fragmentation of random trees · fragmentation of a random tree • nodes are removed one at a...
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Fragmentation of Random TreesEli Ben-Naim
Los Alamos National Laboratory
poster & paper available from: http://cnls.lanl.gov/~ebnZ. Kalay and E. Ben-Naim, J. Phys. A 48, 045001 (2015)
Random Graph Processes, Austin TX, March 11, 2016
with: Ziya Kalay (Kyoto University)
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Formation of a Random Tree• Start with a single node, the root
• Nodes are added one at a time
• Each new node links to a randomly-selected existing node
• A single connected component with N nodes, N-1 links
• Degree distribution is exponential
!
• In-component degree distribution is power-law
nk = 2�k
bs =1
s(s+ 1)
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Fragmentation of a Random Tree
• Nodes are removed one at a time: many previous studies on removal of links [Janson, Baur, Bertoin, Kuba]
• When a node is removed, all links associated with it are removed as well
• Random Forest: a collection of trees formed by the node removal process
• Degree distribution of individual nodes is known (Moore/Ghosal/Newman PRE 2006)
What is the size distribution of trees in the forest?
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Main Result: Size Distribution of Trees in Random Forest
distribution of trees of size s is controlled by one parameter:
fraction m of remaining nodes*
�s =1�m
m2
�(s)�( 1m )
�(s+ 1 + 1m )
size distribution has a power-law tail
�s ⇠ s�1� 1m
for s � 1
*exact result, valid in the infinite N limit
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Removal of a Single Node• Remove a single, randomly-chosen, node from a
random tree with N nodes
• Let be the average number of trees with size s
• Two “conservation” laws !
!
!
• Recursion equation (add node to original random tree)
Ps,N
X
s
Ps,N =2(N � 1)
Nand
X
s
s Ps,N = N � 1
Ps,N+1 =N
N + 1
✓s� 1
NPs�1,N +
N � s
NPs,N
◆+
1
N + 1(�s,1 + �s,N )
existing trees grow in size due to new node
new trees attributed to new node
tree with N nodes has N-1 links every link connects two nodes
removal of a single node reduces total size by 1
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Size Distribution of Trees• Manual iteration of recursion equation gives
!
!
!
• By induction: incredibly simple distribution
!
• Scaling form
Ps,2 =�
11·2 + 1
1·2��s,1
Ps,3 =�
11·2 + 1
2·3�(�s,1 + �s,2)
Ps,4 =�
11·2 + 1
3·4�(�s,1 + �s,3) +
�12·3 + 1
2·3��s,2
Ps,5 =�
11·2 + 1
4·5�(�s,1 + �s,4) +
�12·3 + 1
3·4�(�s,2 + �s,3)
Ps,N =1
s(s+ 1)+
1
(N � s)(N + 1� s)
Ps,N ' 1
N
2 ⇣s
N
⌘ (x) =
1
x
2+
1
(1� x)2
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The Scaling Function
0 0.2 0.4 0.6 0.8 1x
100
101
102
103
104
105
Ψ
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Iterative Removal of Nodes• Remove randomly-selected nodes, one at a time
• Key observation: all trees in the random forest are statistically equivalent to a random tree!
• Treat the number of removed nodes as time t
• Let be the average number of trees with size s at time t
• A single conservation law !
• Recursion equation (represents removal of one node)
Fs,N (t)
X
s
s Fs,N (t) = N � t
Fs(t+ 1) = Fs(t)� sfs(t) +X
l>s
l fl(t)Ps,l with fs(t) =Fs(t)Ps s Fs(t)
loss of trees loss rate = tree size
gain of trees by fragmentation
of larger ones
normalized tree-size distribution
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Rate Equation Approach• Take the infinite tree-size limit:
• Treat time as continuous variable
• Recursion equation becomes a differential equation
!
• Use limiting size distribution, fraction of remaining nodes
!
!
• Problem reduces to the differential equation
N ! 1
dFs
dt= �sfs +
X
l>s
l fl Ps,l
�s(m) = limN!1t!1
Fs,N (t)Ps s Fs,N (t)
and m =N � t
N
(↵� 1)d�s
d↵= (1� s)�s +
X
l>s
l �l
s(s+ 1)+
l �l
(l � s)(l + 1� s)
�↵ = 1 +
1
m
fragmentation kernel = size distribution, single node removal
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100 101 102 103
s10-12
10-10
10-8
10-6
10-4
10-2
100
φs
Theory, N=103
Theory, N=104
Theory, N=105
Simulation, N=103
Simulation, N=104
Simulation, N=105
2/[s(s+1)(s+2)]
• Miraculously, exact solution of the rate equation feasible
• Power-law tail
!
• Special case
The Size Distribution
�s =1�m
m2
�(s)�( 1m )
�(s+ 1 + 1m )
�s ⇠ s�1� 1m
�s =2
s(s+ 1)(s+ 2)
m = 1/2
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Addition and Removal of Nodes
• Addition: Nodes are added at constant rate r
• Removal: Nodes are removed at constant rate 1
• Outcome: random forest with growing number of nodes
• Straightforward generalization of rate equation
!
!
• Normalized distribution of tree size decays exponentially
!
!
• Problem reduces to the differential equation
dFs
dt= r [(s�1)fs�1�sfs]�sfs(t)+
X
l>s
l fl(t)Ps,l.
�s ⇠ s�r�1� e�r
�s
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Summary• Studied fragmentation of a random tree into a random forest
• Nodes removed one at a time
• Distribution of tree size becomes universal in the limit of infinitely many nodes
• Distribution of tree size has a power law tail
• Exponent governing the power law depends only on the fraction of remaining nodes
• Rate equation approach is a powerful analysis tool