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RESILIENCE NOTIONS FOR SCALE-FREE NETWORKSGUNES ERCALJOHN MATTA

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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

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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-Rényi random graphs:• These have Gaussian degree distributions

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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 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

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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

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CONDUCTANCE AS A MEASURE OF RESILIENCE• Combinatorial measure of edge based resilience

• conductance = minimum{S non-majority subset 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

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MORE ON CONDUCTANCE

What does conductance say in the face of node attacks?

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CONDUCTANCETwo three-regular graphs with 10 nodes:

High Conductance Low Conductance

In homogeneous degree graphs, the property of having high conductance maps directly to being resilient against both node and edge attacks.

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MORE ON CONDUCTANCE

What does conductance say in the face of node attacks for heterogeneous degree graphs (e.g. scale-free graphs)?

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CONDUCTANCE IN HETEROGENEOUS DEGREE GRAPHS

A highly heterogeneous degree graph with a high conductance

• An attack against the center node disconnects the entire graph.• Conductance is not a good measure of this graph's resilience.

= 1

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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

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A PROPOSED MEASURE OF NODE-BASED RESILIENCEWhat we really wish to measure is the following function:

where Cmax is the largest connected component that remains in the graph G(V – S)

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CALCULATIONSconductance s(G)

Disconnecting 1 node leaves 9 nodes still connected

Cutting 4 edgesdisconnects 4 nodes

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CALCULATIONSconductance s(G)

Disconnecting 1 node leaves 5 nodes still connected

Cutting 1 edgesdisconnects 5 nodes

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CALCULATIONSconductance s(G)

Disconnecting 1 node leaves a largest connected component of only 1 node

Cutting 1 edgedisconnects 1 node

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CONDUCTANCEVS S(G)

Conductance:

s(G):

1 (high) .2 (low) 1 (high)

1 (high) .2 (low) .1111 (low)

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HOTNET

*As described in Fabrikant, Koutsoupias, Papadimitriou, Heuristically Optimized Tradeoffs: A New Paradigm for Power Laws in the Internet

conductance

s(G) degree = 1.92

18*As described in C. Palmer and J. Steffan, Generating Network Topologies That Obey Power Laws

PLOD

Conductance: .5s(G): .25degree 2.88

cond = = .5

s(G) =