Network Rewiring in the <i>r</i>-<i>K</i> Plane
We generate correlated scale-free networks in the configuration model through a new rewiring algorithm that allows one to tune the Newman assortativity coefficient <i>r</i> and the average degree of the nearest neighbors <i>K</i> (in the range <inline-formula> <math...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-06-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/22/6/653 |
| _version_ | 1797565461105213440 |
|---|---|
| author | Maria Letizia Bertotti Giovanni Modanese |
| author_facet | Maria Letizia Bertotti Giovanni Modanese |
| author_sort | Maria Letizia Bertotti |
| collection | DOAJ |
| description | We generate correlated scale-free networks in the configuration model through a new rewiring algorithm that allows one to tune the Newman assortativity coefficient <i>r</i> and the average degree of the nearest neighbors <i>K</i> (in the range <inline-formula> <math display="inline"> <semantics> <mrow> <mo>−</mo> <mn>1</mn> <mo>≤</mo> <mi>r</mi> <mo>≤</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>K</mi> <mo>≥</mo> <mo>〈</mo> <mi>k</mi> <mo>〉</mo> </mrow> </semantics> </math> </inline-formula>). At each attempted rewiring step, local variations <inline-formula> <math display="inline"> <semantics> <mrow> <mo>Δ</mo> <mi>r</mi> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mo>Δ</mo> <mi>K</mi> </mrow> </semantics> </math> </inline-formula> are computed and then the step is accepted according to a standard Metropolis probability <inline-formula> <math display="inline"> <semantics> <mrow> <mo form="prefix">exp</mo> <mo>(</mo> <mo>±</mo> <mo>Δ</mo> <mi>r</mi> <mo>/</mo> <mi>T</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, where <i>T</i> is a variable temperature. We prove a general relation between <inline-formula> <math display="inline"> <semantics> <mrow> <mo>Δ</mo> <mi>r</mi> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mo>Δ</mo> <mi>K</mi> </mrow> </semantics> </math> </inline-formula>, thus finding a connection between two variables that have very different definitions and topological meaning. We describe rewiring trajectories in the <i>r</i>-<i>K</i> plane and explore the limits of maximally assortative and disassortative networks, including the case of small minimum degree (<inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>k</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>≥</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>), which has previously not been considered. The size of the giant component and the entropy of the network are monitored in the rewiring. The average number of second neighbors in the branching approximation <inline-formula> <math display="inline"> <semantics> <msub> <mover accent="true"> <mi>z</mi> <mo>¯</mo> </mover> <mrow> <mn>2</mn> <mo>,</mo> <mi>B</mi> </mrow> </msub> </semantics> </math> </inline-formula> is proven to be constant in the rewiring, and independent from the correlations for Markovian networks. As a function of the degree, however, the number of second neighbors gives useful information on the network connectivity and is also monitored. |
| first_indexed | 2024-03-10T19:12:31Z |
| format | Article |
| id | doaj.art-dba6b4dcd2ee4e8b85b97dd5e3f35d9f |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-10T19:12:31Z |
| publishDate | 2020-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-dba6b4dcd2ee4e8b85b97dd5e3f35d9f2023-11-20T03:42:45ZengMDPI AGEntropy1099-43002020-06-0122665310.3390/e22060653Network Rewiring in the <i>r</i>-<i>K</i> PlaneMaria Letizia Bertotti0Giovanni Modanese1Faculty of Science and Technology, Free University of Bozen-Bolzano, 39100 Bolzano, ItalyFaculty of Science and Technology, Free University of Bozen-Bolzano, 39100 Bolzano, ItalyWe generate correlated scale-free networks in the configuration model through a new rewiring algorithm that allows one to tune the Newman assortativity coefficient <i>r</i> and the average degree of the nearest neighbors <i>K</i> (in the range <inline-formula> <math display="inline"> <semantics> <mrow> <mo>−</mo> <mn>1</mn> <mo>≤</mo> <mi>r</mi> <mo>≤</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>K</mi> <mo>≥</mo> <mo>〈</mo> <mi>k</mi> <mo>〉</mo> </mrow> </semantics> </math> </inline-formula>). At each attempted rewiring step, local variations <inline-formula> <math display="inline"> <semantics> <mrow> <mo>Δ</mo> <mi>r</mi> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mo>Δ</mo> <mi>K</mi> </mrow> </semantics> </math> </inline-formula> are computed and then the step is accepted according to a standard Metropolis probability <inline-formula> <math display="inline"> <semantics> <mrow> <mo form="prefix">exp</mo> <mo>(</mo> <mo>±</mo> <mo>Δ</mo> <mi>r</mi> <mo>/</mo> <mi>T</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, where <i>T</i> is a variable temperature. We prove a general relation between <inline-formula> <math display="inline"> <semantics> <mrow> <mo>Δ</mo> <mi>r</mi> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mo>Δ</mo> <mi>K</mi> </mrow> </semantics> </math> </inline-formula>, thus finding a connection between two variables that have very different definitions and topological meaning. We describe rewiring trajectories in the <i>r</i>-<i>K</i> plane and explore the limits of maximally assortative and disassortative networks, including the case of small minimum degree (<inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>k</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>≥</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>), which has previously not been considered. The size of the giant component and the entropy of the network are monitored in the rewiring. The average number of second neighbors in the branching approximation <inline-formula> <math display="inline"> <semantics> <msub> <mover accent="true"> <mi>z</mi> <mo>¯</mo> </mover> <mrow> <mn>2</mn> <mo>,</mo> <mi>B</mi> </mrow> </msub> </semantics> </math> </inline-formula> is proven to be constant in the rewiring, and independent from the correlations for Markovian networks. As a function of the degree, however, the number of second neighbors gives useful information on the network connectivity and is also monitored.https://www.mdpi.com/1099-4300/22/6/653scale-free networksnetwork rewiringnetwork assortativitynetwork entropy |
| spellingShingle | Maria Letizia Bertotti Giovanni Modanese Network Rewiring in the <i>r</i>-<i>K</i> Plane Entropy scale-free networks network rewiring network assortativity network entropy |
| title | Network Rewiring in the <i>r</i>-<i>K</i> Plane |
| title_full | Network Rewiring in the <i>r</i>-<i>K</i> Plane |
| title_fullStr | Network Rewiring in the <i>r</i>-<i>K</i> Plane |
| title_full_unstemmed | Network Rewiring in the <i>r</i>-<i>K</i> Plane |
| title_short | Network Rewiring in the <i>r</i>-<i>K</i> Plane |
| title_sort | network rewiring in the i r i i k i plane |
| topic | scale-free networks network rewiring network assortativity network entropy |
| url | https://www.mdpi.com/1099-4300/22/6/653 |
| work_keys_str_mv | AT marialetiziabertotti networkrewiringintheiriikiplane AT giovannimodanese networkrewiringintheiriikiplane |