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Solution HW 3
3. Caley tree
a) For d = 0, the number of nodes reachable from the central node in 0-steps, N (0),
is 1, the starting node. For d = 1, the number of nodes reachable in exactly
1 step is the number of its first neighbors, N (1) = k = 4. For d = 2, the
number of nodes reachable in exactly two steps N (2) is given by the number of
first neighbors of the central node, k = 4, multiplied by all their first neighbors
excluding the central node, that is k
1. Hence N (2) = k ⇥ (k
1) = 4 ⇥ 3. For a
generic d, the number of nodes reachable in d steps N (d) is given by the number
of nodes at d
1 steps multiplied by their first neighbors that are not a (d
neighbor of the central node, that is k
N (1) ⇥ (k
1)d
1
1. Then, N (d) = N (d
= 4 ⇥ 3d 1 . This is valid for d 2 [1, P ].
1) ⇥ (k
2)1) =
b) The farthest nodes from the central node are the leaves, that are at distance
dmax = P . Since we have to go through the central node to reach a leaf on
one side of the network starting from the leaf on the other side, the diameter is
D = 2dmax = 2P .
c) The total number of nodes N in the network can be computed by summing all
PP
the nodes reachable in 0 steps, 1 steps, up to P steps. N =
d=0 N (d) =
P
P
1 + d=1 k ⇥ (k 1)d 1 = 1 + k (k k1) 2 1 = 2 ⇥ 3P 1. From here, inverting the
formula, we have P =
D = 2P , we have D =
log N2+1
.
log 3
N +1
2 log 2
.
log 3
Because from (b) we know that the diameter is
d) The network displays the small-world distance property since from (c) we see that
the diameter scales as the logarithm of the number of nodes.
4. Snob-ism in social networks
a) In order to reach global connectivity we must first make sure that each of the two
sub-communities are connected. Then it would be sufficient for just a few (even
a single) link between the communities, to open a path between all the nodes in
the network. Since a typical red (blue) node is just log N steps away from the rest
of the reds (blues), a sporadic link between a red and a blue will suffice to make
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all the blues just ⇠ 2 log N steps away from a typical red. To satisfy the first
condition of communal connectivity we require that the average degree within a
sub-community is larger than log N , therefore
p>
log N
.
N
(1)
To evaluate the value of q needed to satisfy the second condition we require that
at least one edge exists between the red and the blue communities. There are
altogether ⇠ N 2 potential pairs between the communities, so to ensure intercommunity connectivity we must satisfy
q>
1
.
N2
b) A very snobbish network is a network where p
(1) and we set q =
p
,
N
(2)
q. If p satisfies the condition
q will automatically satisfy (2). For N
highly snobbish network, since p
1, we will have a
q, but at the same it will be still connected,
with pairs of nodes at a typical distance 2 log N .
c) We define the fraction of purple nodes to be f , namely that there are 2f N purple
nodes out of the 2N nodes making up the whole society. Note that this means that
there are N (1
f ) red (blue) nodes, to ensure that altogether there are 2N nodes.
Moreover, since everybody can attach to a purple node, the pool of nodes from
which a red (blue) node selects its friends consists of N (1 + f ) nodes – including
the N (1
f ) nodes of its own color and the 2f N purple nodes. Consider a typical
red node, with k nearest neighbors. These neighbors are distributed such that
there are
kp =
2f N
2f
k=
k
N (1 + f )
1+f
kr =
N (1 f )
1 f
k=
k
N (1 + f )
1+f
purple nodes, and
red nodes. Here we simply assumed that on average the distribution of friends
would reflect the distribution of color identities in society. This arises from the
assumption of the equally probable random connections. The same analysis for
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purple nodes shows that a typical purple node has
kb =
N (1 f )
1 f
k=
k
2N
2
blue friends (where here the pool of nodes to which it could connect is 2N ). Thus
a typical red node has kp purple neighbors, each with kb blue friends, yielding
kp kb blue people just two steps away. This becomes
kp kb =
✓
f (1 f )
1+f
◆
k2.
Interactive communities will exist if the above is at least one. We obtain
f (1 f ) 2
k
1+f
f
2
✓
1
1
k2
◆
1, =)
f+
1
 0.
k2
This quadratic inequality leads to the requirement
b
r
b2
1
f b+
k2
where
b=
r
b2
1
,
k2
k2 1
.
2k 2
Note that there is both an upper bound and a lower bound for f . The lower
bound makes sure that there are enough purple nodes to enable interconnectivity.
The upper bound ensures that there are enough red and blue nodes (and not too
many purple ones), so that each purple node is guaranteed to have both a blue
and a red node as its nearest neighbors.
d) In the case k
1 we have b ⇡ 12 . The upper bound for f approaches then 1,
while the lower bound gets close to 0. This means that we can allow the purple
community to be as large or as small as we wish, and the network will still be
interconnected.
e) These models capture the tendency of social networks to be homophilic – namely
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that nodes tend to connect with their own type. In a sense it is a generalization
of the geographic embedding assumed in the Watts-Strogatz model, where here
the distance measure is based on some mutual virtue, symbolized here (and unfortunately in many real life societies as well) by color. It is shown, that just like
in the other small world models, even a very small amount of mixing is sufficient
to lead to global connectivity.