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)

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)

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?

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

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

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

The Scaling Function

0 0.2 0.4 0.6 0.8 1x

100

101

102

103

104

105

Ψ

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

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

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

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

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