What characteristic defines a weighted network?

A weighted network is specifically characterized by having weights associated with its edges. These weights can represent various quantities, such as distance, cost, capacity, or any measure that describes the relationship between the vertices connected by the edges. This weighting adds an important layer of information to the network, allowing for the analysis of flows, optimal paths, or resource allocations based on the varying costs or capacities indicated by the weights.

For example, in a transportation network, the weights might represent the time it takes to travel from one point to another, which would enable the identification of the quickest route. The concept of weights allows for a more nuanced understanding of the interactions within the network, making it possible to use algorithms that can find optimal solutions based on these weighted values.

Other options present characteristics that don't fundamentally define a weighted network. Multiple paths or undirected networks can exist independently of the edge weights. Furthermore, networks inherently require vertices (or nodes); thus, a network with no vertices cannot exist. Hence, the defining feature of a weighted network remains the presence of weights assigned to its edges.