crime and terror: mathematical exploration and modeling of dark networks

Dark Networks are DifferentCascade Resilience in Dark Networks

Agent-Based Model of Banned Radical Groups

CRIME AND TERRORMATHEMATICAL EXPLORATION AND MODELING OF DARK

NETWORKS

A. “Sasha” Gutfraind

University of Texas at Austin

University of Waterloo 2011

A. “Sasha” Gutfraind Crime and Terror



NETWORK SCIENCE

How did this network emerge?

How is it structured?

How does it change?Newman MEJ. “Physics of Networks”, Physics Today 2008.




UNIFYING IDEAS (1999-2001)

Distribution of node degrees. Newman MEJ, 2003.

Emerged through a “Preferential Attachment” process

Highly-clustered with a many “small-world” shortcuts

When attacked, it’s “robust [spokes] yet fragile [hubs]”




OUTLINE

1 DARK NETWORKS ARE DIFFERENT

2 CASCADE RESILIENCE IN DARK NETWORKS

3 AGENT-BASED MODEL OF BANNED RADICAL GROUPS

Key findings:

Dark networks 6= Other complex networks

Two flavors of dark networks: Designed and Spontaneous

Open problems in formation and dynamics




AL-QAIDA 2001

Xu, J. and Chen, “The topology of dark networks”, Comm. ACM, October 2008.




THE 9/11 NETWORK AND FTP

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Krebs, “Mapping Networks of Terrorist Cells”, Connections 24(3), 2002; AG, “Optimizingtopological cascade resilience based on the structure of terrorist networks,” PLoS ONE,10.1371, 2010;




PARIS 1944




ABSTRACT MODEL FOR DESIGNING DARK NETWORK

Hypothesis: compartmentalization provides cascade resiliencewhile maintaining performance

Approach: Recreate the network design problem

Optimal network G ∈G maximizes a mix of

Resilience R(G)Efficiency W (G)

Optimization problem over “design” space G:

maxG∈G

rR(G)+(1− r)W (G)︸︷︷︸Fitness




CASCADE ILLUSTRATION

τ

τ

Time 1

τ

Time 2




RESILIENCE AND EFFICIENCY

Resilience to cascade of network

R(G) = 1− 1n−1

E[extent of cascade that starts at a single node]

Efficiency (or Value)

W (G) =1

n(n−1) ∑u,v 6=u∈V

1d(u,v)

FitnessF(G) = rR(G)+(1− r)W (G)




FITNESS OF EMPIRICAL NETWORKS (r = .51)

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tnes

s11M9/11CollabNetEmailFTPGnutellaInternet AS




PERCOLATION THEOREM

THEOREM

Let G be a graph on n nodes with edges sampled independently withprobability p. Then in the limit n→ ∞ with probability→1 the graph isconnected if p > logn

n and disconnected otherwise.

FIGURE: Random graph with n = 50. (a) p = 0.02, (b) p = 0.05

Erdos, P and Renyi, A, “On the evolution of random graphs”, Pub. Math.Instit. Hungar.Acad.Sci 1960.




NETWORK DESIGNS

Networks based on Erdos-Renyi (ER) random graph G(n,p)

Networks based on Connected Stars design

Others




IMPORTANCE OF CELLS (r = .51)

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τ

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Fitn

ess

ConnCliquesConnStarsCliquesCyclesERStars

Connected Stars (cells) beats ER!A. “Sasha” Gutfraind Crime and Terror



AVERAGE DEGREE

0.0 0.2 0.4 0.6 0.8 1.0

τ

101

102

Avg D

egre

e+

1


FIGURE: r = .49

0.0 0.2 0.4 0.6 0.8 1.0

τ100

101

102

Avg D

egre

e+

1


FIGURE: r = .51

Bifurcation around r = 0.5: maximize efficiency (high averagedegree) or resilience (low average degree)




EFFICIENT FRONTIER (τ = 0.4)

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Resilience0.0

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1.0Eff

icie

ncy





HOME-GROWN TERROR CELLS

Emerge spontaneously as radicals find one another andmobilize

Most notorious cases:Madrid 2003-03-11 and London 2005-07-07 bombings

Recent cases in Canada:2006 “Toronto 18” plot

Severe challenge to law enforcement!

How do those networks emerge, despite the risk and the low density?




SIZES OF UNDERGROUND CELLS

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Cell Size

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Environmentalist

Right-Wing

Islamist




RANDOM INTERSECTION GRAPHS

DEFINITION

[Sociological] Magnet is any place (physical or online) where newfriendships can form (e.g. pub, pickup volleyball team, book club).

DEFINITION

A Uniform Random Intersection Graph G(n,k ,p) assigns each of then nodes each of the k colours at random with probability p. An edge(i, j) ∈ G iff i and j have at least one shared color.

K1 K3K2 K4

1 2 3 4 5 6




SELF-ASSEMBLY OF TERRORIST CELLS

Genkin, AG: “How Do Terrorist Cells Self-Assemble”. Winner: best paper award AmericanSociological Association, 2008.




SUMMARY OF FINDINGS




CELL SIZE VS. MAGNETS

THEOREM

Let G(n,k ,p) with k ≤ n. Then in the limit n→ ∞ with probability→1the graph is connected if and only if p > logn

k .

Fill, Scheinerman, Singer-Cohen: “Random intersection graphs when m = w(n)”. Rand.Struct. Algorithms, 16(2), 2000.




SIZES OF UNDERGROUND CELLS

1 2 3 4 5 6 7 8 9 10 11 12 13

Cell Size

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0.6

Pro

bab

ility

Empirical

Simulated




OPEN QUESTIONS

Scaling is P(X ≥ x)∼ x2.5 for a variety of regions. What drivesthis?How do networks heal after attacks?

Clauset, A. et al “On the Frequency of Severe Terrorist Events”, Journal of Conflict Resolution,51(1), 2007.




SUMMARY

Dark networks 6= Other complex networks

Two flavors of dark networks: Designed and Spontaneous

Open problems in formation and dynamics

[email protected]

Thanks to Michael Genkin, Rick Durrett, Aric Hagberg




OPTIONAL SLIDES

Optional Slides




NETWORK DESIGNS I

Multiple independent cells:

Cliques, Cycles and StarsCascade-proof edges (not shown) could provide connectivity




NETWORK DESIGNS II

Erdos-Renyi (ER) random graph G(n,p)

Connected Cliques

Connected Stars




11M NETWORK




IMPLICATIONS

Star-like features improve resilience

For some network designs, efforts are best put improvingefficiency not cascade resilience; for others the other wayaround.

For terrorist networks, depending on their design and cascaderecovery (r ), it might be possible to induce phase transitionswhere their fitness drops or tactics turn non-violent.




RESULTS ABOUT FITNESS

LEMMA 1For a given design G, the fitness of the optimal configuration is adecreasing function of τ (cascade risk) and g (attenuation).

LEMMA 2For a given design G, the fitness of the optimal configuration is acontinuous function of r (the weight of resilience).

Notes

Lemma 1 for τ stems from “monotonicity” of cascades.

Lemma 2 - a standard result in multi-objective optimization?

Fitness need not be a continuous function of τ .




SOLUTION PROCESS

Huge solution space:

Brute force is not an option: O(exp(c2 nlogn )) graphs on n nodes

Optimal set could be too large to analyze

Solution idea: search the space of graph-generating programs

Only a few parameters are enough to configuration a programActual networks are similar: constructed through “protocols”

Solving the optimization problem

Grid Search & Derivative-Free OptimizationProcedure: “Design” + Configuration =⇒Instances of networks=⇒evaluation of resilience & efficiencyCascade extent is found by simulation




RESILIENCE R(G) OF OPTIMAL NETWORKS

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Resi

lience


FIGURE: r = .49

0.0 0.2 0.4 0.6 0.8 1.0τ

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Resi

lience


FIGURE: r = .51

Bifurcation around r = 0.5: optimize for resilience or for efficiency




EFFICIENCY W (G) OF OPTIMAL NETWORKS

0.0 0.2 0.4 0.6 0.8 1.0τ

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Eff

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FIGURE: r = .49

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FIGURE: r = .51




CELL SIZE k OF OPTIMAL NETWORKS

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ConnCliquesConnStarsCliquesCyclesStars

FIGURE: r = .49

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ConnCliquesConnStarsCliquesCyclesStars

FIGURE: r = .51




PARAMETER p OF OPT. NETWORKS

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ConnCliquesConnStarsER

FIGURE: r = .49

0.0 0.2 0.4 0.6 0.8 1.0τ

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1.0ConnCliquesConnStarsER

FIGURE: r = .51




ACCURACY OF OPTIMAL k

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80ConnCliquesConnStarsCliquesCyclesStars

FIGURE: r = .49

0.0 0.2 0.4 0.6 0.8 1.0τ

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FIGURE: r = .75




FITNESS LANDSCAPE FOR STARS DESIGN

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FIGURE: g = 0.5

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FIGURE: g = 2.0

at low g, hardest to optimize near r = 0.5Stars is slightly biased towards maximizing resilience.




BINARY & WEIGHTED REPRESENTATION

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11M (binary)11M (weighted)9/11 (binary)9/11 (weighted)

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crime and terror: mathematical exploration and modeling of dark networks

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