Analyzing Kleinberg's (and other) small-world Models

Abstract

We analyze the properties of Small-World networks, where links are much more likely to connect "neighbor nodes" than distant nodes. In particular, our analysis provides new results for Kleinberg's Small-World model and its extensions. Kleinberg adds a number of directed long-range random links to an n × n lattice network (vertices as nodes of a grid, undirected edges between any two adjacent nodes). Links have a non-uniform distribution that favors arcs to close nodes over more distant ones. He shows that the following phenomenon occurs: between any two nodes a path with expected length O(log 2 n) can be found using a simple greedy algorithm which has no global knowledge of long-range links.We show that Kleinberg's analysis is tight: his algorithm achieves θ(log 2 n) delivery time. Moreover, we show that the expected diameter of the graph is θ(log n), a log n factor smaller. We also extend our results to the general kdimensional model. Our diameter results extend traditional work on the diameter of random graphs which largely focuses on uniformly distributed arcs. Using a little additional knowledge of the graph, we show that we can find shorter paths: with expected length O(log 3/2 n) in the basic 2-dimensional model and O(log 1+1/k n) in the general k-dimensional model (for k ≥ 1).Finally, we suggest a general approach to analyzing a broader class of random graphs with non-uniform edge probabilities. Thus we show expected θ(log n) diameter results for higher dimensional grids, as well as settings with less uniform base structures: where links can be missing, where the probability can vary at different nodes, or where grid-related factors (e.g. the use of lattice distance) has a weaker role or is dismissed, and constraints (such as the uniformness of degree distribution) are relaxed.

Navigating Low-Dimensional and Hierarchical Population Networks

“…More generally, our results can be viewed as extending Kleinberg's theorem to a dimension-independent model that allows varying population density (and one that holds in real networks [21]). There have been some recent theoretical results extending and refining Kleinberg's result-for example, considering routing on other types of underlying graphs [28,7,9], among other results [4,23,26]-and we might hope to be able to make analogous improvements to our results.…”

“…A number of partially decentralized algorithms (e.g., [8,20,22,23,27]) have been shown to outperform Greedy theoretically or experimentally; it would be interesting to analyze them in rankbased networks. More generally, our results can be viewed as extending Kleinberg's theorem to a dimension-independent model that allows varying population density (and one that holds in real networks [21]).…”

“…There have been several relevant extensions to Kleinberg's original model, which we review here. In k-dimensional grids, there has also been considerable work on upper and lower bounds for the diameter and the length of the greedy path (e.g., [23,26,4]), and partially decentralized algorithms other than greedy routing have also been considered [8,20,22,23,27]. Kleinberg has extended his model to tree-based structures and group structures [17].…”

Eclecticism shrinks even small worlds

“…1 For d ≥ 2, indirect-greedy routing performs faster than any other greedy algorithm, for any value of c such that the amount of awareness is (log 2 n) bits, i.e., c = log n for Kleinberg's greedy routing and Decentralized 1 The coefficient 1/c 1/d in front of the performance of indirectgreedy routing comes from the fact that if every node has c long-range contacts, then to get an awareness of O(log n) long-range links, every node just needs to be aware of the long-range contacts of all nodes at distance O(( log n c ) 1/d ) from it. (The same holds for [13]). Greedy [7] O( 1 c log 2 n) O(c log n) Greedy [3,13] ( 1 c log 2 n) O(c log n) Greedy [2] ( 1 c log 2 n/ log log n) O(c log n) NoN-greedy [12] O( 1 c log c log 2 n) O(c 2 log n) Decentralized algorithm [10] …”

“…(The same holds for [13]). Greedy [7] O( 1 c log 2 n) O(c log n) Greedy [3,13] ( 1 c log 2 n) O(c log n) Greedy [2] ( 1 c log 2 n/ log log n) O(c log n) NoN-greedy [12] O( 1 c log c log 2 n) O(c 2 log n) Decentralized algorithm [10] …”