Structural cut-off

The structural cut-off is a concept in network science which imposes a degree cut-off in the degree distribution of a finite size network due to structural limitations (such as the simple graph property). Networks with vertices with degree higher than the structural cut-off will display structural disassortativity.

Definition

The structural cut-off is a maximum degree cut-off that arises from the structure of a finite size network.

Let

E

k

k

′

{\displaystyle E_{kk'}}

be the number of edges between all vertices of degree

k

{\displaystyle k}

and

k

′

{\displaystyle k'}

if

k

≠

k

′

{\displaystyle k\neq k'}

, and twice the number if

k

=

k

′

{\displaystyle k=k'}

.

Given that multiple edges between two vertices are not allowed,

E

k

k

′

{\displaystyle E_{kk'}}

is bounded by the maximum number of edges between two degree classes

m

k

k

′

{\displaystyle m_{kk'}}

.

Then, the ratio can be written

r

k

k

′

≡

E

k

k

′

m

k

k

′

=

⟨

k

⟩

P

(

k

,

k

′

)

min

{

k

P

(

k

)

,

k

′

P

(

k

′

)

,

N

P

(

k

)

P

(

k

′

)

}

{\displaystyle r_{kk'}\equiv {\frac {E_{kk'}}{m_{kk'}}}={\frac {\langle k\rangle P(k,k')}{\min\{kP(k),k'P(k'),NP(k)P(k')\}}}}

,

where

⟨

k

⟩

{\displaystyle \langle k\rangle }

is the average degree of the network,

N

{\displaystyle N}

is the total number of vertices,

P

(

k

)

{\displaystyle P(k)}

is the probability a randomly chosen vertex will have degree

k

{\displaystyle k}

, and

P

(

k

,

k

′

)

=

E

k

k

′

/

⟨

k

⟩

N

{\displaystyle P(k,k')=E_{kk'}/\langle k\rangle N}

is the probability that a randomly picked edge will connect on one side a vertex with degree

k

{\displaystyle k}

with a vertex of degree

k

′

{\displaystyle k'}

.

To be in the physical region,

r

k

k

′

≤

1

{\displaystyle r_{kk'}\leq 1}

must be satisfied.

The structural cut-off

k

s

{\displaystyle k_{s}}

is then defined by

r

k

s

k

s

=

1

{\displaystyle r_{k_{s}k_{s}}=1}

.

Structural cut-off for neutral networks

The structural cut-off plays an important role in neutral (or uncorrelated) networks, which do not display any assortativity. The cut-off takes the form

k

s

∼

(

⟨

k

⟩

N

)

1

/

2

{\displaystyle k_{s}\sim (\langle k\rangle N)^{1/2}}

which is finite in any real network.