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Resilience Notions for Scale-Free Networks

Resilience Notions for Scale-Free Networks. Gunes Ercal 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.

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Resilience Notions for Scale-Free Networks

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  1. Resilience Notions for Scale-Free Networks GunesErcal John Matta

  2. 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

  3. 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: • Frequency of nodes with degree d = for a constant α> 1. • Looks linear on a log-log scale • Contrast with Erdős-Rényirandom graphs: • These have Gaussian degree distributions

  4. models for scale-free networks • Two popular generative models: • Preferential attachment: • Dynamic model, “rich get richer” phenomenon • Given parameters m, a, and b • For node varriving 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

  5. 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

  6. conductance as a measure of resilience • Combinatorial measure of edge based resilience • conductance = minimum{S non-majoritysubset of V} • Can think of Cut(S, V-S) as the “attacked edges” that disconnect the vertex set • If conductance is low: • 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

  7. More on Conductance • What does conductance say in the face ofnode attacks?

  8. Conductance Two three-regular graphs with 10 nodes: Low Conductance High Conductance In homogeneous degree graphs, the property of having high conductance maps directly to being resilient against both node and edge attacks.

  9. More on Conductance • What does conductance say in the face ofnode attacks for heterogeneous degree graphs (e.g. scale-free graphs)?

  10. conductance in heterogeneous degree graphs A highly heterogeneous degree graph with a high conductance = 1 • An attack against the center node disconnects the entire graph. • Conductance is not a good measure of this graph's resilience.

  11. 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 • What is needed is a measure of node-based resilience

  12. a proposed measure of node-based resilience • What we really wish to measure is the following function: • where Cmax is the largest connected component that remains in the graph G(V – S)

  13. calculations • conductance • s(G) • Disconnecting 1 node leaves 9 nodes still connected Cutting 4 edges disconnects 4 nodes

  14. calculations • conductance • s(G) • Disconnecting 1 node leaves 5 nodes still connected Cutting 1 edges disconnects 5 nodes

  15. calculations • conductance • s(G) • Disconnecting 1 node leaves a largest connected component of only 1 node Cutting 1 edge disconnects 1 node

  16. conductancevs s(G) Conductance: s(G): 1 (high) .2 (low) 1 (high) 1 (high) .2 (low) .1111 (low)

  17. HOTnet • conductance • s(G) • degree = 1.92 *As described in Fabrikant, Koutsoupias, Papadimitriou, Heuristically Optimized Tradeoffs: A New Paradigm for Power Laws in the Internet

  18. plod Conductance: .5 s(G): .25 degree 2.88 cond = = .5 s(G) = *As described in C. Palmer and J. Steffan, Generating Network Topologies That Obey Power Laws

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