma=3 1000000 100000 10000 1000 100 10 1 0.1 9 1 10 100 1000
10. 10. Erdös-Rényi Random Graphs Connectedness p Threshold log n / nG(n, p) pG(n, m) p p p pEnsembles of Graphs p pWhen describe values of pproperties, we actually the p Pr(Aij = 1) = pexpected value of the propertyd := d = ∑ Pr(G)⋅ d(G) ∝ log n Pr(G) = p m (1− p) () n 2 −m G log k ⎛ n⎞ m =⎜ ⎟ p k = (n − 1)p C = k (n − 1) −1 ⎝ 2⎠ ⎛ n − 1⎞ k k kpk = ⎜ ⎟ p (1− p) n−1−k n→∞ pk = e − k 10 ⎝k ⎠ k!
11. 11. p Watts-Strogatz Model In the modified model, we only add the edges. ki = κ + si ps = e −κ s (κ p ) s C= 3(κ − 2) s! 4(κ − 1) + 8κ p + 4κ p 2Edges inthe lattice # added pk = e −κ s (κ p ) k−κ ≈ log(npκ ) shortcuts ( k − κ )! κ p 2 11
12. 12. Strogatz-Watts Model - 10000 nodes k = 4 1 CPL(p)/CPL(0) C(p)/C(0) 0.8CPL(p)/CPL(0) 0.6 C(p)/C(0) 0.4 0.2 0 0 0.2 0.4 p 0.6 0.8 1 Short CPL Large Clustering Coefficient 12 Threshold Threshold