advertisement

RESILIENCE NOTIONS FOR SCALE-FREE NETWORKS GUNES ERCAL 1 JOHN MATTA THE STRUCTURE OF NETWORKS • A graph, G = (V, E) represents a network. • The degree of a node v in a network is the number of nodes that v is connected to. • The distribution of node degrees in a network is clearly an important structural property of the network. • Homogeneous degree distribution: • all nodes have similar degrees • Heterogeneous degree distribution: node degrees clearly variant 2 • HIGH VARIANCE IN DEGREE DISTRIBUTION • Scale-Free degree distribution: • High variance, heterogeneous degree distribution • Heavy-tailed degree distribution • • Most nodes have small degree, but… For arbitrarily high degrees: non-negligibly many nodes • Power Law: • • 1 Frequency of nodes with degree d = πα for a constant α > 1. Looks linear on a log-log scale • Contrast with ErdΕs-Rényi random graphs: 3 • These have Gaussian degree distributions MODELS FOR SCALEFREE NETWORKS • Two popular generative models: • Preferential attachment: • • • Dynamic model, “rich get richer” phenomenon Given parameters m, a, and b For node v arriving at time t, choose m neighbors of v with probability p(v, u) = probability that u is a neighbor of v • p(v, u) = (degree(u)a+b)/N • Where N = Σ (degree(x)a+b) • Random scale-free: Assume that you have generated a degree distribution D that is scale-free (e.g. power-law) • Randomly choose edges conditional upon D 4 • ROBUSTNESS • Characterizing the robustness of networks: • under various forms of attack • • Nodes vs. Edges Targeted vs. Random • for various generative models of such networks • What is known so far: • Lots of work on edge based resilience • Theoretically: Spectral gap captures resilience • Lots of work on general resilience for homogeneous nets Corollary of edge based resilience 5 • CONDUCTANCE AS A MEASURE OF RESILIENCE • Combinatorial measure of edge based resilience |Cut(S, V−S)| |S| • Can think of Cut(S, V-S) as the “attacked edges” that disconnect the vertex set • If conductance is low: • conductance = minimum{S non-majority subset of V} • • There exists relatively few edges whose removal disconnects two relatively large sets of vertices Bad bottleneck • Otherwise, there is no such set of bad edges i.e. You need to attack proportionally many edges to disconnect large sets from each other 6 • MORE ON CONDUCTANCE 7 What does conductance say in the face of node attacks? CONDUCTANCE Two three-regular graphs with 10 nodes: High Conductance Low Conductance 8 In homogeneous degree graphs, the property of having high conductance maps directly to being resilient against both node and edge attacks. MORE ON CONDUCTANCE 9 What does conductance say in the face of node attacks for heterogeneous degree graphs (e.g. scale-free graphs)? CONDUCTANCE IN HETEROGENEOUS DEGREE GRAPHS A highly heterogeneous degree graph with a high conductance πππππ’ππ‘ππππ = πππ |Cut(S, V−S)| =1 |S| 10 • An attack against the center node disconnects the entire graph. • Conductance is not a good measure of this graph's resilience. EDGE FAILURES VS NODE FAILURES • Conductance captures resilience under a model of edge failures. • This coincides with a measure of resilience under node failures when the graph has a homogeneous degree distribution • Conductance no longer captures resilience under a model of node failures when the graph is highly heterogeneous, and in particular scale free 11 • What is needed is a measure of node-based resilience A PROPOSED MEASURE OF NODE-BASED RESILIENCE What we really wish to measure is the following function: |π| π πΊ = πππ{ππ£ππ πππ π π’ππ ππ‘π π ππ π} |π − π − πΆπππ₯ | + 1 12 where Cmax is the largest connected component that remains in the graph G(V – S) CALCULATIONS |Cut(S, V−S)| = πππ |S| 4 =4=1 Cutting 4 edges disconnects 4 nodes s(G) |π| = πππ |π − π − πΆπππ₯ | + 1 1 = πππ 10 − 1 − 9 + 1 1 1 = =1 Disconnecting 1 node leaves 9 nodes still connected 13 conductance CALCULATIONS = πππ |Cut(S, V−S)| |S| 1 = 5 = .2 Cutting 1 edges disconnects 5 nodes s(G) |π| = πππ |π − π − πΆπππ₯ | + 1 1 = πππ 10 − 1 − 5 + 1 1 5 = = .2 Disconnecting 1 node leaves 5 nodes still connected 14 conductance CALCULATIONS |Cut(S, V−S)| = πππ |S| 1 =1=1 Cutting 1 edge disconnects 1 node s(G) |π| = πππ |π − π − πΆπππ₯ | + 1 1 = 10 − 1 − 1 + 1 1 9 = = .1111 Disconnecting 1 node leaves a largest connected component of only 1 node 15 conductance Conductance: 1 (high) .2 (low) 1 (high) s(G): 1 (high) .2 (low) .1111 (low) 16 CONDUCTANCE VS S(G) HOTNET conductance 1 8 = = .125 s(G) = 1 25 −1 −8 + 1 = 1 17 = .055235 *As described in Fabrikant, Koutsoupias, Papadimitriou, Heuristically Optimized Tradeoffs: A New Paradigm for Power Laws in the Internet 17 degree = 1.92 PLOD Conductance: .5 s(G): .25 degree 2.88 1 cond = 2 = .5 1 *As described in C. Palmer and J. Steffan, Generating Network Topologies That Obey Power Laws 18 s(G) = 25 −1 − 21 + 1 = .25