An overview of the Small-World Phenomenon

1
An overview of the Small-World Phenomenon

• Bernard Lamers

2
?????????small world(1)

• ??????????????????????transivity of
  distances????????
• Transitive distance??? ?????????????
• ?????????????a??b??c???????????????????

d(a,c) ? d(a, b) d(b, c)
3
?????????small world(2)

• ?a??b??????b??c??????????????d(a, b)?d(b,
  c)??????????????????a??c?????????????????d(a,
  c)?????????????
• ???????????????????

4
Small world???????????

• 1950?? Rapoport?Solomonoff?randomly connected
  network?????????????
• ??
• ?connection???connection????
• ?????connection??????
• ???????????????

5
???????????(1)

• ??????P(0) (P(0)ltlt1)??????????????????(1-P(0))????
  ?????????
• ?connection????????????????
• ???????P(t)?????????????

?????????????????
T???? ??? ???? P(t-1)? ??????
P(t) (1 SP(i)) (1 e -kP(t-1) )
1????
6
???????????(2)

• t?????????????????????????????????
• ?1-( 1-P(0) )e-k?
• ???????
• ????
• P(0)?k?????????????????

7
????????(1)

• ????????????????????
• connection????????????????
• ??????????????????connection????????

8
????????(2)

• bias???
• Homophily ????????????
• Symmetry of edges edges?undirected(?????)?
• Triad closure ???????????????????????????
• Social differentiation ??????????????????????????
  ????????????????

9
??????

• Rapoport(1957)?????a?k????????????k??????????????
  ???? ?????(t). ?(t)?k.
• Fararo and Sunshine(1964), Skvoretz(1985)
  ?(t)???????bias???????? (p. 13, ?(2.3)).

10
p.13, ?(2.3)

• Triad closure bias ???????????????????????????
• Strong ties/weak ties
• ?a??b
• a????b????a?b????????????????
• a?b????????????????????????a?b?????????

11
Weak ties?strong ties????(1)

• ??s(1 - qS)
• ?s(1 - q(1 ?w/?s)))
• ?s(1 q q(?w/?s))
• ?s- q?sq?w
• (1 - q)?s q?w

12
Weak ties?strong ties????(2)

• ?s 0.7, ?w 0.2, q0.7, k10???
• ?0.35, ?3.48
• ?s 0.7, ?w 0.2, q0.9, k10???
• ?0.21, ?5.80 ??????

13
Density(??)

• ??????????triadic closure???????triadic
  closure??????????

14
Density?block modelling?????

• Barnes(1969)
• Density(local property)??????????property?????????
  block model? Block model?local(e.g.density)?non-l
  ocal(e.g.weak ties)????????????????

???v????????????????connection??
Density
v????????????v?connection?????????
15
????????????????Multidimensional scaling

• ????????????????
• ?????????????????????
• ??????????????????????????????????????????????????
  ??????????
• ????????

16
??????

• Metric methods, ?2.4
• Non-metric methods, ?2.5
• Non-metric method?metric??????Euclidean
  metric?????
• ??????????????????????????????4??????????????

17
???????

• Block model???????
• ??????
• ???????????????????????
• ???????renormalisationnetworks are viewed as
  meta-networks of highly clustered or equivalent
  subnetworks.

18
???????????????

• Milgram(1967)Nebraska??Kansas????????????Massachu
  setts???????????????????????????????????????????
• ??????????5????????

19
????????????? vs. ???????????

• ???????????????????
• ????????????????
• ??????????????
• ??????????????????local??????????????????????????
  ???????????????????? ?????????????

20
?????????????????????

• ??????????bridges???????non-local????????????local
  ????????????????????
• ????????????????????????????????????????connection
  ??????????????????????????
• ??????????????????????????????????????????????????
  ???????

21
????????????????

• ????????????????????????????
• ????????????????????????
• ?????????????????
• ???????????? ???
• ???metric???????
• ????????

22
??????

• ????????????????????????????????????
• ?????????????????????(1)????bias(2)??????(3)??????
  ????????????????????????????????????????????????

23
Network distance

• ?????????????connection?????????? network
  distance?
• ???a???????????b n.d. 1
• ???a????b??????c n.d. 2
• Network distance?????????nonmetric,
  non-Euclidian?????????????

24
??????

• ??????????????
• ???????????????small world????????????????????????
  ??????????????????????????????????????????????????
  ??
• ?????????????????

25
??????????

• ????????Definition 2.2.1 (p.25)
• ????order V(G)
• ????size E(G)
• Edge????vertices?????????????

26
????????????????(1)

• ???????????????????????
• Undirected edge???-????-????????????????????
• Unweightededge??????????
• Simple?????????edges??????vertex?????vertex???edg
  e??????

27
????????????????(2)

• Sparseundirected(and simple?)??????????M?n(n-1)/2
  ????Sparseness????????????ltltn(n-1)/2???????
• Connected????????vertex????vertex??????????????

28
Adjacency matrix/adjacency list

• Adjacency matrix n x n?matrix.
  Mi,j?i?j???edge??(??????1???0)
• Adjacency list????vertices??????vertex?????vertex
  ???????vertices?????????????

29
Degree of a vertex/average degree

• Vertex v???????edge??degree of v??? kv
• ????????vertices????degree average degree
• ????vertices???degree??????????k-regular?????regul
  ar????

30
Characteristic path length

• Characteristic path length L(G)?vertex??????verte
  x???????????
• ??network distance
• Characteristic path length??????????????????

31
Characteristic path length???(1)

• Characteristic path length??d(i,
  j)???????????n???????????????????????
• Random sampling???????????
• ??????median shortest path??????????????????????me
  dian??????

32
Characteristic path length???(2)

• Median??????
• Vertices??n??????characteristic path
  length??????????????????n?several orders of
  magnitude??????characteristic path
  length?order?????????????????c.p.l.???????? Def.
  2.2.2

33
Scaling

• Scaling n???k?????c.p.l.???????
• Scaling??????????(???????edge???)???????

34
Scaling??

• One-parameter family of graphs(3?), parametrized
  by some parameter p.
• ?????p????????????
• 1?c.p.l.?8
• Scaling properties????????
• p???????????????scaling????,p??????????????

35
Neighbourhoods and distribution sequences

• Neighbourhood and distribution sequences??????????
  ?????????????????????????
• ???2.2.5(p.31)-2.2.9(p.32)

36
Clustering

• Clustering coefficient G(v)????????vertices??????
  ???
• ??2.2.10 (p.33)
• ??????????lattice graphs(d-lattices)?random
  graphs?????????

37
Lattice graphs (d-Lattices)

• ??2.2.12
• ?figure 2.3? 2.4? 2.5
• D-lattices?length-scaling?clustering????????????
• 1-lattice???
• L scales linearly with respect to n and inversely
  with respect to k
• ? is independent of n, and for large k- also
  independent of k.