A Spatial Small-World Graph Arising from Activity-Based Reinforcement

Abstract

In the classical preferential attachment model, links form instantly to newly arriving nodes and do not change over time. We propose a hierarchical random graph model in a spatial setting, where such a time-variability arises from an activity-based reinforcement mechanism. We show that the reinforcement mechanism converges, and prove rigorously that the resulting random graph exhibits the small-world property. A further motivation for this random graph stems from modeling synaptic plasticity.

Extremal linkage networks

“…On the other hand, a rapidly evolving fitness distribution could substantially influence the graph distances. In Heydenreich and Hirsch (2019), we could illustrate that working with inhomogeneous layers is indeed feasible.…”

“…The situation changes dramatically when looking at a slightly different setup. In an earlier work (Heydenreich and Hirsch 2019), we investigated a WARM-type model on a layered network. On this layered network, we proved rigorously that sufficiently strong reinforcement is responsible for logarithmic distances, and thus the small-world property applies for the resulting random graph.…”

“…On this layered network, we proved rigorously that sufficiently strong reinforcement is responsible for logarithmic distances, and thus the small-world property applies for the resulting random graph. However, one may criticize that many of the final findings in Heydenreich and Hirsch (2019) were already hard-wired exogenously into the model from the beginning. In particular, each layer is assigned a (fixed) scope, which grows exponentially in the number of layers, and this scope determines how far a vertex can connect.…”

“…We now outline the model definition in the present work and then showcase how extreme value theory enters the stage in the form of the max-stability of the Fréchet distribution. Instead of activity-based reinforcement, like in Heydenreich and Hirsch (2019), we simplify the interaction by assigning each vertex an a-priori fitness, which has two functions: first it measures the node attractiveness in comparison with the fitnesses of neighboring nodes; secondly, it encodes how far a connection from this vertex may reach. Then, we draw one directed edge from vertex (i, h) ∈ ℤ × ℤ with fitness F i,h > 0 (where i is the spatial location and h is the layer) to the vertex with highest fitness in the set see Fig.…”

Shape of shortest paths in random spatial networks

“…Shortest paths form an important aspect of their study. Consider for example the appearance of bottlenecks impeding traffic flow in a city [3,4], the emergence of spatial small worlds [5,6], bounds on the diameter of spatial preferential attachment graphs [7][8][9], the random connection model [10][11][12][13], or in spatial networks generally [14,15], as well as geometric effects on betweenness centrality measures in complex networks [11,16], and navigability [17].…”