A complex network is a graph model of an interacting system whose connectivity has nontrivial structure relevant to its behavior. The phrase does not specify one mathematical graph class: degree heterogeneity, clustered neighborhoods, community organization or changing connectivity may be important depending on the application.
For example, two graphs can have the same numbers of vertices and edges but very different shortest-path graph distances. A six-cycle has average degree two and diameter three; two disjoint triangles also have average degree two but lack paths between the two components. Their equal mean degree conceals a decisive connectivity difference.
Network science therefore uses several measurements and compares them with explicit random graph models. Labels such as small-world network or scale-free network express additional quantitative hypotheses; they should be tested rather than inferred merely because a diagram looks complicated.
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A complex network is a network characterized by a non-trivial topological structure and features that are not present in simple or regular networks. These networks are typically composed of multiple interconnected nodes (or vertices) and edges (or links) that can reflect a variety of systems found in nature, society, and technology. Complex networks are often studied to understand the underlying principles that govern their structure and dynamics.