10. Milgram’s experiment really demonstrated
two striking facts about large social networks:
first, that short paths are there in abundance;
and second, that people, acting without any sort of global “map” of the network, are effective at collectively finding these short paths.
It is easy to imagine a social network where the first of these is true but the second isn’t — a world where the short paths are there,
but where a letter forwarded from thousands of miles away might simply wander from one acquaintance to another,
lost in a maze of social connections.

11. the numbers are growing by powers of 100 with each step, bringing us to 100 million after four steps.

12. As we’ve seen, social networks abound in triangles
— sets of three people who mutually know each other —
and in particular, many of your 100 friends will know each other.
many of these edges go from one friend to another, not to the rest of world
So the effect of triadic closure in social networks works to limit the number of people you can reach by following short paths.

14. Duncan Watts and Steve Strogatz argued that such a model follows naturally from a combination of
two basic social-network ideas that we saw in Chapters 3 and 4:
Homophily
(the principle that we connect to others who are like ourselves)
and
weak ties
(the links to acquaintances that connect us to parts of the network that would otherwise be far away).
Homophily creates many triangles,
while the weak ties produce the kind of widely branching structure that reaches many nodes in a few steps.

15. Start with a ring of n vertices, each connected to its k nearest neighbors by edges (in the figure, n=20, k=4)
Choose a vertex and the edge that connects it to its nearest neighbor in a clockwise sense
With a probability, reconnect this edge to a vertex chosen uniformly at random over the entire ring
Repeat this process, considering each vertex in turn until one lap is completed
Next consider the edges that connect vertices to their second nearest neighbors (as there are nk/2 edges in the entire graph, the rewiring process stops after k/2 laps)

17. Recall that the clustering coefficient C is defined as follows:
Suppose that a vertex v has kv neighbors; then
at most kv (kv – 1)/2 edges can exist between them
Let Cv denote the fraction of these allowable edges that actually exist
Define C as the avegare of Cv over all v
For a regular lattice shown
C = 3 (k – 2)/[4 (k – 1)]
Cv reflects the extent to which friends of v are also friends of each other
C measures the cliquishness

18. The characteristic path length L, is defined as the number of edges in the shortest path between two vertices, averaged over all pairs of vertices
For a regular lattice shown
L = n (n + k – 2)/[2k (n – 1)]
L is the average number of friendships in the shortest chain connecting two people

19. Watts and Strogatz, Nature, 393, 440 (1998)

21. Note the log horizontal scale to resolve the rapid drop in L,
The data shown in the figure are averages over 20 random realizations of the rewiring
Note the log horizontal scale to resolve the rapid drop in L,
corresponding to the onset of small world phenomenon
caused by the introduction of a few long-range edges
During this drop for L,
C remains almost constant at its value for the regular lattice,
indicating that the transition to a small world is almost undetectable at the local level

22. At time t=0, a single infective individual is introduced into an otherwise healthy population
Infective individuals are removed permanently after a period of sickness that lasts one unit of time
During this time, each infective can infect each of healthy neighbors with probability r
The critical infectiousness at which the disease infects half of the population decreases rapidly for small p
On subsequent time steps, the disease spreads along the edges of the graph until it either infects the entire population, or it dies out, having infected some fraction of the population in the process.
The critical infectiousness rhalf, at which the disease infects half the population, decreases rapidly for small p

23. For a disease that is sufficiently infectious to infect the entire population regardless of its structure, the time T(p) required for global infection resembles the L(p) curve

24. Recapitulation: Small-world network
Small-world networks interpolate between two limiting cases of regular lattices with high local clustering and of random graphs with short distances between the nodes
High clustering implies that, if node A is linked to node B, and B is linked to C, there is an increased probability that A will also linked to C
Two randomly chosen nodes in the network are likely to be connected through only a small number of links
Experimental study of the phenomenon revealed that it has two fundamental components:
such short links are ubiquitious, and
individuals operating with purely local information are very adept at finding these links

25. Erdős–Rényi Random Graph model is used for generating random networks
in which links are set between nodes with equal probabilities
Starting with n isolated nodes and
connecting each pair of nodes with probability p
As a result, all nodes have roughly the same number of links
(i.e., average degree, <k>)
In a random graph, the presence of a connection
between A and B as well as a connection
between B and C will not influence the
probability of a connection between A and C.

26. Random Graphs (Cont’d)
Average path length:
Clustering coefficient:
Degree distribution
Binomial distribution for small n and Poisson distribution for large n
Probability mass function (PMF)

27. Clustering Coefficient (C) Degree Distribution (P(k))
Topology
Average Path Length (L)
Clustering Coefficient (C)
Degree Distribution (P(k))
Random Graph
Poisson Dist.:
Small World
(Watts & Strogatz, 1998)
Lsw  Lrand
Csw  Crand
Similar to random graph
Scale-Free network
LSF  Lrand
Power-law Dist.:
P(k) ~ k-

28. To really find the shortest path from a starting person to the target,
one would have to instruct the starter to forward a letter to
all of his or her friends,
who in turn should have forwarded the letter to all of their friends,
and so forth.
This “flooding” of the network would have reached the target as rapidly as possible.

29. So the success of the experiment raises fundamental questions about the power of collective search:
even if we posit that the social network contains short paths, why should it have been structured so as to make this type of decentralized search so effective?
Clearly the network contained some type of “gradient” that helped participants guide messages toward the target.

31. The Watts-Strogatz network is thus effective at capturing the density of triangles and the existence of short paths, but not the ability of people, working together in the network, to actually find the paths.
Essentially, the problem is that the weak ties that make the world small are “too random” in this model:
– since they’re completely unrelated to the similarity among nodes that produces the homophily-based links, they’re hard for people to use reliably.

32. Is it easy to find short links?
The network is rich in structured short-range connections and has a few random long-range connections
Long-range connections are added to a two-dimensional lattice controlled by a clustering exponent that determines the probability of a connection between two nodes as a function of their lattice distance
At each step, the holder of the message must pass it across one of its short- or long-range connections
This current holder, however, does not know the long-range connections of nodes that have not touched the message
The primary variable is the expected delivery time, which represents the expected number of steps needed to forward a message between a random source a target in a network
It is crucial to constrain the algorithm to use only local information
Mandatory reading may stall here!

33. Kleinberg, Nature, 406, 845 (2000)

35. Navigation in a small world
A characteristic feature of small-world networks is that their diameter (average shortest path between two random nodes) is exponentially smaller than their size
bounded by logN 
there is always a very short path between any two nodes
There is infact a unique value of exponent at which a decentralized algorithm will be able to discover such short paths
Kleinberg, Nature, 406, 845 (2000)

36. The navigability of small-world networks
The network model is derived from an nxn lattice
Each node, u, has a short range connection to its nearest neighbours (a, b, c, and d) and a long-range connection to a randomly chosen node, where node v is selected with probability proportional to r-a, where r is the lattice distance between u and v

38. The navigability of small-world networks (cont’d)
Each message holder forwards the message across a connection that brings it as close as possible to the target in lattice distance
a = 2 is the only exponent at which any decentralized algorithm can achieve a delivery time bounded by logN
The expected delivery time
T > c nb

39. The navigability of small-world networks (cont’d)
Efficient navigability is a fundamental property of only some small-world structures
The results also generalize to d-dimensional lattices with the critical value of the clustering coefficient equal to d
Simulations of the greedy algorithm yield results that are qualitatively consistent with the analytical bounds