Diagonal degree correlations vs. epidemic threshold in scale-free networks
Diagonal degree correlations vs. epidemic threshold in scale-free networks
复制标题
无标度网络中的对角度相关性与流行病阈值
DOI：
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发表时间：
2021
期刊：
影响因子：
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通讯作者：
G. Modanese
中科院分区：
文献类型：
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作者：
M. Bertotti;G. Modanese
我们证明了对角线分类度相关性的存在，即使很小，也会显著降低大型无尺度网络的流行阈值。考虑的相关矩阵为<jats:inline-formula>
<jats:p>We prove that the presence of a diagonal assortative degree correlation, even if small, has the effect of dramatically lowering the epidemic threshold of large scale-free networks. The correlation matrix considered is <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M1">
<mi>P</mi>
<mfenced open="(" close=")" separators="|">
<mrow>
<mrow>
<mi>h</mi>
<mi>|</mi>
</mrow>
<mi>k</mi>
</mrow>
</mfenced>
<mo>=</mo>
<mfenced open="(" close=")" separators="|">
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>r</mi>
</mrow>
</mfenced>
<msubsup>
<mi>P</mi>
<mrow>
<mi>h</mi>
<mi>k</mi>
</mrow>
<mi>U</mi>
</msubsup>
<mo>+</mo>
<mi>r</mi>
<msub>
<mrow>
<mi>δ</mi>
</mrow>
<mrow>
<mi>h</mi>
<mi>k</mi>
</mrow>
</msub>
</math>
</jats:inline-formula>, where <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M2">
<msup>
<mrow>
<mi>P</mi>
</mrow>
<mrow>
<mi>U</mi>
</mrow>
</msup>
</math>
</jats:inline-formula> is uncorrelated and <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M3">
<mi>r</mi>
</math>
</jats:inline-formula> (the Newman assortativity coefficient) can be very small. The effect is uniform in the scale exponent <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M4">
<mi>γ</mi>
</math>
</jats:inline-formula> if the network size is measured by the largest degree <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M5">
<mi>n</mi>
</math>
</jats:inline-formula>. We also prove that it is possible to construct, via the Porto–Weber method, correlation matrices which have the same <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M6">
<msub>
<mrow>
<mi>k</mi>
</mrow>
<mrow>
<mi>n</mi>
<mi>n</mi>
</mrow>
</msub>
</math>
</jats:inline-formula> as the <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M7">
<mi>P</mi>
<mfenced open="(" close=")" separators="|">
<mrow>
<mrow>
<mi>h</mi>
<mi>|</mi>
</mrow>
<mi>k</mi>
</mrow>
</mfenced>
</math>
</jats:inline-formula> above, but very different elements and spectra, and thus lead to different epidemic diffusion and threshold. Moreover, we study a subset of the admissible transformations of the form <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M8">
<mi>P</mi>
<mfenced open="(" close=")" separators="|">
<mrow>
<mrow>
<mi>h</mi>
<mi>|</mi>
</mrow>
<mi>k</mi>
</mrow>
</mfenced>
<mo>⟶</mo>
<mi>P</mi>
<mfenced open="(" close=")" separators="|">
<mrow>
<mrow>
<mi>h</mi>
<mi>|</mi>
</mrow>
<mi>k</mi>
</mrow>
</mfenced>
<mo>+</mo>
<mi mathvariant="normal">Φ</mi>
<mfenced open="(" close=")" separators="|">
<mrow>
<mi>h</mi>
<mo>,</mo>
<mi>k</mi>
</mrow>
</mfenced>
</math>
</jats:inline-formula> with <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M9">
<mi mathvariant="normal">Φ</mi>
<mfenced open="(" close=")" separators="|">
<mrow>
<mi>h</mi>
<mo>,</mo>
<mi>k</mi>
</mrow>
</mfenced>
</math>
</jats:inline-formula> depending on a parameter which leaves <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M10">
<msub>
<mrow>
<mi>k</mi>
</mrow>
<mrow>
<mi>n</mi>
<mi>n</mi>
</mrow>
</msub>
</math>
</jats:inline-formula> invariant. Such transformations affect in general the epidemic threshold. We find, however, that this does not happen when they act between networks with constant <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M11">
<msub>
<mrow>
<mi>k</mi>
</mrow>
<mrow>
<mi>n</mi>
<mi>n</mi>
</mrow>
</msub>
</math>
</jats:inline-formula>, i.e., networks in which the average neighbor degree is independent from the degree itself (a wider class than that of strictly uncorrelated networks).</jats:p>