**What is Random Graph?**

Random Graph, denoted by Gn,p, consists of n vertices and each possible edge between two vertices is present
with independent probability p. Often one wishes to
express properties of Gn,p not in terms of p but in terms of the average degree z of a vertex. The
average number of edges on the graph as a whole is 1/2*n(n-1)p, and the average number of
ends of edges is twice this, since each edge has two ends. So the average degree of a
vertex is

where the last approximate equality is
good for large n.

**Random Graph Vs Real-world Network**

*Clustering*

Real-world networks show strong
clustering or network
transitivity, whereas Random graph model does
not. A network is said to show
clustering if the probability of two
vertices being connected by an edge is higher when
the vertices in
question have a common neighbor. Watts and Strogatz measured this
clustering by defining a clustering coefficient C, which is the
average probability that two neighbors of a given vertex
are also neighbors of one another. In many
real-world networks the
clustering coefficient is found to have a high value, anywhere
from
a few percent to 50 percent or even more. In the random graph on the
other hand, the probabilities of vertex pairs being connected by
edges
are by definition independent, so that there is no greater
probability of two vertices
being connected if they have a mutual neighbor than if they do not. This means that the clustering coefficient for a
random graph is simply C = p, or equivalently C ~ z/n.

*Degree Distribution*

The probability pk that a vertex in a random graph has
degree
exactly k is given by the binomial distribution

In the limit where n >> kz, this
becomes

which is the well-known Poisson
distribution. Both binomial and Poisson distributions
are strongly
peaked about the mean z, and have a large-k tail that decays rapidly
as
1/k!. We can compare these predictions to the degree
distributions of real networks by
constructing histograms of the
degrees of vertices in the real networks.

As the figure shows, in most cases the degree distribution
of the real network is power-law degree distributions which is very
different from the
Poisson distribution.

**Small World Effect**

Random graphs having non-Poisson degree distributions can be generated by specifying a degree sequence, i.e., a
specified set {ki} of the
degrees of the vertices i = 1...n. Typically this set will be chosen in such a way that the fraction of vertices having degree k
will tend to the
desired degree distribution pk as n becomes large. Once one has one's degree sequence, the
method for generating the graph is as follows: one gives each vertex i a
number {ki} of “stubs",ends of edges emerging
from the
vertex, and then one chooses pairs of these stubs uniformly at random
and
joins them together to make complete edges. When all stubs have
been used up, the
resulting graph is a random graph with the desired degree
sequence.

The mean number of neighbors of a
randomly chosen vertex A in a graph with degree distribution pk is

The
probability distribution of the
degree of the vertex to which an
edge leads is proportional to kpk

and the number of edges emerging from the vertex
other than
the one we arrived along is
one less than the total degree of
the vertex and its correctly normalized distribution
is therefore

or, equivalently,

The average degree of such a vertex is
then

We can extend this calculation to
further neighbours also. The average number
of edges leading from
each second neighbour, other than the one we arrived along, is
also
given by (1), and indeed this is true at any distance m away from
vertex A. Thus
the average number of neighbours at distance m is

where z1 = z = <k> and z2 is given by
Eq. (2). Iterating this equation we then
determine that

Depending on whether z2 is greater than z1 or not, this expression will either diverge
or converge
exponentially as m becomes large, so that the average total number
of neighbors of vertex A at all distances is finite if z2 < z1 or
infinite if z2 > z1 (in the limit of infinite n).

Thus the graph shows a
phase transition similar to that of the
random graph precisely at
the point where

z2 = z1.

Making use of Eq. (2) and rearranging, we find that this condition is also equivalent to

or, as it is more commonly written,

The quantity of interest in many
networks is the typical distance through the network between pairs of vertices.
If
we are “below" the phase transition of Eq. (4), in the
regime where there is no giant
component, then most pairs of
vertices will not be connected to one another at all, so
vertex-vertex distance has little meaning. On the other hand, "above" the transition,
where there is a giant component, all vertices in
this giant component are connected
by some path to all others. Eq.
(3) tells us the average number of vertices a distance
m away from a
given vertex A in the giant component. When the total number of
vertices within distance m is equal to the size n of the whole graph,
m is equal to
the so-called “radius" r of the network around
vertex A. Indeed, since z2/z1>>1 well
above the transition, the
number of vertices at distance m grows quickly with m in this
regime, which means that most of the vertices in the
network will
be far from A, around distance r, and r is thus also
approximately equal to the average vertex-vertex distance l. Well above
the transition therefore, l is given approximately by, or,

as noted above, this expression reduces
to the well-known standard formula for this

case:

The important point to notice about Eq. (5) is that the vertex-vertex distance increases logarithmically with the graph size n, i.e., it grows rather slowly. Even for very large networks we expect the typical distance through the network from one vertex to another to be quite small. In social networks this effect is known as the

**, and was famously observed by the experimental psychologist Stanley Milgram in the letter-passing experiments he conducted in the 1960s (Milgram, 1967; Travers and Milgram, 1969; Kleinfeld, 2000).**

*small-world effect*References-

[1] M.E.J Newman, Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, U.S.A.

"

*Random graphs as models of networks*"

## No comments:

## Post a Comment