Notes on popularity versus similarity model

1. Notes on Popularity versus Similarity Network Model
Peiyuan Sun
April 13, 2017
Model 0. PA with initial attractiveness
• reference
Structure of Growing Networks with Preferential Linking
S.N.Dorogovtsev, J.F.F.Mendes, and A.N.Samukhin
Physical Review Letters, 2000
• generative process:
1. at equally discrete time step, a new site appears,
2. simultaneously, m new directed links coming out from nonspecified sites are in-
troduced,
3. the m new links are distributed proportional to the attractiveness of node s:
As = A + qs, where A is each node’s initial attractiveness and qs incoming degree
of node s.
• Master Equation
P(q, s, t + 1) =
m
l=0
P(ml)
s P(q, s, t + 1)
=
m
l=0
m
l
q − l + am
(1 + a)mt
l
1 −
q − l + am
(1 + a)mt
m−l
p(q − l, s, t)
(1)
P(q, s, s) = δ(q) (2)
connectivity distribution of the entire networks:
P(q, t) =
t
u=1
P(q, s, t)/t (3)
1

2. summing up E.q 1 over s from 1 to t
t
s=1
P(q, s, t + 1) =
t
s=1
m
l=0
P(ml)
s P(q − l, s, t + 1)
=
t
s=1
m
l=0
m
l
q − l + am
(1 + a)mt
l
1 −
q − l + am
(1 + a)mt
m−l
P(q − l, s, t)
=
m
l=0
m
l
q − l + am
(1 + a)mt
l
1 −
q − l + am
(1 + a)mt
m−l t
s=1
P(q − l, s, t)
=
m
0
1 −
q − l + am
(1 + a)mt
m
tP(q, t)
+
m
1
q − l + am
(1 + a)mt
1 −
q − l + am
(1 + a)mt
m−1
tP(q − 1, t)
= t 1 −
q − l + am
(1 + a)mt
m
P(q, t)
+ mt
q − l + am
(1 + a)mt
1 −
q − l + am
(1 + a)mt
m−1
P(q − 1, t)
= t −
q − l + am
1 + a
P(q, t) +
q − l + am
1 + a
P(q − 1, t) + O(
P
t
)
(4)
t
s=1
P(q, s, t + 1) =
t+1
s=1
P(q, s, t + 1) − P(q, t + 1, t + 1)
= (t + 1)P(q, t + 1) − P(q, t + 1, t + 1)
(5)
(t + 1)P(q, t + 1) − P(q, t + 1, t + 1) = t −
q − l + am
1 + a
P(q, t)
+
q − l + am
1 + a
P(q − 1, t) + O(
P
t
)
(6)
(1 + a)t (P(q, t + 1) − P(q, t)) + (1 + a)P(q, t + 1) + (q + am)P(q, t) − (q − 1 + am)P(q − 1, t)
= (1 + a)δ(q)
(7)
at long times t 1:
(1 + a)t
∂P(q, t)
∂t
+ (1 + a)P(q, t) + (q + am)P(q, t) − (q − 1 + am)P(q − 1, t) = (1 + a)δ(q)
(8)
assume the limit P(q) = P(q, t → ∞) exists:
(1 + a)P(q) + (q + am)P(q) − (q − 1 + am)P(q − 1) = (1 + a)δ(q) (9)
2

3. assume the generating function:
Φ(z) =
∞
q=0
P(q)zq
(10)
we first derive some trivial cases for the generating function:
Z[P(q)] =
∞
q=0
P(q)zq
= Φ(z)
Z[qP(q)] =
∞
q=0
qP(q)zq
= z
∞
q=0
qP(q)zq−1
= z
dΦ
dz
Z[qP(q − 1)] =
∞
q=1
qP(q − 1)zq
= z2
∞
q=1
(q − 1)P(q − 1)zq−2
+ z
∞
q=1
P(q − 1)zq−1
= z2
∞
q =0
q P(q )zq −1
+ z
∞
q =0
P(q )zq
= z2 dΦ
dz
+ zΦ
Z[P(q − 1)] = z
∞
q=1
P(q − 1)zq−1
= z
∞
q =0
P(q )zq
= zΦ
(11)
then E.q 9 transforms to the following differential equation:
(1 + a)Φ + z
dΦ
dz
+ maΦ − z2 dΦ
dz
− zΦ − (ma − 1)zΦ = 1 + a (12)
after some simple algebra operations:
z(1 − z)
dΦ
dz
+ ma(1 − z)Φ + (1 + a)Φ = 1 + a (13)
this is a first order nonhomogeneous linear differential equation with following formula:
dy
dx
+ P(x)y = Q(x) (14)
and the general solution to this equation has the following forms:
y = Ce− P(x)dx
+ e− P(x)dx
Q(x)e− P(x)dx
dx (15)
3

4. comparing E.q 12 with E.q 14:
P(z) =
ma(1 − z) + 1 + a
z(1 − z)
Q(z) =
1 + a
z(1 − z)
(16)
do some simple integration operations:
P(z)dz = [(m + 1)a + 1] lnz − (1 + a)ln(1 − z)
Q(z)e P(z)dz
dz = (1 + a)
z
0
x(m+1)a
(1 − x)a+2
dx
e− P(z)dz
=
(1 − z)1+a
z(m+1)a+1
(17)
then we get the general formula of Φ:
Φ = Cz−1−(m+1)a
(1 − z)1+a
+ (1 + a)z−1−(m+1)a
(1 − z)1+a
z
0
dx
x(m+1)a
(1 − x)2+a
(18)
from E.q 17, we can get that the constant term C = 0, then:
Φ = (1 + a)z−1−(m+1)a
(1 − z)1+a
z
0
dx
x(m+1)a
(1 − x)2+a
=
1 + a
1 + (1 + m)a2
F1[1, ma; 2 + (m + 1)a; z]
(19)
using hypergeometric function’s expansion in z and comparing with the z’s generating
function:
P(q) = (1 + a)
Γ [(m + 1)a + 1]
Γ(ma)
Γ(q + ma)
q + 2 + (m + 1)a
(20)
E.q 20 is the 1st main result which is the analytic solution of the degree distribution
of the growing network. We then discuss 2 special cases:
1. when a = 1, which corresponding to the case As = m + qs
P(q) =
2m(m + 1)
(q + m)(q + m + 1)(q + m + 2)
(21)
2. when ma + q 1
P(q) ≈ (1 + a)
Γ [(m + 1)a + 1]
Γ(ma)
(q + ma)−(2+a)
= (1 + a)
Γ [(m + 1)a + 1]
Γ(ma)
(q + ma)−γ
(22)
with γ = 2 + a = 2 + A
m
4

5. then let us derive the distribution of P(q, s, t), for t 1 we expand the Master Equation
to 2nd order:
P(q, s, t + 1) = 1 −
q + am
(1 + a)t
P(q, s, t) +
q − 1 + am
(1 + a)t
P(q − 1, s, t) + O(
p
t2
) (23)
(1 + a)tP(q, s, t + 1) = [(1 + a)t − (q + am)] P(q, s, t) + (q − 1 + am)P(q − 1, s, t) + O(
p
t
)
(24)
(1 + a)tP(q, s, t + 1) − (1 + a)tP(q, s, t) = (q − 1 + am)P(q − 1, s, t) − (q + am)P(q, s, t)
(25)
replace the difference with derivative for large t:
(1 + a)t
∂p
∂t
(q, s, t) = (q − 1 + am)P(q − 1, s, t) − (q + am)P(q, s, t) (26)
define the generating function G(x, t) by means of:
G(x, t) =
+∞
q=0
P(q, s, t)xq
(27)
5

6. then we begin by taking the time derivative:
∂G(x, t)
∂t
=
+∞
q=0
∂P(q, s, t)
∂t
xq
=
+∞
q=0
[
q
(1 + a)t
P(q − 1, s, t) +
am − 1
(1 + a)t
P(q − 1, s, t)
−
q
(1 + a)t
P(q, s, t) −
am
(1 + a)t
P(q, s, t)]xq
=
1
(1 + a)t
+∞
q=1
qP(q − 1, s, t)xq
+
am − 1
(1 + a)t
+∞
q=1
P(q − 1, s, t)xq
−
1
(1 + a)t
+∞
q=0
qP(q, s, t)xq
−
am
(1 + a)t
+∞
q=0
P(q, s, t)xq
=
1
(1 + a)t
+∞
q=1
(q − 1)P(q − 1, s, t)xq
+
1
(1 + a)t
+∞
q=1
P(q − 1, s, t)xq
+
(am − 1)x
(1 + a)t
+∞
q=1
P(q − 1, s, t)xq−1
−
x
(1 + a)t
+∞
q=0
qP(q, s, t)xq−1
−
am
(1 + a)t
+∞
q=0
P(q, s, t)xq
=
1
(1 + a)t
+∞
q=0
qP(q, s, t)xq+1
+
1
(1 + a)t
+∞
q=0
P(q, s, t)xq+1
+
(am − 1)x
(1 + a)t
+∞
q=0
P(q, s, t)xq
−
x
(1 + a)t
+∞
q=0
qP(q, s, t)xq−1
−
am
(1 + a)t
+∞
q=0
P(q, s, t)xq
=
x2
(1 + a)t
+∞
q=0
qP(q, s, t)xq−1
+
x
(1 + a)t
+∞
q=0
P(q, s, t)xq
+
(am − 1)x
(1 + a)t
+∞
q=0
P(q, s, t)xq
−
x
(1 + a)t
+∞
q=0
qP(q, s, t)xq−1
−
am
(1 + a)t
+∞
q=0
P(q, s, t)xq
=
x2
(1 + a)t
∂G(x, t)
∂x
+
x
(1 + a)t
G(x, t) +
(am − 1)x
(1 + a)t
G(x, t)
−
x
(1 + a)t
∂G(x, t)
∂x
−
am
(1 + a)t
G(x, t)
=
x(x − 1)
(1 + a)t
∂G(x, t)
∂x
+
am(x − 1)
(1 + a)t
G(x, t)
(28)
6

7. then we get the partial differential equation for the generating function G(x, t):
∂G(x, t)
∂t
=
x(x − 1)
(1 + a)t
∂G(x, t)
∂x
+
am(x − 1)
(1 + a)t
G(x, t) (29)
with initial condition:
G(x, s) =
+∞
q=0
P(q, s, s)xq
=
+∞
q=0
δ(q)xq
= 1 (30)
To get the analytic solution of E.q 29, we resort to the method of characteristics.
By the following characteristic equation:
dx
−x(x−1)
(1+a)t
=
dt
1
=
dG
am(x−1)
(1+a)t
G(x, t)
(31)
which is equivalent to:
dx
x(x − 1)
=
dt
(1 + a)t
=
dG
am(x − 1)G
(32)
we consider the following two pairs:
dx
x(x − 1)
=
dt
(1 + a)t
(33)
dx
x(x − 1)
=
dG
am(x − 1)G
(34)
do integration on both sides of E.q 33 and E.q 34:
x
x − 1
t− 1
1+a = C1 (35)
G(x, t) =
1
x
C2
(36)
we could view C2 as a function of C1, since C1 = x
x−1
t− 1
1+a , we can write the general
solution of the differential equation as:
G(x, t) =
1
f( x
x−1
t− 1
1+a )
1
x
(37)
plug in the initial condition:
f(
x
x − 1
t− 1
1+a ) =
1
x
(38)
replace x
x−1
t− 1
1+a with r:
f(r) =
r − s− 1
1+a
r
(39)
7

8. insert f( x
x−1
t− 1
1+a ) into 37, we get the analytic solution of the partial differential equa-
tion E.q 29:
G(x, t) = 1 −
s
t
− 1
1+a
x +
s
t
− 1
1+a
−am
(40)
to get the analytic result of P(q, s, t), we use Taylor expansion of E.q 40. The coefficient
of xq
is:
∂G(n)
(x = 0, t)
∂x
1
q!
=
(−am)(−am − 1) · · · (−am − q + 1)
q!
1 −
s
t
− 1
1+a
q
s
t
amq
1+a
=
(−1)q
am(am + 1) · · · (am + q − 1)
q!
(−1)q
1 −
s
t
1
1+a
q
s
t
am
1+a
=
am(am + 1) · · · (am + q − 1))
q!
1 −
s
t
1
1+a
q
s
t
am
1+a
=
Γ(am + q)
Γ(am)q!
1 −
s
t
1
1+a
q
s
t
am
1+a
(41)
then:
P(q, s, t) =
Γ(am + q)
Γ(am)q!
1 −
s
t
1
1+a
q
s
t
am
1+a
(42)
the average connectivity of a given site is:
∂G(x = 1, t)
∂x
=
+∞
q=0
qP(q, s, t)xq−1
x=1
= am
s
t
− 1
1+a
− 1
(43)
¯q(s, t) = am
s
t
− 1
1+a
− 1 (44)
let β = 1/(1 + a) and comparing with γ = 2 + a, we get the relation between single
site’s age and the whole network:
β(γ − 1) = 1 (45)
8

9. Model 1. Popularity versus Similarity (m closest nodes)
• reference
Popularity versus Similarity in growing networks
Fragkiskos Papadopoulos, Maksim Kitsak, M. ´Angeles Serrano
Mari´an Bogu˜n´a & Dimitri Krioukov
Nature, 2012
• generative process:
1. initially the network is empty;
2. at time t ≥ 1, new node t appears having coordinates (rt, θt), where rt = lnt,
while θt is uniformly distributed on [0, 2π], and every existing node s, s < t,
moves increasing its radial coordinate according to rs(t) = βrs + (1 − β)rt with
parameter β ∈ [0, 1];
3. node t connects to the m hyperbolically closest nodes s, s < t; at early times
t ≤ m, node t connects to all the existing nodes. The hyperbolic distance between
two points (rs, θs) and (rt, θt) is given by:
xst =
1
2
arccosh(cosh2rscosh2rt − sinh2rssinh2rtcosΘst)
≈ rs + rt + ln(θst/2), whereθst = π − |π − |θs − θt||
(46)
• Degree Distribution
we first derive the probability that node s attracts a link from the m new links intro-
duced by node t:
1. Since the generative process assume that each new node connects to the hyper-
bolically m closest existing nodes. The only requirement that need to meet is that
the hyperbolic distance xst is less than the largest hyperbolic distance between
node t and its m closest neighbors. We could image this area as a disc centered
at node t. Then:
(a) the radius of this disc Rt must satisfy the following condition:
¯N(Rt) =
t
1
p(s, t)ds = m (47)
(b) the connection probability between node s and t:
p(s, t) = p(xst < Rt)
= p(rs(t) + rt + ln
θst
2
< Rt)
= p(θst < 2e−(rs(t)+rt−Rt)
)
=
π
2
e−(rs(t)+rt−Rt)
(48)
9

10. (c) plug in this probability to the first equation:
¯N(Rt) =
t
1
π
2
e−(rs(t)+rt−Rt)
ds = m
=
2
π
e−(rt−Rt) 1 − e−(1−β)rt
1 − β
= m
(49)
and get Rt:
Rt = rt − ln
2(1 − e−(1−β)rt
)
πm(1 − β)
(50)
(d) then the probability that node s attracts a link from node t is:
p(s, t) = p(xst < Rt)
= m
e−rs(t)
1
1−β
(1 − e−(1−β)rt )
= m
e−rs(t)
t
1
e−ri(t)di
= m
(s
t
)−β
t
1
(i
t
)−βdi
(51)
during step 2 and 3, the author uses a very important transformation:
t
1
e−rs(t)ds
=
t
1
e−βrs−(1−β)rt
ds
= e−(1−β)rt
t
1
elns−β
ds
= e−(1−β)rt
t
1
s−β
ds
=
1
1 − β
(1 − e−(1−β)rt
)
(52)
2. from Model 0, we can deduce that the probability that an existing node s attracts
a link from the new node t is:
p(s, t) = m
q(s, t) + A
(m + A)t
(53)
the degree distribution for a single site is:
q(s, t) = A
s
t
−β
− 1 (54)
the author makes another assumption that the total attractiveness of the network
can be written as:
(m + A)t =
t
1
(q(s, t) + A)ds (55)
10

11. then 53 can be transformed to:
p(s, t) = m
s
t
−β
t
1
i
t
−β
di
(56)
which can be found equivalent to Model 0’s result.
It is worth noting that β in Model 0 comes from the expectation degree of single
node and equals to 1/(1 + A
m
). And it has been proven that the whole network’s
exponent and single node’s degreee distribution follow the rule:β(γ − 1) = 1. We
recall that this relation is derived based on the probability that existing node s
attracts a link from the m new links introduced by node t.For the PSO model the
basic master equation still holds:
t
s=1
P(q, s, t + 1) =
t
s=1
m
l=0
P(ml)
s P(q − l, s, t + 1) (57)
So the network’s degree distribution is the same as in Model 0.
11

12. Model 2. Popularity versus Similarity (select randomly m nodes according to Fermi-dirac
distribution)
• reference
Popularity versus Similarity in growing networks
Fragkiskos Papadopoulos, Maksim Kitsak, M. ´Angeles Serrano
Mari´an Bogu˜n´a & Dimitri Krioukov
Nature, 2012
• generative process:
1. initially the network is empty;
2. at time t ≥ 1, new node t appears having coordinates (rt, θt), where rt = lnt,
while θt is uniformly distributed on [0, 2π], and every existing node s, s < t,
moves increasing its radial coordinate according to rs(t) = βrs + (1 − β)rt with
parameter β ∈ [0, 1];
3. node t picks a randomly chosen existing node s, s < t, and connects to it with
probability p(xst) = 1
1+e(xst−Rt)/T . Repeat this procedure until t get m links
• Degree Distribution
we again derive the probability that node s attracts a link from the m new links
introduced by node t:
1. the probability that node t selects randomly a node s, s < t, and connect with it:
p(s, t) =
1
t
1
π
t
1
1
1 + X(s, t)θst
2
1
T
dθst (58)
I think the term 1/t stands for selecting randomly a node from [1 · · · t]. Since the
model assumes that each node appears uniformly on the discrete time interval.
And the integration term stands for the average probability for node t connecting
with node s. Since node t appears with coordinate (rt, θt) within which rt is fixed
and the θt is a random variable. The author sums up all the possible angles and
divides it by the interval [0 · · · π].
The integration is feasible by residue theorem which has the following classic
conclusion:
+∞
0
1
1 + xβ
dx =
π
β
sinπ
β
, for β > 1 (59)
2. Then the probability that node s attracts a link from node t is:
p(s, t) = m
p(s , t)
p(t)
= m
s
t
−β
t
1
i
t
−β
di
(60)
which is equivalent to Model0 and Model1
12

13. Model 3. Popularity versus Similarity (try to connect with each node according to Fermi-
dirac distribution)
• reference
Popularity versus Similarity in growing networks
Fragkiskos Papadopoulos, Maksim Kitsak, M. ´Angeles Serrano
Mari´an Bogu˜n´a & Dimitri Krioukov
Nature, 2012
• generative process:
1. initially the network is empty;
2. at time t ≥ 1, new node t appears having coordinates (rt, θt), where rt = lnt,
while θt is uniformly distributed on [0, 2π], and every existing node s, s < t,
moves increasing its radial coordinate according to rs(t) = βrs + (1 − β)rt with
parameter β ∈ [0, 1];
3. node t tries to connect with each existing node s (s < t) with probability p(xst) =
1
1+e(xst−Rt)/T .
• Degree Distribution
we again derive the probability that node s attracts a link from the m new links
introduced by node t:
1. Intuitively, the probability that node t connects with node s with probability:
p (s, t) = tp(s, t) (61)
we still have an useful condition that the average number of nodes that node t
connects to is:
¯N(Rt) =
t
1
p (s, t)ds = t
t
1
p (s, t)ds = tp(t) (62)
if we set ¯N(Rt) as m then we get the following relationship:
t =
m
p(t)
(63)
then we can get the equivalent probability formula:
p (s, t) = tp(s, t) =
m
p(t)
p(s, t) = m
s
t
−β
t
1
i
t
−β
di
(64)
which is equivalent to Model0, Model1 and Model2
13

14. Model 4. Popularity versus Similarity with internal links
• reference
Network Mapping by Replaying Hyperbolic Growing
Fragkiskos Papadopoulos, Constantinos Psomas, and Dmitri Krioukov
IEEE/ACM Transactions on Networking, 2015
• generative process:
1. initially the network is empty;
2. at time t ≥ 1, new node t appears having coordinates (rt, θt), where rt = lnt,
while θt is uniformly distributed on [0, 2π], and every existing node s, s < t,
moves increasing its radial coordinate according to rs(t) = βrs + (1 − β)rt with
parameter β ∈ [0, 1];
3. node t tries to connect with each existing node s (s < t) with probability p(xst) =
1
1+e(xst−Rt)/T .
4. the radius Rt is adjusted to make the expected number of connections that i
establishes is ¯mi(t) = m + ¯Li(t), where ¯Li(t) is the expected number of internal
links between node i and existing nodes j < i by time t.
• Network Mapping
1. ¯Li(t)
we first derive p(i, j, l) which means node i and j establishes an internal link at
time l. The trick is that we need to integrate out the angular random variable
θst and divide it by π to get the averaged connection probability.
p(i, j, l) =
1
l2
1
π
π
0
1
e
ri(l)+rj(l)+ln
θij
2
−Rl /T
dθij
let X(i, j, l) = e(ri(l)+rj(l)−Rl)
=
1
l2
1
π
π
0
1
1 + X(i, j, l)θ
2
1
T
dθ
ref to E.q 59
=
2T
l2sinTπ
·
1
X(i, j, l)
(65)
there is an hidden approximation (trick) here that E.q 59 holds for the +∞ in-
tegration upper bound which is not true here.
Then we derive Π(i, j, l) which stands for the probability that node i and j are
selected randomly and get connected. Since L internal links are introduced when
14

15. each new node appears:
Π(i, j, l) = L ·
2T
l2sinTπX(i,j,l)
1
2
·
l
0
l
0
2T
l2sinTπX(i,j,l)
didj
= 2L ·
e−(ri(l)+rj(l)−Rl)
l
0
l
0
e−(ri(l)+rj(l)−Rl)didj
= 2L ·
e−(ri(l)+rj(l))
l
0
e−ri(l)di
l
0
e−rj(l)dj
= 2L ·
e−(ri(l)+rj(l))
l
0
e−ri(l)di
2
= 2L ·
(ij)−β
l−2(1−β)
I2
l
where Il =
1
1 − β
· (1 − e−(1−β)rl
)
(66)
then we have all the ingredients for deriving ¯Li(t):
¯Li(t) =
i
1
t
i
Π(i, j, l)dl dj
=
i
1
2L ·
(ij)−β
l−2(1−β)
I2
l
dl dj
assume Il ≈ It for large l
=
2L(1 − β)
(1 − t−(1−β))2(2β − 1)
(
t
i
)2β−1
− 1 (1 − i−(1−β)
)
(67)
2. Node Degree Distribution ¯ki(t)
Recall from Model 1 that the probability that an existing node i attracts a link
from one of m links introduced by node l is:
Π(i, l) = m
e−ri(l)
l
1
e−ri(l)di
(68)
In this Model, m is replaced with mi(t) and during the derivation process the au-
thor makes use of the approximation Il ≈ It for large t again. This approximation
is necessary but not elegant since the result is still very complex. The explicit
formula is omitted here. But the conclusion is as previous Model:
¯ki(t) ∝
i
t
β
¯k = 2(m + L)
(69)
15

16. 3. Node Appearance Time
The author employs two stage process to inference the coordinates of nodes. The
first step is to deduce the node appearance time. Denote the nodes coordinates
likelihood as:
L1 = L(ri(t), θi|αij, Φ) (70)
(a) Using the following Bayes’ Rule:
L({ri(t), θi}|αij, Φ)
L1
· L(αij|Φ)
L3
= P({ri(t), θi}|Φ) · L(αij|{ri(t), θi}, Φ)
L2
(71)
It should be noted that we need only solve {ri(t), θi}, so maximizing L1 is
equivalent to maximizing P({ri(t), θi}|Φ) · L2.
(b) P({ri(t), θi}|Φ)
which stands for the joint probability of nodes’ coordinates. We first derive
node’s radius probability:
p(ri(t) < r) ⇒ βri + (1 − β)rt < r
⇒ i < e− 1
β
r−1−β
β
rt
(72)
since radius coordinate is uniformly distributed:
p(ri(t) < r) =
e− 1
β
r−1−β
β
rt
t
(73)
This is the cumulative distribution, so the probability distribution is the
derivation of p(ri(t) < r) with respect to r:
ft(i) =
1
β
e
1
β
(r−rt)
(74)
Then
P({ri(t), θi}|Φ) =
1
2π
t t
i
ft(i) (75)
(c) L2
L2 =
1≤j<i≤t
˜p(xij(t))αij
{1 − ˜p(xij(t))}1−αij
(76)
(d) maximize L1
the derivative of L1’s logarithm with respect to ri(t):
∂L1
∂ri(t)
=
1
T
− T ·
t
j=1,j=i
αij −
1
1 + e(xij(t)−Rt
(77)
16

17. set the derivative equals to 0:
˜¯ki(t)
expected degree
= ki
empirical degree
−
T
β (78)
using ¯ki(t)’s mean-field approximation:
˜¯ki(t)
expected degree
≈ ¯ki(t)
mean filed approximation
∝
i∗
t
−β
= ki
empirical degree
−
T
β (79)
then the MLE of i∗
:
i∗
MLE of appearance time
∝ k
− 1
β
i = k
−(γ−1)
i (80)
we can find that the higher the empirical degree of the node the earlier its
MLE appearance time.
4. node’s angular coordinate
maximizing the following likelihood per node by sampling different θ in [0, 2π]:
Li
2 =
1≤j<i
p(xij)αij
[1 − p(xij)]1−αij
(81)
17