1. MIS 644 Social Newtork Analysis 2017/2018 Spring
Chapter 6-C
The Small World Model

2. Outline The Small world model Degree Distribution
Clustering Coefficient
Average Path Length

3. The Small World Model One of the least well-understood - real networks
trnsitivity: propensty of two neighbors of a vertex being neighbors of one another
Neither RG or CM nor network growth models
generate significant lvvel of transitivity
Measured by clustering coefficients
E.g. As n becomes large – CC vanishes
Orders of magnitudes smaller than observed for real nets

4. A simple triangular latice has a trnsitivity
# of triangles = 2 n
C(6,2) =15 connected triples for each vertex
0.4 competible with many real social networks
Not depends on size of the network

5. Fig of N-N
A triangular lattice. Any vertex in a triangular lattice, such as the one highlighted here, has six neighbors and hence pairs of neighbors, of which six are connected by edges, giving a clustering coefficient of = 0.4 for the whole network, regardless of size

6. Another model with high CC
Fig. 15.2a of N-N
Vertices – on a one dimensional line
connectd to c nearest vertices – c being even

7. A simple one-dimensional network model.
(a) Vertices are arranged on a line and each is connected to its c nearest neighbors, where c = 6 in this example.
(b) The same network with periodic boundary conditions applied, making the line into a circle

8. Traversing a “triangle” in our circle model means taking two steps forward around the circle and one step back.
C(c/2,2) total # of possibilities for two steps
# ways of observing the target c/2 steps
forward = (1/2)(c/2)(c/2-1) = (1/4)c(c/2-1)

9. # of connected triples centered on each vertex C(c,2)=c(c-1)/2
Total # connected triples: nc(c-1)/2 C:
As c varied from 2 to infinite
CC varies from 0 to ¾
Net depends on n
Unrealistic
Regular networks with DD = c

10. A more serious problem
Large worlds: - do not display small world effect observed in most real networks
The shortest path distances between most pairs of vertices is small (a few steps) even for networks of billions of nodes
acquaintance network of entire World population

11. The shortest distance between two vertices in the circle model:
fastest move in a ring c/2 spacings
Two vertices m spaces apart can be connected by a path of 2m/c
Averaging over all possible m from 0 to n/2
gives: n/2c
e.g.: for the acquaintance net of world pop
n - O(109), c - O(103), l - O(109-3) - millions
measured l – 6 to 10

12. By contrast the RG capture the SW effect rather well
As with many other network models CM pass
Average shortest path – ln(n)/ln(c)
For the acquaintance network – 9/3 =3
But has an unrealistically low CC
The two models: RG and simple circle (SC)
Each capture one property of real nets
RG – SW effect , SC – CC
Small world (SM) by Wattsw and Stogatz

13. SWM interpolate between SC and RG
By moving or rewiring edges from circle to random possitions
Starting from a CM of n vertices each having c edges
Go through each of the edges in turn
With some prob p:
remove that edge and
Replace with one joining two vertices choosen uniformly at random - shortcuts

14. The parameter p controls the interpolation between the CM and RGM
p=0: no edges are rewired – original CM
P=1: all edges are rewired - RGM
For intermediate values of p:
Networks in between
For p=0, the SWM shows clustering (c>2) but not SW effect
For p=1: reverse -RGM
For modest values of p, both high CC and SW efect

15. For analytical tracability
Variant of the original model:
Edges are added randomly
No edges are removed from the circle
p – same as in the original model
For every edge with independent prob p
add a shorcut between vertices choosen uniformly at random

16. Downside:
for p=1 – not RG
GR+original SC
Most interst p small
The two models are hardly difer
Small # edges around the circle
absent in the orignal model

17. Two versions of the small-world model.
(a) In the original version of the small world model, edges are with independent probability p removed from the circle and placed between two vertices chosen uniformly at random, creating shortcuts across the circle as shown. In this example n = 24, c = 6, and p = 0.07, so that 5 out of 72 edges are “rewired” in this fashion.
(b) In the second version of the model only the shortcuts are added and no edges are removed from the circle.

18. Fig of N-N

19. Degree Distribution In CM - egree of a vertex: c
# shortcut edges in the SWM
For each of non-shortcut edges
#: nc/2
Add a shortcut with prob. p
There are npc/2 shortcuts on average
ncp ends of shortcuts
cpn/n = cp shortcuts end in any vertex on average
s vertex Poisson distributed
with a mean of cp

20. Total degree of a vertex: k = c+s
Putting s = k-c
Fig 15.4 of N-N
c=6,p=1/2
Not mimic the DD of real networks
But the model does not intend to mimic that

21. Fig of N-N
The degree distribution of the small-world model. The frequency distribution of vertex degrees in a small-world model with parameters c = 6 and p = 1/2

22. Clustering Coefficient
The CC is given by
# of triangles and connected triples in the metwork
# triangles:
The circle does not change
# triangles in the cicle: nc(c/2-1)/4
Some new triangles by the shortcuts:
Vertex pairs c/2+1 to c by one or more paths of length two - if by shortcuts also - triangles

23. # of such paths of length two  n
average #of shortcuts in the SWM: ncp/2
C(n,2):n(n-1)/2 places they can fall
any particular pair of vertices is connected with prob:
just cp/n whn n ,
# paths of length two compleated by shortcuts to form trianges  n x cp/n = cp – constant
For large network size they can be neglected compaird to triangles from CM – O(n)

24. triangles can be formed from two or three shortcuts
Those turn out to be neglegible in number
Leading order in n
# of triangles in the SWM - # of triangles in the CM = (nc)(c/2-1)/4

25. # of connected triples:
All connected triples in the CM present in SWM
nc(c-1)/2
Triples from shortcuts combining with an edge in the circle
average # shortcuts: ncp/2 ,
c edges – they can form a triple – each of two ends
Total: (ncp/2) x c x 2 = nc2p: - connected triples

26. triples by pairs of shortcuts:
If a vertex connected to m shortcuts
# triples of shortcuts: C(m,2) = m(m-1)/2
averaging over Poisson distribution of m
With a mean cp
expected # of conected triples centered at a vertex: c2p2/2
total of nc2p2/2 for all vertices
Expected # of triples of all kind:
nc(c-1)/2 + nc2p + nc2p2/2

27. Substituting into clustering eq
for p=0, same as for the CM
as p increases - becomes smaller
when p=1 – minimum value of :
(3/4)(c-2)/(4c-1)
e.g., when c=6 CC: 3/23 = 0.130

28. Contrast with original WS-SWM
edges are removed from the circle
CC 0 as n  
when p=1
Since the network – RG
Fig 15.5 of N-N shows
CC as a function of p for SWM, for c=6

29. Clustering coefficient and average path length in the small-world model.
The solid line shows the clustering coefficient, Eq. (15.7), for a small-world model with c = 6 and n = 600, as a fraction of its maximum value ,CCmax:0.6, plotted as a function of the parameter p.
The dashed line shows the average geodesic distance between vertices for the same model as a fraction of its maximum value ℓmax = n/2c = 50, calculated from the mean-field solution, Eq. (15.14). Note that the horizontal axis is logarithmic.

30. Fig of N-N

31. Average Path Lengths
Calculating the average path length or mean geodesic path distances between pairs of vetrices in SWM -
harder than DD or CC
No exact expression
Some approximate expressins
by simulations – reasonaby accurate

32. Known about path lenghts
How they scale with model parameters
Simple SWM with c=2:
each vertex is connected to its immediate neighbors
Argument:
distane covered by an edge – one unit: meter
What other quantities in terms of – length
distance around the whole cicle :n

33. Mean distance between the ends of the shortcuts
suppose s shortcuts – 2s ends
s= ncp/2
average distance  between end around the circle:  =n/2s
Once specifiy n and  - specify the entire model
from n, to s to p, when c=2

34. Ratio of l to n
l: average shortest path
as function of n and  - specify the model
ratio of two distances - dimensionless
one such dimensionless combination n/ 
F(x) does not depends on any of the parameters – universal function in scaling
Hence mean geodesic path in SWM wih c=2

35. For larger c
c  lenghts of shortest paths (SP)between vertices
Keep everything the same – n, s
s from 2 to 4  halve SPs
If the SP includes a shortcuts – distance does not change
If density of shortcuts low – most paths are on ciicles
For c length of the path  by c/2
For low densty of shortcuts

36. Alternatively
s = npc/2
l= 2(n/c)F(npc)
Absorbing the leading factor of 2 into the functional form
definng f(x)= 2F(x)
l= (n/c)f(npc)
How average path lengh depends on parameters
n,p,c for low values of shortcut density

37. We do not know the form fo f(x)
Numerical simulation
generate SW metworks
measure mean distance l – between vertices
Many runs with different parameters
the combination cl/n same function of npc
Fig 15.6 of N-N
Many networks – all roughly the same function

38. Scaling function for the small-world model.
The points show numerical results for cℓ/n as a function of ncp for the small-world model with a range of parameter values n = 128 to and p = 1 × 10−6 to 3 × 10−2, and two different values of c as marked.
Each point is averaged over 1000 networks with the same parameter values. The points collapse, to a reasonable approximation, onto a single scaling function ƒ(ncp) in agreement with Eq. (15.13).
The dashed curve is the mean-field approximation to the scaling function given in Eq. (15.14)

39. Fig of N-N

40. Another approach Calculate f(x) approximately
Series, distributional or mean-field methods
A mean field approximation :
becomes exact when # shortcuts is very small or very large
In between around x=1 – approximate
Show as the dashed line in Fig 15.6 of N-N
agrees well with numerical resutls at the ends
less in the middle
Enough to prove that SWM expalins SW effect

41. for large velues of x: x >>1 – npc >>1
npc : 2 x # of shortcuts
average of l  logarithmically with n – very slowly
for fixed p and c
n very large
average l: remains small
SW effect

42. # shortcuts not (# shortcuts per vertex) large
small density of random shortcutrs to a large networks – SW behavior
Why most real networks show SW effects
Have long range connections with some randomness
Very few are regular or with short range connctions

43. SWM shows not only
SW effect but displaying clustering
# shortcuts = npc/2
 when n holding p and c fixed
CC independent of n
retains (non-zere) as n
In this limit - simultaneously
non-zere clustering and SW effect
The two are not with odds with one antoher

44. Fig 15.5 of N-N
plot of approximate values of l s a function of p
for a SWM n=600,c=6
Along with CC
substantial values of p
the value of l is low
the value of CC is high

45. Clustering coefficient and average path length in the small-world model.
The solid line shows the clustering coefficient, Eq. (15.7), for a small-world model with c = 6 and n = 600, as a fraction of its maximum value ,CCmax:0.6, plotted as a function of the parameter p.
The dashed line shows the average geodesic distance between vertices for the same model as a fraction of its maximum value ℓmax = n/2c = 50, calculated from the mean-field solution, Eq. (15.14). Note that the horizontal axis is logarithmic. 45

46. Fig of N-N 46