Understanding Terrorist Network Topologies and Their Resilience Against Disruption

Abstract

This chapter investigates the structural position of covert (terrorist or criminal) networks. Using the secrecy versus information tradeoff characterization of covert networks it is shown that their network structures are generally not small-worlds, in contradistinction to many overt social networks. This finding is backed by empirical evidence concerning Jemaah Islamiyah’s Bali bombing and a heroin distribution network in New York. The importance of this finding lies in the strength such a topology provides. Disruption and attack by counterterrorist agencies often focuses on the isolation and capture of highly connected individuals. The remarkable result is that these covert networks are well suited against such targeted attacks as shown by the resilience properties of secrecy versus information balanced networks. This provides an explanation of the survival of global terrorist networks and food for thought on counterterrorism strategy policy.

Appendix

A covert network is modeled by a graph $g {=} (N,E)$, where N represents the set of members (terrorists or terror cells) of the organization and E represents the links among these members. For instance, such links may represent the exchange of bomb making material or the communication over the internet. We set $|N|=n$ and $|E|=m$. The set of all such networks is indicated by $\mathbb{G}(n,m)$.

1.1 The information measure I

The information measure of a graph $g\in \mathbb{G}(n,m)$ is defined by the normalized reciprocal of the total distance in g, i.e.,

$$I(g)=\frac{n(n-1)}{T(g)}.$$

(A.1)

Here, $T(g)$ equals the total geodesic distance, i.e., $T(g)=\sum_{(i,j)\in N^2}l_{ij}(g)$ with $l_{ij}(g)$ the geodesic distance between vertex i and vertex j. It follows that $0\leq I(g) \leq 1$. Thus, the information measure captures the ability of the terrorist organization to exchange information, i.e., to coordinate and control. The higher the value for I the better the organization can do so.

1.2 Secrecy measure S

The secrecy measure of a graph $g\in \mathbb{G}(n,m)$ is defined by

$$S(g)=\frac{ 2m(n-2)+n(n-1)-\sum_{i\in N}d_i^2(g)}{(2m+n)n}.$$

(A.2)

Here, $d_i(g)$ equals the degree of vertex i in g. It follows that $0\leq S(g) \leq 1$. It can be seen that the secrecy measure equals the expected fraction of the organization that survives given that members of the organization are exposed according to a realistically chosen probability distribution.

1.3 Balanced trade-off performance measure $\mu$

For $g\in \mathbb{G}(n,m)$ it holds that,

$$\mu(g)=S(g)I(g)=\frac{ (n-1)(2m(n-2)+n(n-1)-\sum_{i\in N}d_i^2(g)) }{(2m+n)T(g) }.$$

(A.3)

Following multi-objective optimization theory the terrorist organization, faced with trading off secrecy versus information, adopts those values of S and I that maximize their product. For a more thorough motivation of this measure, see Lindelauf et al. (2009a).

1.4 Small-world indicators

For $g\in \mathbb{G}(n,m)$ the characteristic path length is defined by

$$L(g)=\frac{1}{2}\frac{T(g)}{n(n-1)}=\frac{1}{2I(g)},$$

(A.4)

and the clustering coefficient is defined by,

$$C(g)=\frac{1}{n}\sum_{i\in N}C_i,$$

(A.5)

where

$$C_i=\frac{|N_i(g)|}{ |\Gamma_i(g)|(|\Gamma_i(g)|-1)}.$$

(A.6)

Here, $\Gamma_i(g)=\{j\in N|l_{ij}(g)=1\}$ is the set of neighbors of vertex i in network g, and $N_i(g)=\{\{k,l\}\in \Gamma_i(g)|l_{kl}(g)=1\}$ is the set of neighbor pairs of vertex i that are connected in g. Small-world networks are characterized by low L and high C. When compared to random networks a small-world network satisfies $L\approx L_{\rm random}$ and C is of a different order of magnitude than $C_{\rm random}$.

1.5 Use of normalization

Since only relative comparison plays a role we normalized the indicators I, S, L, C and $\mu$ by dividing them by the maximum they attained at each relevant instance. This avoids scaling differences in the corresponding figures but does not affect the resulting analysis.

1.6 Generating an approximate optimal covert network

A theoretically optimal covert network was approximated on $n=100$ individuals as follows. We let $p\in \{0.3, 0.4, 0.5, 0.6, 0.7\}$ and for each fixed p we generated 100.000 random graphs with each possible edge present independently and identically distributed with probability p. Among these 500.000 networks the one that attained the highest value for $\mu$ was selected.