Week 5 - Models of Complex Networks I
Dr. Anthony BonatoRyerson University
AM8002Fall 2014
Key properties of complex networks
1. Large scale.
2. Evolving over time.
3. Power law degree distributions.
4. Small world properties.
• in the next two lectures, we consider various models simulating these properties
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Why model complex networks?
• uncover and explain the generative mechanisms underlying complex networks
• predict the future• nice mathematical challenges• models can uncover the hidden reality of
networks
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“All models are wrong, but some are more useful.” – G.P.E. Box
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G(n,p) random graph model(Erdős, Rényi, 63)
• p = p(n) a real number in (0,1), n a positive integer
• G(n,p): probability space on graphs with nodes {1,…,n}, two nodes joined independently and with probability p
51 2 3 4
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Degrees and diameter
• an event An happens asymptotically almost surely (a.a.s.) in G(n,p) if it holds there with probability tending to 1 as n→∞
Theorem 5.1: A.a.s. the degree of each vertex of G in G(n,p) equals
• concentration: binomial distribution
Theorem 5.2: If p is constant, then a.a.s diam(G(n,p)) = 2.
pnonpnOpn ))1(1()log(
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Aside: evolution of G(n,p)
• think of G(n,p) as evolving from a co-clique to clique as p increases from 0 to 1
• at p=1/n, Erdős and Rényi observed something interesting happens a.a.s.:– with p = c/n, with c < 1, the graph is disconnected with all
components trees, the largest of order Θ(log(n))– as p = c/n, with c > 1, the graph becomes connected with a giant
component of order Θ(n)
• Erdős and Rényi called this the double jump• physicists call it the phase transition: it is similar to
phenomena like freezing or boiling
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G(n,p) is not a model for complex networks
• degree distribution is binomial
• low diameter, rich but uniform substructures
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Preferential attachment model
Albert-László Barabási Réka Albert
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Preferential attachment
• say there are n nodes xi in G, and we add in a new node z
• z is joined to the xi by preferential attachment if the probability zxi is an edge is proportional to degrees:
• the larger deg(xi), the higher the probability that z is joined to xi
|)(|2
deg
)deg(
deg
1
GE
x
x
x i
nii
i
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Preferential attachment (PA) model(Barabási, Albert, 99), (Bollobás,Riordan,Spencer,Tusnady,01)
• parameter: m a positive integer• at time 0, add a single edge• at time t+1, add m edges from a new node vt+1 to
existing nodes forming the graph Gt
– the edge vt+1 vs is added with probability
)1(2
deg
mt
vsGt
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Wilensky, U. (2005). NetLogo Preferential Attachment model. http://ccl.northwestern.edu/netlogo/models/PreferentialAttachment.
Preferential Attachment Model(Barabási, Albert, 99), (Bollobás,Riordan,Spencer,Tusnady,01)
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• Theorem 5.3 (BRST,01) A.a.s. for all k satisfying 0 ≤ k ≤ t1/15
• Theorem 5.4 (Bollobás, Riordan, 04) A.a.s. the diameter of the graph at time t is
.))1(1( 3, kot
N tk
Properties of the PA model
.loglog
log)1(1
t
to
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Idea of proof of power law degree distribution
1. Derive an asymptotic expression for E(Nk,t) via a recurrence relation.
2. Prove that Nk,t concentrates around E(Nk,t). – this is accomplished via martingales or using
variance
Azuma-Hoeffding inequality
If (Xi:0 ≤ i ≤ t) is a martingale satisfying the c-Lipschitz condition, then for all real λ > 0,
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.2
exp2)|)Pr(|2
2
0
c
tXX t
Sketch of proof of (2), when m=1
• let A = Nk,t and Zi = Gi
• define Xi = E[A| Z1,…, Zi]
• It can be shown that (Xi) is a martingale (ie a Doob martingale)
• a new vertex can affect the degrees of at most two existing nodes, so we have that
|Xi – Xi-1| ≤ 2
• now apply Azuma-Hoeffding inequality with
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tlog
).1(2)log|)Pr(| 8/10 otttXX t
ACL PA model
• (Aeillo,Chung,Lu,2002) introduced a preferential attachment model where the parameters allow exponents to range over (2,∞)
• Fix p in (0,1). This is the sole parameter of the model.• At t=0, G0 is a single vertex with a loop.
• A vertex-step adds a new vertex v and an edge uv, where u is chosen from existing vertices by preferential attachment.
• An edge-step adds an edge uv, where both endpoints are chosen by preferential attachment.
• To form Gt+1, with probability p take a vertex-step, and with probability 1-p, an edge-step.
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ACL PA, continued
• note that the number of vertices is a random variable; but it concentrates on 1+pt.
• to give a flavour of estimating the expectations of random variables Nk,t we derive the following result. The case (2) for general k>1 follows by an induction.
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Power law for expected degree distribution in ACL PA model
Theorem 5.5 (ACL,02).
1)
2) For k sufficiently large,
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.4
2lim ,1
p
p
t
NE t
t
.lim 22
,
p
p
tk
tkO
t
NE
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Copying models
• new nodes copy some of the link structure of an existing node
Motivation:
1. web page generation (Kumar et al, 00)
2. mutation in biology (Chung et al, 03)
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u
v
x
yN(u)
N(v)
Copying model (Kumar et al,00)
• Parameters: p in (0,1), d > 0 an integer, and a fixed digraph G0 = H with constant out-degree d
• Assume Gt has out-degree d.
• At time t+1, an existing vertex, ut, is chosen u.a.r. The vertex ut is called the copying vertex.
• To form Gt+1 a new vertex vt+1 is added. For each of the d-out-neighbours z of ut, add a directed edge (vt+1,z) with probability 1-p, and with probability p add a directed edge (vt+1,z), where z is chosen u.a.r. from Gt
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Properties of the copying model
• power laws:– Kumar et al: exponent in interval (2,∞)– Chung, Lu: (1,2)
• bipartite subgraphs:– Kumar et al: larger expected number of
bicliques than in PA models– simplified model of community structure
Properties of the copying model
Theorem 5.6 (Kumar et al, 00) If k > 0, then the copying model with parameter p satisfies a.a.s.
In particular, the in-degree distribution follows a power law with exponent (2-p)/(1-p)
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.1
2,
p
pintk kt
N
Properties of the copying model
Theorem 5.7 (Kumar et al, 00) A.a.s. with parameter d >0 and for i ≤ log t,
where Nt,i,d is the expected number of Ki,i which are subgraphs of Gt.
• indicates strong community structure in copying model
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)),exp((,, itN dit
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