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. Thus, if vertices of degree k ≥ k s {\displaystyle k\geq k_{s}} exist, it is physically impossible to attach enough edges between them to maintain the neutrality of the network.
Structural disassortativity in scale-free networks In a scale-free network the degree distribution is described by a power law with characteristic exponent γ {\displaystyle \gamma } , P ( k ) ∼ k − γ {\displaystyle P(k)\sim k^{-\gamma }} . In a finite scale free network, the maximum degree of any vertex (also called the natural cut-off), scales as
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