a small world network from scratch is given in Barmpoutis et al., where a network with very small average distance and very large average clustering is constructed. A fast algorithm of constant complexity is given, along with measurements of the robustness of the resulting graphs. Depending on the application of each network, one can start with one such "ultra small-world" network, and then rewire some edges, or use several small such networks as subgraphs to a larger graph. Small-world properties can arise naturally in social networks and other real-world systems via the process of dual-phase evolution. This is particularly common where time or spatial constraints limit the addition of connections between vertices The mechanism generally involves periodic shifts between phases, with connections being added during a "global" phase and being reinforced or removed during a "local" phase. Small-world networks can change from scale-free class to broad-scale class whose connectivity distribution has a sharp cutoff following a power law regime due to constraints limiting the addition of new links. For strong enough constraints, scale-free networks can even become single-scale networks whose connectivity distribution is characterized as fast decaying. It was also shown analytically that scale-free networks are ultra-small, meaning that the distance scales according to . Applications Applications to sociology The advantages to small world networking for social movement groups are their resistance to change due to the filtering apparatus of using highly connected nodes, and its better effectiveness in relaying information while keeping the number of links required to connect a network to a minimum. The small world network model is directly applicable to affinity group theory represented in sociological arguments by William Finnegan. Affinity groups are social movement groups that are small and semi-independent pledged to a larger goal or function. Though largely unaffiliated at the node level, a few members of high connectivity function as connectivity nodes, linking the different groups through networking. This small world model has proven an extremely effective protest organization tactic against police action. Clay Shirky argues that the larger the social network created through small world networking, the more valuable the nodes of high connectivity within the network. The same can be said for the affinity group model, where the few people within each group connected to outside groups allowed for a large amount of mobilization and adaptation. A practical example of this is small world networking through affinity groups that William Finnegan