^{*}

Conceived and designed the experiments: AG. Analyzed the data: AG. Contributed reagents/materials/analysis tools: AG. Wrote the paper: AG.

The author has declared that no competing interests exist.

Complex socioeconomic networks such as information, finance and even terrorist networks need resilience to cascades - to prevent the failure of a single node from causing a far-reaching domino effect. We show that terrorist and guerrilla networks are uniquely cascade-resilient while maintaining high efficiency, but they become more vulnerable beyond a certain threshold. We also introduce an optimization method for constructing networks with high passive cascade resilience. The optimal networks are found to be based on cells, where each cell has a star topology. Counterintuitively, we find that there are conditions where networks should not be modified to stop cascades because doing so would come at a disproportionate loss of efficiency. Implementation of these findings can lead to more cascade-resilient networks in many diverse areas.

Cascades are ubiquitous in complex networks and they have inspired much research in modeling, prediction and mitigation

Dark networks are therefore designed to operate in conditions of intense cascade pressure. As such they might serve as useful prototypes of networks that are cascade-resilient because of their connectivity structure (topology) alone. Their nodes are often placed in well-defined cells - closely-connected subnetworks with only sparse connections to the outside (for an example from World War II see

Its organizational unit was the combat group (A). In an idealized case, nor always followed, this was divided into two “teams” of three fighters, where leader L1 was in overall command and in command of team

To represent networks from different domains, this paper will use simple unweighted graphs. This approach offers simplicity and can employ tools from the well-developed field of graph theory. A simplification is also unavoidable given the lack of data on networks, especially on dark networks where only the connectivity is known, if that. Ultimately through, models of networks, especially dark networks must consider their evolving nature, fuzzy boundaries and multiplicities of node classes and diverse relationships.

Fortunately, the loss of information involved in representing networks as simple rather than as weighted graphs could be evaluated. In the

Our preliminary task is to compare the cascade resilience of networks from different domains. We will see that dark networks are indeed more successful in the presence of cascades than other complex networks. Their success stems not from cascade resilience alone but from balancing resilience with efficiency (a measure of their ability to serve their mission).

We will consider a particular type of cascade resilience and a particular definition of efficiency. For resilience we will use a probabifolistic process known as “SIR” (susceptible-infected-recovered). In SIR any failed (captured) node leads to the failure of each neighboring node independently with probability

Observe that the most cascade-resilient network is the network with no edges (hence no cascades can propagate), but it is also the least efficient kind of network. It is expected that resilience and efficiency will be in opposition, requiring trade-offs. Just as disconnected networks are resilient and inefficient, highly-efficient networks such as densely-connected graphs are likely to have low resilience (for a historic example see

Define the overall “fitness”,

We will compare the fitnesses of several complex networks, including communication, infrastructure and scientific networks to the fitnesses of dark networks. The class of dark networks will be represented by three networks: the 9/11, 11M and FTP networks. The 9/11 network links the group of individuals who were directly involved in the September 11, 2001 attacks on New York and Washington, DC

11M is the network responsible for the March 11, 2004 attacks in Madrid (

The 9/11 and the 11M networks are very successful for low values of

The success of dark networks must be due to structural elements of those networks, such as cells. If identified, those elements could be used to design more resilient networks and to upgrade existing ones. Thus, by learning how dark networks organize, it will be possible to make networks such as communication systems, financial networks, and others more resilient and efficient.

Those identification and design problems are our next task. We propose to solve both using an approach based on discrete optimization. Let a set of graphs

Cliques (A), Stars (B), Cycles (C), Connected Cliques (D), Connected Stars (E), and Erdos-Renyi “ER” (F). Each design is configured by just one or two parameters (the number of individuals per cell and/or the random connectivity). This enables rapid solution of the optimization problem. In computations the networks were larger (

In the first step we will find the most successful network within each design. Namely, consider an optimization problem where the decision variable is the topology

This optimization problem could be used more broadly: It introduces a method for designing cascade-resilient networks for applications such as vital infrastructure networks. To apply this to a given application, one must make the design

A complementary approach is to consider the multi-objective optimization problem in which

The first set of experiments compares the designs against each other under different cascade risks (

The Connected Stars design is the best design at all cascade risks,

Comparing designs to each other reveals that Connected Stars is superior to all others in fitness (

It has been long conjectured that cells provide dark networks with high resilience. Indeed, this is probably the reason why we found that dark networks have higher fitnesses than other networks. But cells also reduce the efficiency of a network since they isolate nodes from each other. To rigorously determine the net effect of cells, we compare the ER design (random graphs) to the Connected Stars design. ER is a strict subset of Connected Stars but only Connected Stars has cells. Therefore it is notable that Connected Stars has a higher fitness than ER, often significantly so. Indeed, cells must be the cause of higher fitness because cells are the only feature in Connected Stars that ER lacks.

Many properties of the optimal networks such as resilience, efficiency and edge density show rapid phase transitions as

At

Intuition may suggest that the networks grow more sparse as cascade risk grows. Instead, the trend was non-monotonic (

A complementary perspective on each design is found from its Pareto frontier of resilience and efficiency (

The configurations of the Connected Stars design dominate over other designs when the network must achieve high resilience. However, designs based on cliques are dominant when high efficiency is required. Several designs show sharp transitions where at a small sacrifice of efficiency it is possible to achieve large increases in cascade resilience.

The sharp phase transitions discussed earlier are seen clearly: along most of the frontiers, if we trace a point while decreasing resilience, there is a threshold at which a small sacrifice in resilience gives a major gain of efficiency. More generally, consider the points where the frontier is smooth. By taking two nearby networks on the frontier one can define a rate of change of efficiency with respect to resilience:

The analysis above considered both empirical networks and synthetic ones. The latter were constructed to achieve structural cascade resilience and efficiency. In contrast, in many empirical networks the structure emerges through an unplanned growth process or results from optimization to factors such as cost rather than blocking cascades. Without exception the synthetic networks showed higher fitness values despite the fact that they were based on very simple designs. This suggests that network optimization can significantly improve the fitness and cascade resilience of networks. It means that such an optimization process can indeed be an effective method for designing a variety of networks and for protecting existing networks from cascades.

Many empirical networks also have power-law degree distributions

In some successful synthetic networks the density of edges increased when the cascade risk

There are other important applications of this work, such as the design of power distribution systems. For power networks, the definition of resilience and efficiency will need to be changed. It would also be necessary to use much broader designs and optimization under design constraints such as cost. Furthermore, this work could also be adapted to domains of increasing concern such as financial credit networks, whose structure may make them vulnerable to bankruptcies

Research on graph theory has led to the development of a variety of metrics of robustness or resilience

For many applications the distance between pairs of nodes in the network is one of the most important determinants of the network's efficiency (see e.g.

An appendix (

Optimizing topological cascade resilience based on the structure of terrorist networks.

(1.11 MB PDF)