| Summary: | Modeling data transmission problems in graph theory is internalized to the existence of fractional flows, and thus can be surrogated to be characterized by a fractional factor in diversified settings. We study the fractional factor framework in the network environment when some sites are damaged. The setting we focus on refers to the lower and upper fractional degrees described by two functions on the vertex set. It is determined that <i>G</i> is fractional <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>f</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> critical if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>δ</mi><mrow><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><mo>≥</mo><mrow><mo stretchy="false">⌊</mo><mfrac><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>a</mi><mi>b</mi><mo>+</mo><mn>2</mn><mi>a</mi><mo>+</mo><mn>2</mn><mi>b</mi><mo>−</mo><mn>3</mn></mrow><mrow><mn>4</mn><mi>a</mi></mrow></mfrac><mo stretchy="false">⌋</mo></mrow><mo>+</mo><mi>n</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mrow><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><mo>></mo><mfrac><mrow><mi>n</mi><mo>+</mo><mo stretchy="false">⌊</mo><mfrac><msup><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><mi>b</mi><mo>−</mo><mn>1</mn></mrow><mi>a</mi></mfrac><mo stretchy="false">⌋</mo></mrow><mn>2</mn></mfrac></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>a</mi><mo>≤</mo><mi>b</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>.
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