Structural cut-off

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

Let ${\displaystyle E_{kk'}}$ be the number of edges between all vertices of degree ${\displaystyle k}$ and ${\displaystyle k'}$ if ${\displaystyle k\neq k'}$, and twice the number if ${\displaystyle k=k'}$. Given that multiple edges between two vertices are not allowed, ${\displaystyle E_{kk'}}$ is bounded by the maximum number of edges between two degree classes ${\displaystyle m_{kk'}}$.

Then, the ratio can be written

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

where ${\displaystyle \langle k\rangle }$ is the average degree of the network, ${\displaystyle N}$ is the total number of vertices, ${\displaystyle P(k)}$ is the probability a randomly chosen vertex will have degree ${\displaystyle k}$, and ${\displaystyle P(k,k')=E_{kk'}/\langle k\rangle N}$ is the probability that a randomly picked edge will connect on one side a vertex with degree ${\displaystyle k}$ with a vertex of degree ${\displaystyle k'}$.

To be in the physical region, ${\displaystyle r_{kk'}\leq 1}$ must be satisfied.

The structural cut-off ${\displaystyle k_{s}}$ is then defined by ${\displaystyle r_{k_{s}k_{s}}=1}$. [1]

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

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

which is finite in any real network.

Thus, if vertices of degree ${\displaystyle k\geq k_{s}}$ exist, it is physically impossible to attach enough edges between them to maintain the neutrality of the network.

In a scale-free network the degree distribution is described by a power law with characteristic exponent ${\displaystyle \gamma }$, ${\displaystyle P(k)\sim k^{-\gamma }}$. In a finite scale free network, the maximum degree of any vertex (also called the natural cut-off), scales as

${\displaystyle k_{\text{max}}\sim N^{\frac {1}{\gamma -1}}}$.

Then, networks with ${\displaystyle \gamma <3}$, which is the regime of most real networks, will have ${\displaystyle k_{\text{max}}}$ diverging faster than ${\displaystyle k_{s}\sim N^{1/2}}$ in a neutral network. This has the important implication that an otherwise neutral network may show disassortative degree correlations if ${\displaystyle k_{\text{max}}>k_{s}}$. This disassortativity is not a result of any microscopic property of the network, but is purely due to the structural limitations of the network. In the analysis of networks, for a degree correlation to be meaningful, it must be checked that the correlations are not of structural origin.

• Assortativity
• Degree distribution
• Complex network
• Rich-club coefficient