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Characterizing the topology and random walk of a random network is difficult because the connections in the network are uncertain. We propose a class of the generalized weighted Koch network by replacing the triangles in the traditional Koch network with a graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>R</mi><mi>s</mi></msub></semantics></math></inline-formula> according to probability <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>p</mi><mo>≤</mo><mn>1</mn></mrow></semantics></math></inline-formula> and assign weight to the network. Then, we determine the range of several indicators that can characterize the topological properties of generalized weighted Koch networks by examining the two models under extreme conditions, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>, including average degree, degree distribution, clustering coefficient, diameter, and average weighted shortest path. In addition, we give a lower bound on the average trapping time (ATT) in the trapping problem of generalized weighted Koch networks and also reveal the linear, super-linear, and sub-linear relationships between ATT and the number of nodes in the network.