A Place-based Model for the Internet Topology Xiaotao Cai Victor T.-S. Shi William Perrizo NDSU {Xiaotao.cai, Victor.shi,

1. Introduction Graphs are commonly used to model the topological structure of inter-networks in order to study problems ranging from routing to resource reservation. it is generally more efficient to assess solutions using analysis or simulation, provided the model is a “good” abstraction of the real network and application.

2. Graph Generation Methods Flat Random Methods, Regular Graphs, and Hierarchical Methods.

2.1 Four Flavors of Flat Random Methods Idea: A set of nodes is distributed in a plane, and an edge is added between each pair of vertices with some probability.

A.Pure Random Method P(u, v) = p Where P(u, v) is the probability of an edge from node u to node v and 0 <= p <= 1.

B. Waxman Method P(u, v) =  * e -d/(  * L) where 0 < ,  < I, d is the Euclidean distance from u to v, and L is the maximum distance between any two nodes.

C. Exponential Method P(u, v) =  * e -d/(L - d) where 0 <  < I, d is the Euclidean distance from u to v, and L is the maximum distance between any two nodes.

D. Locality Method , if d < r P(u, v) = , if d >= r Where r is a positive number and 0 <= ,  <= I.

2.2 Regular Graphs Regular graphs are often used in analytic studies of algorithm performance because their structure makes them tractable.

2.3 Hierarchical Methods A. N-Level Method B. Transit-Stub Method

A. N-Level Method Idea: constructs a graph by iteratively expanding individual nodes into graphs as follows: Begin with a connected graph Iterate the following process as you want. Each node in the graph is replaced by a connected graph. The edges of the original graph are then reattached to nodes in the replacement.

B. Transit-Stub Method Idea: Internet can be viewed as a collection of interconnected routing domains, which are groups of nodes that are under a common administration and share routing information. Each routing domain in the Internet can be classified as either a stub domain or a transit domain. A domain is a stub domain if the path connecting any two nodes u and v goes through that domain only if either u or v is in that domain. A transit domains do not have this restriction.

Algorithm Step 1 Generate a connected random graph using any one of the methods discussed earlier; Step 2 Each node in that graph is then replaced by another connected random graph, representing one transit domain. Step 3 For each node in each transit domain, generate a number of connected random graphs representing the stub domains attached to that node. Each of these stub domains has an edge to its transit node. Step 4 Add some “extra” edges between pairs of nodes, one from a transit domain and one from a stub or one from each of two different stub domains.

3. Metrics Introduce some metrics in order to compare and evaluate graphs. For a graph with n nodes and m edges,  Average Node Degree -- (2*m/n).  Number of Biconnected Components -- the number of bicomponents. A biconnected component (or bicomponent) is a maximal set of edges such that any two edges in the set are on a common simple cycle.

Diameter--The diameter of a graph is the length of the longest shortest path between any two nodes - The hop diameter is the length of the Iongest shortest path between any two nodes, where the Iongest shortest paths are computed and evaluated using hot count as the metric. - The length diameter is the length of the longest shortest path between any two nodes where the Iongest shortest paths are computed and evaluated using Euclidean Length as the metric..

4. Evaluation Methodology To compare methods, normalize by fixing a value for the number of nodes n, the number of edges m, and the maximum distance between two nodes, L. Specifically, select n= 100 and m = 175; this corresponds to an average node degree of 3.5. The scale of the Euclidean plane in which the nodes of each graph are placed is 100 by 100, thus L = l00 *2 1/2 = 141.

Generate100 connected graphs for each combination and then measure the metrics described earlier.

Next slide shows the comparison using a box plot to indicate 1) the median over the 100 values (with a white Iine), 2) the 25 and 75% range boundaries (with a box), 3) the 5 and 95% range boundaries (with “whiskers”), and 4) any outliers with single tines.

5. Internet The internet is a loose collection of networks organized into a multilevel hierarchy using a wide variety of interconnection technologies. Consider a packet from NYC to Beijing. It travels along the following path in under a second.

Network of world level Network of nation level Net. of region level (MAN) Network of nation level Net. of region level (MAN) Net. of local level (LAN) Sender in NYCReceiver in Beijing

AT & T Network Backbones of US (2000)

UUNET Network Backbones of US (3/6/2001)

GoodNet Network Backbones of US (2000)

Some Properties of MAN Several nodes forms a clique where every pair of nodes has a edge. Some of remains of nodes are connected each to at least two nodes above. Some of remains of nodes are connected each to only one other nodes. After removing nodes with degree 1 MAN is a bicomponent (e.g. at least 2-connected). Notes: 1) Flat Random Methods and Transit-Stub Method often generate disconnected graphs, for example, using exponential Method, only about three out of 100 of graphs with 100 nodes and 175 edges are connected. 2) Flat Random Methods are not hierarchical.

Network of Northwestern Polytechnic Univ. (1998)

Our idea and future work Our method is a multilevel hierarchy, which consists of networks of world level, networks of nation level, MANs, and LANs (using random fractal); Construct a network of world level, network of nation level, and MAN according to first three properties above in MAN(using random fractal); Construct a LAN like tree Add some “extra” edges between world level, networks of nation level, MANs, and LANs.

END Thanks!