A graph is defined by vertices and edges. Its structure does not depend on how it is drawn on a page. The same network can be represented as a circle, a map, a grid or a cloud without its topological properties changing.
Small-world networks are characterized by two tendencies: the neighbors of a vertex often have links among themselves, and the average number of steps between any two vertices remains small. The Watts–Strogatz model shows that by adding or rewiring a few long-distance links in a regular network, global distances are greatly reduced while much local clustering is preserved.
This combination appears in social, biological and technical networks, but with nuances. In a social network, dense communities represent groups; a few inter-group relationships shorten contact chains. In the brain, long-range connections cost space and energy, so the organization results from a trade-off between integration and cost.
Scale-free networks constitute a different notion. Their degree distribution can exhibit a heavy tail, with a few highly connected nodes. The Barabási–Albert model produces this behavior through growth and preferential attachment. Small-world and scale-free are not synonymous, even though they can coexist.
Graph drawing algorithms often seek to reduce crossings or to bring strongly linked vertices closer together. A layout algorithm can produce arms, circles or spirals, but these shapes are visualization choices. They do not prove a property of the network.
To analyze a graph, one measures instead the clustering coefficient, path length, centrality, modularity and robustness. The propagation of information or an epidemic also depends on the direction, weight and dynamics of the links.
This article embodies an essential distinction between geometry and topology. In a graph, conceptual proximity matters more than visual distance. The convergence phenomenon lies in the local-global organization, not in a spiral imposed by the drawing.