The Barabasi and Albert model ´ • A discrete time network evolution process, relating the graph G(t + 1) to G(t). • Start at t=0 with a single isolated node. • At each discrete time step, a new node arrives. • This new node makes m edges to already existing nodes. (Why m edges? i.e., what happens if m = 1?) • The likelihood of a new edge to connect to an existing node j is proportional to the degree of node j, denoted dj. • We are interested in the limit of large graph size, n → ∞. Visualizing a PA graph (m = 1) at n = 5000 Probabilistic treatment (kinetic theory) • Start at t = 0 with one isolated node (or a small core set). – At time t the total number of nodes added n = t. – At time t the total number of edges added is mt. • Let dj(t) denote the degree of node j at time t. • Probability an edge added at t + 1 connects to node j: P r(t + 1 → j) = dj(t)/ P j dj(t). • Normalization constant easy (but time dependent): P j dj(t) = 2mt (Each node 1 through t, contributes m edges.) (Each edge augments the degree of two nodes.) Network evolution Process on the degree sequence • The probability a new edge connects to a particular node j of degree k at time t: dj(t)/ P j dj(t) = k/2mt • Also, when a node of degree k gains an attachment, it becomes a node of degree k + 1. • When the new node arrives, it increases by one the number of nodes of degree m. The “Master Equation” / “rate eqns” / “kinetic theory” Evolution of the typical graph (Let nk,t ≡ expected number of nodes of degree k at time t, and nt ≡ total number of nodes added by time t: Note nt = t) For each arriving link: • For k > m : nk,t+1 = nk,t + (k−1) 2mt nk−1,t − k 2mt nk,t • For k = m : nm,t+1 = nm,t − m 2mt nm,t The “Master Equation” / “rate eqns” / “kinetic theory” Evolution of the typical graph For each arriving link (from last page): • For k > m : nk,t+1 = nk,t + (k−1) 2mt nk−1,t − k 2mt nk,t • For k = m : nm,t+1 = nm,t − m 2mt nm,t Each new node contributes m links (and one new node). Assuming n → ∞ there are no multi-edges: • For k > m : nk,t+1 = nk,t + m(k−1) 2mt nk−1,t − mk 2mt nk,t • For k = m : nm,t+1 = nm,t + 1 − m2 2mt nm,t Translating from number of nodes nk,t to probabilities pk,t pk,t = nk,t/n(t) = nk,t/t → nk,t = t pk,t For each arriving node, after m edges added: • For k > m : (t + 1) pk,t+1 = t pk,t + (k−1) 2 pk−1,t − k 2 pk,t • For k = m : (t + 1) pm,t+1 = t pm,t + 1 − m 2 pm,t Steady-state distribution We want to consider the final, steady-state: pk,t = pk. • For k > m : (t + 1) pk = t pk + (k−1) 2 pk−1 − k 2 pk • For k = m : (t + 1) pm = t pm + 1 − m 2 pm Rearranging and solving for pk: • For k > m : pk = (k−1) (k+2) pk−1 • For k = m : pm = 2 (m+2) Recursing pk to pm pk = (k−1)(k−2)···(m) (k+2)(k+1)···(m+3) · pm = m(m+1)(m+2) (k+2)(k+1)k · 2 (m+2) pk = 2m(m+1) (k+2)(k+1)k For k 1 pk ∼ k −3 Based on this info,can you solve the question below?
The Barabasi and Albert model ´
• A discrete time network evolution process,
relating the graph G(t + 1) to G(t).
• Start at t=0 with a single isolated node.
• At each discrete time step, a new node arrives.
• This new node makes m edges to already existing nodes.
(Why m edges? i.e., what happens if m = 1?)
• The likelihood of a new edge to connect to an existing node j
is proportional to the degree of node j, denoted dj.
• We are interested in the limit of large graph size, n → ∞.
Visualizing a PA graph (m = 1) at n = 5000
Probabilistic treatment (kinetic theory)
• Start at t = 0 with one isolated node (or a small core set).
– At time t the total number of nodes added n = t.
– At time t the total number of edges added is mt.
• Let dj(t) denote the degree of node j at time t.
• Probability an edge added at t + 1 connects to node j:
P r(t + 1 → j) = dj(t)/
P
j
dj(t).
• Normalization constant easy (but time dependent):
P
j
dj(t) = 2mt
(Each node 1 through t, contributes m edges.)
(Each edge augments the degree of two nodes.)
Network evolution
Process on the degree sequence
• The probability a new edge connects to a particular node j of degree k at
time t:
dj(t)/
P
j
dj(t) = k/2mt
• Also, when a node of degree k gains an attachment, it becomes a node of
degree k + 1.
• When the new node arrives, it increases by one the number of nodes of
degree m.
The “Master Equation” / “rate eqns” / “kinetic theory”
Evolution of the typical graph
(Let nk,t ≡ expected number of nodes of degree k at time t,
and nt ≡ total number of nodes added by time t: Note nt = t)
For each arriving link:
• For k > m : nk,t+1 = nk,t +
(k−1)
2mt nk−1,t −
k
2mt nk,t
• For k = m : nm,t+1 = nm,t −
m
2mt nm,t
The “Master Equation” / “rate eqns” / “kinetic theory”
Evolution of the typical graph
For each arriving link (from last page):
• For k > m : nk,t+1 = nk,t +
(k−1)
2mt nk−1,t −
k
2mt nk,t
• For k = m : nm,t+1 = nm,t −
m
2mt nm,t
Each new node contributes m links (and one new node). Assuming n → ∞
there are no multi-edges:
• For k > m : nk,t+1 = nk,t +
m(k−1)
2mt nk−1,t −
mk
2mt nk,t
• For k = m : nm,t+1 = nm,t + 1 −
m2
2mt nm,t
Translating from number of nodes nk,t to probabilities pk,t
pk,t = nk,t/n(t) = nk,t/t
→ nk,t = t pk,t
For each arriving node, after m edges added:
• For k > m : (t + 1) pk,t+1 = t pk,t +
(k−1)
2
pk−1,t −
k
2
pk,t
• For k = m : (t + 1) pm,t+1 = t pm,t + 1 −
m
2
pm,t
Steady-state distribution
We want to consider the final, steady-state: pk,t = pk.
• For k > m : (t + 1) pk = t pk +
(k−1)
2
pk−1 −
k
2
pk
• For k = m : (t + 1) pm = t pm + 1 −
m
2
pm
Rearranging and solving for pk:
• For k > m : pk =
(k−1)
(k+2) pk−1
• For k = m : pm =
2
(m+2)
Recursing pk to pm
pk =
(k−1)(k−2)···(m)
(k+2)(k+1)···(m+3) · pm =
m(m+1)(m+2)
(k+2)(k+1)k
·
2
(m+2)
pk =
2m(m+1)
(k+2)(k+1)k
For k 1
pk ∼ k
−3
Based on this info,can you solve the question below?
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