Small and Other Worlds: Global Network Structures from Local Processes

Abstract

Using simulation, we contrast global network structures-in particular, small world properties-with the local patterning that generates the network. We show how to simulate Markov graph distributions based on assumptions about simple local social processes. We examine the resulting global structures against appropriate Bernoulli graph distributions and provide examples of stochastic global "worlds," including small worlds, long path worlds, and nonclustered worlds with many four-cycles. In light of these results we suggest a locally specified social process that produces small world properties. In examining movement from structure to randomness, parameter scaling produces a phase transition at a "temperature" where regular structures "melt" into stochastically based counterparts. We provide examples of "frozen" structures, including "caveman" graphs, bipartite structures, and cyclic structures.

Collaboration structures in Slovenian scientific communities

“…Yet the final structure has noteworthy features that are apparent in many real world networks if we allow that the mechanisms for this topology are likely to be different. Instead of random rewiring there is actor agency regarding the creation and deletion of ties rather than having the implicit mechanism residing in the ties themselves (Robins et al 2005). They show also that smallworld models are fully compatible with ERGMs, in the form of stochastic actor-oriented models, where the micro-mechanisms are found in some local structural configurations of ties.…”

Constrained Markovian Dynamics of Random Graphs

“…In extending this theory to evolving graphs we are motivated by the necessity to bridge an existing gap between the static and the dynamical treatment of graphs. While many statistical aspects of random graph topology are now understood (like the influence of topology on processes occurring on graphs, percolation and critical phenomena, loop statistics, or the entropies of different topologies in various random graph ensembles [3][4][5][6][7][8][9][10]), much less work has been invested in the mathematical study of the dynamics of graphical structures (see [11][12][13] for recent examples). Besides their mathematical interest, dynamical problems are prominent in application areas where the issue of sampling uniformly the space of graphs with certain prescribed macroscopic properties is vital.…”

Methodological Transactionalism and the Sociology of Education

“…This has entailed a growing interest in network change as well as in identifying the ongoing interactional micro-mechanisms that give rise to the formal properties of global level networks (e.g., Robins et al 2005). We think what is especially exciting about these developments is that as SNA becomes more sophisticated it allows us to quantitatively study the classroom in ways that match our current qualitative and theoretical view of them as interdependent and processual social contexts.…”