Diagonal degree correlations vs. epidemic threshold in scale-free networks

<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>