On Power-Law Relationships of the Internet Topology

Benefits of topology studying §Protocols: l more effective §Simulations: l more accurate artificial models §Analysis: l Estimates for topological parameters (e.g. the average number of neighbors within h hops)

Definitions and concepts Domain - connected subnetwork under separate administrative authorities Internet graph nodes : Routers in router-level graph (a) Domains in inter-domain level graph (b)

Symbol definitions

Background § Main metrics used are based on l node outdegree (max, min, average) l distances between nodes (max, min, average) Not good for skewed data distributions or data comparison. § Inter-domain graph topology: sparse graph (for 75% of nodes d v  2) l in average one hop for two in Router-level graph

 Power Law: x  y a, a = const l x, y - measures of interest  - proportional to Used to describe traffic patterns § Heavy-tail distribution: P[X>x] = k a x -a L(x), lim t  L(tx)/L(x) = 1 § Pareto distribution: P[X>x] = k a x -a § Heavy-tailed behavior of the traffic due to: l HT distribution of the size of data files l HT characteristics of the human-computer interaction

Internet instances used § Inter-domain level: l Int N=3015E=5156 d avg =3.42 l Int N=3530E=6432 d avg =3.65 l Int N=4389E=8256 d avg =3.76 Growth of 45% § Router level: l Route-95 N=3888E=5012 d avg =2.57

Novel definitions

Rank exponent R d v  r v R l d N =1 : d v  r v /N) R l E = kN/m, k = 1-1/ N R+1 m = 2(R+1)

Outdegree exponent O f d  d O

Hop-plot exponent H P(h)  h H P(1) = N+2E : P(h) = c h H, h<<  N 2, h  c = P(1) = N+2E

Applications - effective diameter §Effective diameter:  def = (N 2 /c) 1/H  Any two nodes are within  def hops of each other with high probability (about 0.8) § Effective diameter is useful for protocol improvements such as broadcast extent selection.

Applications: neighborhood size § NN(h) = P(h) / N - 1, thus NN(h) = c h H / N - 1, c = N+2E (estimate) § Neighborhood is considered as a sphere of radius h in H-dimensional space §Commonly-used estimate: NN’(h) = d avg (d avg -1) h-1

Eigen exponent E  i E

§ Eigenvalues of a graph are closely related to such basic topological properties as l diameter l number of edges l number of spanning trees l number of connected components l number of walks of the certain length between vertices.

Power-Laws: Exponent values Inter-domain evolutions

Detected features: Each graph consists of l Tree component - nodes, belonging only to trees (40-50% of all nodes) l Core components - all the rest and tree roots § Depth of 80% of the trees is 1 § Maximum tree depth is 3

Discussion points § Describing Graphs: exponents vs. averages l Single number capturing topological property l Not implying uniform distribution § Protocol performance l Estimating useful graph metrics § Predictions and extrapolations § Graph generations and Selection l Qualifying characteristics for generated graphs