Strategic Network Disruption

B. Hoyer*,a, K. De Jaegherb

aUtrecht University School of Economics, Utrecht University, Janskerkhof 12, 3512BL Utrecht, The

Netherlands b Utrecht University School of Economics, Utrecht University, Janskerkhof 12, 3512BL Utrecht, The

Netherlands

Abstract Competition among firms is usually modeled as being based on prices alone. However, firms are increasingly made up of internal networks. We thus study, what happens if instead of price competition, firms compete by means of luring away crucial players or links from each others networks. We thus study the problem of a network designer designing a network who is faced with an impending attack by an intelligent network disruptor in a two-stage full information game. We study cases where either the attack is directed on the nodes of the network or on the links of the network. In the benchmark model, a network designer designs a network when there is no threat of an attack. We show that the optimal strategy for the network designer in that case is to build a minimally connected network. We then show that if linking costs are low, the network designer can form a maximally robust network, by building a symmetric network. For permissively high linking costs, not only the topology of the network matters but also if the attack is directed on links or nodes of the network.

* Corresponding Author Emailadresses: b.hoyer@uu.nl (Britta Hoyer) k.dejaegher@uu.nl (Kris De Jaegher)

1. Introduction

Traditionally the competition among firms is modeled as being governed solely by

prices. In other approaches, competition between firms is modeled by means of

quality, product differentiation, raising rivals costs, and so on. However, all of these

approaches do not consider how the firm is structured internally. But not only are

firms internally increasingly structured as networks of collaborators, but in recent

years there has also been an incline in firms externally operating in networks. Thus

competition among firms is becoming synonymous with network competition.

Particular about such competition is that network A may compete with network B by

luring away crucial players from network B, in order to cause maximal disruption in

network B. Such disruptive network competition, however, is not taken into account,

when modeling competition among firms as solely governed by prices. In this paper

we show that taking disruptive network competition into account, leads to a new focus

on structure of the internal firm network, to minimize the damage that a potential

disruption can cause. By using methods and results from graph theory, we can show

which requirements a robust network topology in case of an impending attack from a

network disruptor needs to fulfill.

In the model, we will make the simplifying assumption that the network is designed

by a network designer instead of using an evolutionary approach and that the network

disruptor is a single entity simply deriving his utility out of causing maximal damage

to the network, while the network designer designs the network in the most efficient

way. We say that a given network is a best response if it maximizes the network

designer’s payoffs. There is no friction involved in the network, so that the length of

the path between two nodes does not matter. These assumptions enable us to derive a

relatively simple model of a strategic full information two-stage game between

network designer and disruptor, on which future work can build.

For now, we are therefore ignoring the aspect of endogenous link formation in

networks that has received lots of attention after the seminal papers of Jackson &

Wolinsky (1996) and Bala & Goyal (2000). Network disruption so far has not

received a lot of attention in the economics literature, but has been studied in other

areas of research such as in crime network studies, for example, the paper by Enders

& Su (2007) studying terrorist networks, internet security studies and studies on the

reliability of road networks. Internet networks have most often been studied as an

example for so-called scale free networks. In an influential paper by Albert, Jeong &

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Barabasi (2000) it is shown that while these networks show a great tolerance towards

random errors, they are highly vulnerable to targeted attacks. The literature on

transportation- and road networks is mainly focused on random attacks and it deals in

most parts with measures of accessibility of certain areas or towns and reliability of

networks. A good, if a little outdated, review of what has been done can be found in

Berdica (2002). A paper that is related to what we are doing here, in that it allows for

extra links as a safety measure, is a study on rural Australian road networks by Taylor,

Sekhar & D'Este (2006). They are, however, more interested in vulnerability measures

of certain links and the effects on generalized costs of travel between two nodes, if a

link should be disrupted.

Another paper that deals with network disruption, albeit in the field of engineering, is

the work of Dekker & Colbert (2004). Out of the perspective of the military sphere,

and what they call Network Centric Warfare, they discuss the robustness of networks

based on their graph topology. They reach similar conclusions about the topology of

robust networks as we do, however, they do not consider a linking budget, and

therefore only deal with the extreme case, of wanting to keep the network as safe as

possible, while having an unlimited supply of links. This is reasonable for their

model, however, we deal with costly links and therefore come to some different

conclusions, once costs for links are included in the model.

In economics, to the best of our knowledge, only few papers have dealt with the topic

of network disruption by an intelligent adversary so far. One exception is the work by

Baccara & Bar-Isaac (2008) in which they model a sequential move model between a

legal authority and an organization in which links serve the purpose to transmit

personal information about the members of the organization. Their model, however,

deals with directed links and a repeated game and is therefore quite dissimilar to our

work in terms of analysis and results. A paper that is closer in terms of model set up is

by Goyal & Vigier (2009). They consider a strategic interaction game between a

network designer and an intelligent adversary. In the model, there is a group of

homogeneous nodes that aims at performing a given task. Links between the nodes

serve as a means of communication between the nodes. However, the group is facing

an adversary that can detect nodes. Once detected, nodes are rendered unable to

perform or communicate and also all nodes directly or indirectly linked to the detected

node are now detectable to the adversary. Goyal and Vigier show that when the

designer has a possibility to defend certain nodes, highly connected hub-and-spoke

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networks are the most efficient solution. By contrast, in our model, while using a

similar set up, we do not focus on the spread of infection but look for a robust

network topology. Unlike them, we also do not limit ourselves to the deletion of

nodes, but also deal with link deletion. We therefore reach different conclusions and

also use different methods than they do.

2. Model

In the description of the model, we follow the notation used by Bala & Goyal (2000).

There is a set of 1,...,N n homogeneous nodes. The nodes are homogeneous in

terms of value to the network as well as costs of linking them to each other. Nodes in

N are linked through a network of interaction g. Let G denote the set of graphs

defined on N. If two nodes i and j are linked, we say that 1ijg . If they are not linked

to one another 0ijg . Nodes are indirectly linked to each other if a path exists

between them. We assume that there exists a path between two nodes i and j if there

exists a sequence of nodes kii ,...,1 such that 1...1211

jiiiiiii kkk

gggg . We

denote the set of links by . Links between two nodes are assumed to be undirected,

i.e. ij jig g jiij ggNji :, . We denote g-i as network g with node i removed, and

g-ij as network g with link ij removed. Define as Ni(g) the set of nodes with whom

node i maintains a (direct or indirect) link. Given a network g, a set is called a

component of g if for every distinct pair of nodes i and j in C we have , and

there is no strict superset C' of C for which this is true. The degree of connection of

each node

NC

Nj )(gi

)(gi is defined as the number of direct links the node has, so in effect the

number of nodes it is directly linked with. The minimum degree of the graph, ηmin, is

the smallest degree of any of the individual nodes within the network. The maximum

degree of a graph, ηmax, is the largest degree of any of the nodes within the network.

Contrary to what is the case in Bala & Goyal (2000), the nodes in our networks are

not individual decision makers. Instead, there is a single decision maker, called the

network designer, who benefits from the number of nodes connected into one

network, and who plays a sequential zero-sum game with a network disruptor. The

network designer constructs a (pre-disruption) network g, where each link formed

comes at a cost to him. A second player, called the network disruptor, is worse off the

more nodes the network designer is able to connect. We study two ways in which the

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disruptor can disrupt the network designer’s pre-disruption network, simply reflecting

the two elements of which the network is made up off. In the link deletion model, the

disruptor can take out links from the designer’s network, in the node deletion model,

the disruptor can take out nodes. The game proceeds as follows. At stage 1, the

network designer constructs any (pre-disruption) network g. At stage 2, depending on

the model of network disruption, the disruptor either deletes a number of links or

nodes from the pre-disruption network. At stage 3, both players obtain their payoffs,

which depend on the number of nodes in the post-disruption network. It follows that

at stage 1, the network designer constructs a pre-disruption network in the knowledge

that some of its links or nodes will be deleted, and will do so in a manner that the

damage that can be done by the disruptor is minimized.

We specifically model the network designer’s payoff as depending only on the size of

the largest connected component in the post-disruption g, and an increasing function

of this component’s size. The rationale for this is that the network designer has

increasing returns to scale of network size. When the benefits of a network consist of

information exchange, then adding one node to a network lets each node that was

already in the network benefit from this new information. A simplified way of

modeling such increasing returns to scale is to assume that only the largest connected

component counts. Since links are costly, this means that the network designer’s

payoffs are convex in the size of the post-disruption network.

Figure 1: Convex payoffs

Given that we have a zero-sum game, the network disruptor’s benefit is a decreasing

function of the largest connected component in the post-disruption g. The network

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designer’s costs are an increasing function of the number of links used in the pre-

disruption network, and the network disruptor’s costs, depending on the model of

disruption are an increasing function of the number of nodes or links deleted.

A strategy by the network disruptor consists of deleting a number of nodes or links for

each possible pre-disruption network observed. For each such disruption strategy,

there are one or more optimal response pre-disruption networks that the network

designer may construct, where the network designer optimizes the structure of the

network, and further weighs off the benefits of the size of the network against the

linking costs. The designer’s total payoff depends on the number of links used in the

pre-disruption network, and the size of the largest component in the post-disruption

network. This total payoff is represented in Figure 2, for one particular disruption

strategy of the disruptor. As indicated by the lines fixing the number of links used, for

any given number of links, depending on the network constructed, any size of the

largest component may be achieved, from size 1 (where e.g. links are only used to

connect nodes to themselves), to some maximal size. As indicated, intuitively the

more links that are used, the bigger the size of the largest component in the best

possible network will be. Also, for any fixed size of the largest component, as

indicated by the lines fixing the size, the network designer will have to use at least a

minimal number of links. For this fixed size of the largest component, as links are

costly, this minimal number of links will then also yield the designer the highest

payoff. Also, the bigger the largest component, the larger the minimal number of links

needed will be.

Note that (given a certain disruption strategy), if for a given size of the largest

component s, the minimal number of links needed is l, then necessarily with this

number of links l, s is also the biggest size one can achieve for the largest component.

Oppositely, if s is the largest component one can achieve with l links, it may still be

the case that one can achieve the same with a few links less, as indicated in the figure.

Still, for the top of the hill in Figure 2, it will be the case that it minimizes the number

of links used to achieve a certain size of the largest component, and that it maximizes

the size of the largest component given the numbers of links used – each time given a

certain number of links or nodes that the disruptor chooses to disrupt in equilibrium.

Using this fact, we simplify our model by not explicitly modeling the disruptor’s and

the designer’s benefit and cost functions. Instead, we assume that the disruptor has a

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fixed disruption budget. In case of node deletion, this is denoted by , and defined as

the number of nodes the network disrupter can disrupt in any given pre-disruptoin

network. For the case of link deletion, the disruption budget is denoted by and

defined as the number of links that the network disrupter can disrupt in any given pre-

disruption network. In order to tell in a tractable manner as much as possible about the

optimal pre-disruption network that the designer chooses, we use two approaches,

reflected in Figure 2. In the first approach, for a fixed defense budget Def, which

equals the number of links that the designer can add to his minimal linking budget B,

which includes (n – 1) links, the designer tries to achieve a maximal size of the largest

component in the post-disruption network. In the second approach, for a fixed size of

the largest component that the designer targets, the designer tries to achieve this size

using the minimal number of links. Thus he targets a (n-x)-proof network, which we

can define as a network which biggest component upon strategic deletion by the

disruptor (under a given disruption budget D) connects at least (n-x)-nodes The

reasoning is that we can always find benefit and cost functions for the disruptor and

designer such that it is optimal to maintain respectively such a disruption budget, and

such a linking budget or target size of the largest component.

nD

lD

Designer’s payoff

size

# links

Figure 2: Designer’s payoff

3. Graph Theoretic Preliminaries

To further describe the model and the types of graphs used later on, it is necessary to

introduce here some widely used graph theoretic concepts, that can be found for

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example in Chartrand (1977). Firstly we need to distinguish between two general

types of graphs, the simple graph and the multigraph:

Definition 1:

i) A simple graph is an undirected graph containing no multiple links

between two nodes and no links beginning and ending at the same

node (commonly referred to as loops).

ii) A multigraph is a graph in which neighboring nodes are linked by an

integral number of links.

As is conventionally done, when using the term “graph”, we are referring to a simple

graph, unless explicitly stated otherwise.

Definition 2: A cycle in a network is defined as any path

1...1211

iiiiiiii kkk

gggg , where kk iiiii 121 ... .

So, a cycle is a path in which no links or nodes are repeated, with the exception of the

first and last node, which is the same.

Definition 3: A star network has a central node i, such that 1ijg for all \j N i

and no other links.

Next to the degree of connectivity of each single node ( )i g , connectivity of the

whole network is also measured with two different concepts in graph theory1. Both

will be used here in defining a robust network.

Definition 4:

i) The node connectivity κ is the smallest number of nodes whose

removal results in at least one (non-removed!) node being disconnected

from all other nodes; example in complete network, κ = (n – 1).

ii) The link connectivity λ is the smallest number of links whose removal

results in at least one node being disconnected from all other nodes;

example in complete network, λ = (n – 1).

For any graph, it is the case that κ≤λ≤ηmin holds, as has been shown for example by

Gibbons (1994). Intuitively, this can be seen as it cannot be that one needs to remove

more links or nodes than the degree of a node to disconnect that node: one can always

remove all the direct links or direct neighbors of the node in question. That κ≤λ≤ min ,

is also intuitive, since the removal of a node can do more damage than the removal of

1 Here we will use the definition given in Dekker & Colbert (2004), Definition 2.1.

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a link, as all the links of this node are deleted as well. Thus if several paths between

nodes pass through this single node, the removal of this node, destroys all of these

paths, whereas the removal of a link, would only destroy one path.

Definition 5: A node i in a connected graph g, is called a cut-node, if g-i is

disconnected. Thus at least one node, next to the one that is disrupted, is disconnected

from the network.

In a cycle for example there is no cut-node, as the removal of no single node

disconnects any other nodes from the remaining component. Any two non-

neighboring nodes in a cycle can be interpreted as a node cut set though. Their

removal disconnects at least one node from the remaining connected component. In

the same line of argument, we can define links that are crucial for the network.

Definition 6: A link ij in a connected graph g, is called a bridge, if g-ij is

disconnected. Thus at least one node can be disconnected from the remaining

connected component by the deletion of a bridge.

Again in a cycle, there is no single bridge, however, any two links can be defined as a

bridge-set. Thus using these definitions we can define a network in which each link

and node are crucial for the network.

Definition 7: A minimally connected network2 is one in which each link is a bridge

and each node is a cut-node.

Another widely used type of network, is a regular network. In an r-regular network,

all nodes are linked of exactly degree r.

4.1 Benchmark Case

As long as there is no threat of attack on either the nodes or the links within a

company, competition among firms will simply be based on prices. Therefore the

structure of the within-firm network does not matter. As a benchmark case, we will

therefore analyze the network, that the network designer will build in case that there is

no threat of attack at all.

In this benchmark case, the network designer has a linking budget B=(n-1) and a

defense budget of Def=0. He, therefore, can build a minimally connected network, as

by definition, a minimally connected network uses exactly (n-1) links. Since we

assume that the network designer’s net payoffs are convex in the size of the group that

2 Minimally connected networks are also known as trees. If the minimally connected network contains all nodes, it is known as a spanning tree.

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is connected, we can conclude that the network designer will always aim to form

networks that consist out of one minimally connected component, as can be seen in

figure 3 below. However, there are more than one way of building a minimally

connected network, but in case of no attack they are all equally efficient.

Lemma 1: Every minimally connected network is equally efficient if there is no threat

of attack.

Proof: Every minimally connected network uses exactly n-1 links to connect n nodes.

In case of no impending attack, it does not matter in which way the nodes are linked,

as distance or decay does not play a role in the model. Therefore all minimally

connected networks are equally efficient.

The minimality of the efficient network is due to the assumption that no friction is

involved in the network, so that the benefit of a connection does not depend on the

length of the path between two agents. That the networks be minimally connected is,

however, a very permissive requirement since it includes a range of network

architectures even in the simplest cases with only a limited number of players.

Networks (a) – (f) in figure 3 below are just examples of possible forms that a

minimally connected network with 6 nodes can take. But in the case with no threat of

an attack, these networks are all equally efficient.

Figure 3: Minimally connected networks

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Thus, structure does not matter if there is no threat of an attack. But if there is such a

threat on either the nodes or the links, it does, as we will show in the following

section.

4.2 Network Disruptor and low linking costs

Introducing a network disruptor into the model, makes structure a decisive topic in the

node deletion as well as in the link deletion case. Thus for firms that are facing an

attack, the internal structure of the firm can be a decisive factor on how well they will

deal with the attack. Before we turn to any results, it is necessary to introduce one

more graph theoretic concept, and that is Menger’s Theorem (Menger, 1927). We

introduce the theorem here, rather than in the modeling section with the other graph

theoretic concepts, due to it’s importance for the rest of the paper. Menger’s theorem

(undirected internally node-disjoint version) states that in any finite undirected graph

G, the minimum node cutset between two distinct non-adjacent nodes i and j is equal

to the maximum number of internally pairwise node-independent3 paths gij in the

network. In the undirected internally link-disjoint version, the theorem states that in

any finite undirected graph G, the minimum link-cutset between two non-adjacent

nodes i and j is equal to the maximum number of internally pairwise link-

independent4 paths gij in the network. For a proof of the theorem, see for example

Gibbons (1994).

Since we are in a two-stage full information game, the network designer can

anticipate a disruption of his network. In case linking costs are low, he can afford a

large defense budget Def, to aim for a maximally robust network, while minimizing

the number of links he needs to achieve this. Thus within a firm extra link between

single nodes may help to prevent the total disruption of the network due to the fact

that a limited number of nodes has been taken out. To come up with a robust network

topology, we can use the formulation proposed by Dekker & Colbert (2004). They

rewrite Menger’s theorem, in terms of link and node connectivity, as:

For any graph:

(i) the node connectivity κ is the smallest number of node-distinct paths

between any two nodes 3 Node-independent paths, are path that do not share any of the same nodes. Internally node independent paths, do not share any nodes except for the first and last node. 4 Internally link independent paths, do not share any links within the network.

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(ii) the link connectivity λ is the smallest number of link-distinct paths

between any two nodes.

Thus to separate any node from the network, at least κ nodes need to be disrupted or λ

links. Now this makes it clear that for a network to be robust against an attack by a

network disruptor it is best to have both values as large as possible, given their natural

boundary of ηmin .

While the attack on nodes of a network and the attack on links of a network, lead to

the same most robust network topology, we will still treat them apart from one

another to avoid confusion.

4.2.1 Attack on the nodes of a network if linking costs are low

An attack on the nodes of the network can be interpreted as the network disruptor

trying to lure away crucial players from the network designer’s network, towards his

own. For the sake of simplicity, we abstract from the fact, that those players will be

incorporated in the network disruptor’s firm network somehow. To achieve this, the

network disruptor has a disruption budget of nodes, meaning that he can disrupt

exactly D nodes.

nD

Even though we assume that linking costs are low, links are still costly. Thus, here we

are, also trying to figure out how to achieve a robust network topology using a

minimal number of links. Since the designer cannot keep the whole network safe

because of the positive disruption budget, we aim for the highest possible safety of the

network, namely (n-D) proofness, using a minimal number of links. Thus we first

define robust networks that are minimal, in that they use the fewest links for achieving

the highest degree of κ possible. What is apparent from Menger’s theorem is that for

any D<κ, no damage can be done to the network because κ gives the smallest number

of node-distinct paths between any two nodes in the network. Thus except for the

disrupted nodes, no additional node can be separated from the network. We also know

that κ=ηmin is the maximal value of κ. Thus if each node receives at least ηmin links, κ

can obtain its highest value. By definition, in a regular network, each node receives

exactly r links. Thus setting r= ηmin, we have an r-regular network in which each node

receives exactly ηmin links. The number of links needed to build an r-regular network

can be calculated in a quite simple way, as all nodes need to have exactly r links and

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the easiest way to achieve this is to link them up pairwise, thus using n/2 extra links

for each increase in r by 1. To put it into a simple mathematical formula:

nrr *2 5. Therefore all r-regular networks, for a given r, use exactly the same

amount of links, in relation to the number of nodes included in the network. Knowing

this relation, the proof of existence is straight forward.

Lemma 2: For every even number of nodes n, there exists an r-regular network, for

any given r6.

Proof: Given that the number of links needed to create an r-regular network, given n,

can be calculated as nrr *2 , as long as n is even, 2

n . Since as well,

we also know that

r

*2rr n , since the product of two positive integers is also

always a positive integer. Thus a r-regular network exists for all even numbers of

nodes n for any given r.

Then, assuming that an r-regular network exists that is exactly n-D proof, given a

disruption budget of D, we can show that it is also using only the minimum amount of

links necessary to do so.

Lemma 3: If an r-regular network is exactly n-D7 proof given D, then it is also

minimal n-D proof given D.

Proof: Step 1: In order for a network to be (n-D) proof, each node must have at least

(D+1) links and therefore (D+1) direct neighbors. To see why, suppose a nodes has

x<(D+1) neighbors, then by taking out these x neighbors, the disruptor can take out

(x+1) node. After that he can still disrupt at least (D-x) nodes, so that he can in total

disconnect at least (D+1) nodes from the network. Step 2: By step 1, each (n-D)-proof

network has at least n(D+1)/2 links. And each regular network has exactly

nrr *2 links. Step 3: If a (D+1)-regular network exists for n, and if this network

is (n-D) proof, then by Step 2 it is (n-D)-proof with a minimal number of links.

Thus, we know by Lemma 4, that using an r-regular network, where r=D+1, is a

minimal way to make a network n-D proof. However, we still need to show, that these

networks are indeed n-D proof. However, nothing in general can be said about the

5 This holds only for an even number of n. 6 For an uneven number of nodes, this holds for every even r. 7 Note that not all r regular networks are n-D proof, even for r≥D+1.

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connectivity of networks that are only regular8. And not all r-regular networks are

also (n-D)-proof. We therefore need to refine the concept of regularity to define the

networks that will be n-proof.

Definition 8: Call any subset of connected nodes of size b with 2nb within the

network a b-set.

Lemma 4: To make a r-regular network n-D-proof, given r=D+1, the network

designer needs to link nodes in such a way that every b-set of nodes has at least D+1

neighbors in the rest of the network.

Proof: If b-sets of nodes are linked to the rest of the network with less than D+1

direct neighbors in the rest of the network, the network disruptor with a disruption

budget of D, can disconnect the b-set from the rest of the network by taking out those

neighbors, making the network therefore no longer (n-D)-proof.

This shows that there should not be any local cliques or clusters between nodes in a

network, as this will make them easy targets to be separated from the rest of the

network. In local clusters, links are wasted in the sense that would those links lead to

the rest of the network, the topology of the whole network would be made safer.

Proposition 1: Any r-regular network in which every b-set of nodes has at least D+1

links to the rest of the network is a robust network if the network designer aims to

keep his network (n-D)-proof.

Proof: By Lemma 3 we know that all r-regular networks that are n-D-proof are also

minimal n-D-proof. By Lemma 4, we know that to make an r-regular network n-

proof, we need to make sure that every b-set of nodes has at least D+1 links to the rest

of the network. Combining the two shows that any r-regular network adhering to

lemma 1 will be n-D-proof and minimal.

That such networks exist can be seen from the class of symmetric networks. All

symmetric networks fulfill these conditions. However, since there are also networks

that are non-symmetric but still robust, symmetry is only a sufficient, not a necessary

condition for (n-D)-proofness. A symmetric graph is a regular graph that is both link-

8 For an example of a 3-regular graph, that has κ=λ=1, see figure 1a in Dekker & Colbert (2004)

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and node transitive. Node (link) transitivity refers to the property of the graph that all

nodes (links) have the same surrounding, so that no node (link) can be distinguished

from one another based on their location within the network. This means that all

nodes (links) are similar to one another9. Therefore by definition all symmetric

networks need to be regular networks, however, as we will see in the example below,

not all regular networks are necessarily symmetric. In a symmetric network, all nodes

have the same number of neighbors at the same distance10. Because symmetry is such

an important concept, and it is somewhat different, from the daily use of the world,

we will here show with some examples what is meant by symmetry in a graph

theoretic sense of the word.

Figure 4: Symmetric Networks

As can be seen in the figure above, the original graph can be permutated in several

different ways, while still keeping all the links intact. In this graph, all nodes have the

same number of neighbors at the same distance. Thus each nodes has 3 neighbors at

distance 1, and 4 neighbors at distance 2, and this remains the same also in the

different permutations of the graph. You cannot distinguish the nodes from one

another by means of their position in the network. Therefore, the graph is symmetric.

As can also be seen here, the symmetric graphs also fulfill the b-set condition from

Lemma 4. The following graph also looks symmetric in the conventional sense of the

word. However, when studying the graph more closely, we can see that it is not

symmetric in the graph theoretic sense of the word.

9 Cf Weisstein (2010) 10 By distance we mean the shortest path between any two nodes.

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Figure 5: Non-symmetric networks

The nodes here can be separated into two groups (cycles and squares). While we can

also draw permutations of the network, we can see that the roles (cycles and squares)

remain the same, also in the permutations. This is due to the fact that you can

distinguish between the two groups of nodes, simply by their position in the network.

While the squares have 3 nodes at distance 1, 3 nodes at distance 2 and 1 node at

distance 3, the cycles have also 3 nodes at distance 1, however only 2 nodes at

distance 2 and also 2 nodes at distance 3. Thus the network is not symmetric. While

these difference seem small at first sight, it is already apparent that while both types

of networks in this example use the same number of links, the symmetric one has

κ=λ=ηmin=3, whereas the non-symmetric one is not, with κ=λ=2 and ηmin=3. We can

also see here that the b-set condition is not fulfilled, as the set of nodes 1,6,8 and 7 for

example only has two neighbors in the rest of the network and can thus be separated

by the disruption of nodes 2 and 5 from the rest of the network.

4.2.2 Attack on the links of a network if linking costs are low

Link deletion in this context can be interpreted as an attack on the communication

channels within a firm for example. Again we will abstract from the fact that the

network disruptor might include the links he disrupts in his own network, and assume

that he simply tries to cause maximal damage to the network designer’s network. To

achieve this, the network disruptor has a disruption budget of nodes, meaning that

he can disrupt exactly D links.

lD

Unlike for the node deletion case, we know that the maximally proof network that can

be achieved for the case of link deletion actually includes all nodes. Thus, the network

designer tries to aim for a n-proof network. Since Menger’s Theorem also holds for

link deletion, we can deduce from Lemma 3 that every regular network that is n-proof

is also minimal. However, just as for the node deletion case, we need to make some

15

further adjustments, as not all regular networks are also n-proof, as can be seen from a

simple example:

Figure 6: 3-regular graphs

Whereas all three graphs contain 16 nodes and are 3 regular, only the last one is n-

proof for a disruption budget of 2 links. The first one can be cut in half, and also in the

second one the largest remaining component would only encompass 6 nodes. We

therefore again need to use the definition of b-sets to make the network robust against

an attack.

Lemma 5: To make a r-regular network n-proof, given r=D+1, the network designer

needs to link nodes in such a way that every b-set of nodes has at least D+1 links to

the rest of the network.

Proof: If b-sets of nodes are linked to the rest of the network with less than D+1 links,

the network disruptor with a disruption budget of D, can disconnect the b-set from the

rest of the network, making it therefore no longer n-proof.

This shows that networks that are n-proof under link deletion also do not contain any

clusters. Just as in the node deletion case, clusters are easily targetable. Clustering

makes the links between the clusters more vulnerable to attacks, where as the links

within the network are safe. For a random attack, clustering might therefore be a valid

technique (see for example Albert, Jeong & Barabasi (2000) ), for targeted attacks of

an intelligent network disruptor it does not work though, since the network disruptor

would delete those weak links between clusters.

Proposition 2: Any r-regular network in which every b-set of nodes has at least D+1

links to the rest of the network is a robust network if the network designer wants to

keep all his nodes save.

Proof: By Lemma 3 we know that all r-regular networks that are n-proof are also

minimal n-proof. By Lemma 5, we know that to make a r-regular network n-proof, we

16

need to make sure that every b-set of nodes has at least D+1 links to the rest of the

network. Combining the two shows that any r-regular network adhering to lemma 1

will be n-proof and minimal.

The cycle is still a special case, as the cycle is the only possible 2-regular network that

can be build. In all other cases (except for the complete network), for a sufficiently

large n, there are a number of variants to build regular networks that also conform to

the b-set rule and are therefore minimal n-proof, as can be seen from the example in

figure 4.

So we can conclude that the network designers defense budget needs to contain at

least nrr *2 (and r=D+1) links, to make his network n-proof. As we see here,

however, the number of links used to make a network n-proof is linearly increases

increasing by (n-1)/2, for each increase of 1 in the disruption budget. For a given D,

the network designer needs to use nrr *2 links in total to make a network n-

proof. In terms of D, this means the network designer needs to increase his defense

budget by n/2 for every increase in D by one, for D>1. However, this means that the

costs for keeping the network safe increase exponentially. For even a relatively small

D, the number of links needed to make the network n-proof is very large.

Another more intuitive way to see why regular networks are robust as well as minimal

can be seen from the following example:

In a typical hierarchically structured company, the network of social interaction in

might look something like this:

Figure 6: Firm network

Now to make this network proof against an attack of a disruption of one node, we

know by Menger’s Theorem that κ needs to be at least D+1. From the principal of

17

optimal connectivity, we can then deduce that ηmin=D+1. Thus each node needs to

have at least 2 links. In the case introduced here, we can see that a hierarchical

structure of the firm is not good, as the firm collapses into different pieces as soon as

the boss is removed. Even if the boss cannot be removed, a significant part of the

network is lost as soon as one of the heads of department is disconnected. If the

nodes would instead be arranged in a cycle no node could be disconnected from the

network. Therefore in case of low linking costs, it is important to not build a

hierarchical structure within the firm, but rather an unranked structure, without any

local cliques or clustering. Thereby all people within the firm fill out one social role.

To the network disruptor, they are made anonymous, as he cannot distinguish from

their position in the network who they are. This also prevents that nodes are so central

to the network that the structure completely collapses if this node (or set of nodes) is

removed. The same holds for the case of link deletion. If the link between the boss

and one of the heads of departments is disrupted, a significant part of the network is

disconnected. If the nodes would instead be arranged in a regular network that has no

local cliques, no harm could be done to the network, by the disruption of one link.

Thus summarizing one can see that while for low linking costs both link and node

disruption lead to the same general result of building a network, the post-disruption

network after a disruption of the links is naturally larger than after a disruption of the

nodes, since no nodes are taken out.

4.3 High linking costs

Looking at the other extreme point, we consider permissively high linking costs,

which leads to a defense budget of Def=0. Thus adding additional links next to the

original linking budget of B=(n-1), is permissively expensive; Leaving the network

designer to structure his network as robustly as possible with this limited amount of

links.

As in the previous case, the structure within the firm now becomes decisive again.

However, while in the case of low linking costs, there was no difference in structure

needed between the cases of link deletion and node deletion, here it becomes quickly

apparent that link deletion and node deletion do not lead to the same results. This is

obvious by simply looking at the minimally connected networks introduced in figure

3. While the network disruptor can completely separate the star network (a), by taking

out the central node, the maximum damage he can cause to the chain network (c) is to

18

separate two additional nodes from the network, thus leaving a largest remaining

component of n/2. For link deletion, however, the star network (a) seems like a good

option, since the maximal damage the network disruptor can cause by taking out a

single link, is to disconnect one node from the network, whereas the chain network (c)

can be cut in half. Thus, here again, we will start by analyzing the node deletion case,

and then turning to the link deletion case. In both cases we will first deal with the

special case of a disruption budget of 1, and then turn to more general results.

4.3.1 Attack on the nodes of the network with a disruption budget of 1nD

From Menger’s Theorem we know that to keep a network safe from an attack on any

one of its nodes, there need to be at least two internally pairwise node-independent

paths between any two non-adjacent nodes in the network. However, in a minimally

connected network there is only one path between any two non adjacent nodes in the

network11. Thus it is always possible to separate the network by taking out one node.

Lemma 6: In any minimally connected network, the largest remaining component

after an attack by a network disruptor with a disruption budget of =1, will be

maximally of size (n-1)/2.

nD

12

Proof: Since, by definition there is no cycle in any minimally connected network, it

can always be cut into at least two separate components by disrupting a single node.

Step 1: A disruptor can always do better than to take out a node such that the largest

remaining component has size larger than (n – 1)/2. Let the disruptor take out node x

with degree ( )i g =d, which has links to nodes y1, y2,…, yd. Given that only (n – 1)

links are used, taking out node x leads to d separated components, which we can

denote as g1, g2,…,gd. Let the node labeled yd and the corresponding component gd

have size s, with s > (n – 1)/2. Then it follows that g – gd, meaning the network

obtained when component gd is removed, has size smaller than (n – 1)/2.

By instead disconnecting yd in gd, the disruptor can assure that the component of

nodes connected to a neighbour zd of yd has a size of at most (s – 1), so that the size of

this component is at most (n – 1)/2. At the same time, we have already seen that the

size of component g – gd is smaller than (n – 1)/2. It follows that the disruptor is better

dzg

11 For a proof of this see for example, Chartrand (1977) Theorem 4.1. 12 This holds for an odd number of nodes. For an even number of nodes the size will maximally be n/2.

19

of by disrupting yd. It follows that a disruption strategy where a largest component

larger than (n – 1)/2 is left can never be optimal for the disruptor. Step 2: Given that

by Step 1 a network disruptor never leaves a largest component larger than (n – 1)/2,

the best that the designer can possible do is to leave a largest component of exactly

size (n – 1)/2.

x

yd

y1

y2

y3

g1 g2

g3

(…)

gd zd

dzg

Figure 7: Minimally connected network

That such networks exist for which the most harm a network designer can cause is

actually leaving a largest remaining component of n/2 nodes, can be seen from the

example of the line network. While the line network certainly is not the only

minimally connected network for which this lemma holds, it shows in a very straight

forward way, that this kind of networks do exist.

Generalizing this result, so that for any disruption budget D the maximum size of the

largest remaining component can be determined follows along the same argument and

can also be seen in the line network.

Lemma 7: In any minimally connected network, the largest remaining component

after an attack by a network disruptor on the nodes of the network with a disruption

budget D, will be maximally of size (n-D)/(D+1)13.

Proof: By Lemma 6, we know that this holds for =1, since (n-D)/(D+1)=(n-1)/2.

Since there is no cycle in a minimally connected network, we know that with a

disruption budget of =x, we can split the network into at least (x+1) separate

components. Suppose now that the network disruptor has already used (x-1) nodes.

By Lemma 6, we know that he will use the last node in his disruption budget to split

the largest remaining component in half. Now, if he revokes the disruption of 1 of the

prior nodes, he will again use this last node he has to cut the largest remaining

nD

nD

13 Again for odd numbers of nodes.

20

component in half. He can continue this process until the network is split into (x+1)

equally large components, where he cannot do any better because by making one

component smaller he will automatically make another component larger. Since he

takes out D nodes by definition, and he can split the network into (D+1) pieces, each

component will consist maximally of (n-D)/(D+1) nodes.

However, if we do not include all nodes into the connected component, the network

designer can build a stronger network, by leaving one node out of the connected

component, and building a cycle network of the remaining (n-1) nodes.

Lemma 8: Every 2-regular graph consists of a single cycle.

Proof: Step 1: Take a graph with n nodes. Since it is 2-regular we know that it has n

links. Let there be a node i and a node j in this graph that are directly linked to one

another. Since r=2 and the graph is a simple graph, both nodes then have 1 more link

to a different node. Let i be linked to node k and j be linked to node m. Step 2: If k=m,

then the cycle is complete and we have a network with n=3. If k≠m, by r=2 we know

that they each have 1 more link to another node. Thus we have n-4 nodes left to use

and n-5 links left to use. Since each node has exactly degree 2, we can go on by this

means until we use exactly n links. At this point, because it is a 2-regular network,

these links need to lead to the same node, as we have only the nth node left and no

more additional links.

Lemma 9: In a cycle network, the network disruptor with a disruption budget of D

will cause maximal damage by cutting the network into D separate components, each

maximally of size (n-D)/D14.

Proof: Suppose we have a cycle network of size n. Then by Lemma 8 we know that it

is a 2 regular network with exactly n links. For a disruption budget of D=1, we know

that the network disruptor can disconnect only the node he disrupts, leaving a post-

disruption network of n-1 nodes connected in one component. Step 2: For a disruption

budget of D=2, the network disruptor can disconnect all nodes that are in between the

two nodes he disturbs, because as we have shown in lemma 8 all nodes are on a cycle.

Thus taking out 2 nodes that are not directly linked leads to the separation of the

network. Knowing this, the network disruptor will cause maximal damage by making

the separated part as large as possible, however, not larger than n/2-1 because then the

remaining component would be smaller than that, and we are only interested in the

14 Abstracting from issues of divisibility.

21

largest remaining component. Thus the network disruptor will separate n/2-1 nodes

from the network, leaving a post disruption network of two separate components of

size n/2-1. Step 3: Taking any positive disruption budget D, the network disruptor can

separate any nodes from the network that lie between any two of the nodes he

disrupts, thus separating the network into D parts. Since we are only concerned with

the largest remaining component, making all parts equally large is the best option

because otherwise, by making one part smaller, he would automatically make another

one bigger, since the pre-disruption network was a cycle. Thus, to cause maximal

damage, he will cut the network into D equally sized parts. Step4: The size of the

parts is then determined by the number of nodes that are disrupted. Since if D nodes

are disrupted, only (n-D) nodes are left in the network, the parts will be of size (n-

D)/D.

Thus we can conclude that a network designer with a linking budget of B=(n-1) will

in any case build a cycle network, including (n-1) nodes within the network, when

facing an impending attack by a network disruptor with a disruption budget of D=1.

This implies that the network designer will leave nodes out of his original network to

increase the robustness of his network against an attack. In fact, a cycle network is

also a symmetric 2-regular network. Thus this result is in line with the result from the

previous section. However, to achieve it, we need to leave one node out of the

connected component, due to the limited number of links that we can use.

4.3.2 Attack on the nodes of a network with a disruption budget of n D

We have shown above that building a component connecting all nodes in a minimally

connected network, is not a good idea, since this network can be split in half by

simply removing one node. As we can already see for the case of a disruption budget

of 1, this leads to a tradeoff for the network designer between building a large but

weak component, and building a smaller, stronger component. If we increase the

complexity of the model more and allow for the network disruptor to have a positive

disruption budget up to the size of D=B/2, we see this trade off even more explicitly.

Thus to see that it is never a good idea, to build a network encompassing all nodes,

given a linking budget of B=(n-1), we need to show that (n-D)/(D+1)<(n-D-1)/D.

Here it is (n-D-1) and not simply (n-D) because in a cycle network for a linking

budget of n-1, we can only use n-1 nodes and not all n nodes. If we solve this

22

inequality for D, we receive that for any D smaller (n-1)/2=B/2, building a cycle

network is the better solution. This means that it is a better solution to build a cycle

network than a minimally connected network if facing a network disruptor, for all

interesting cases, because any disruption budget larger or equal to B/2, will lead to a

complete disruption of the network into all single nodes.

Proposition 3: When facing a network disruptor with a positive disruption budget

smaller B/2, a network designer with a linking budget of B=(n-1), will never build a

network encompassing all n nodes.

Proof: By Lemma 6 we know that the largest remaining component in a minimally

connected network after the disruption by a network disruptor with a disruption

budget of D, will be maximally of size (n-D)/(D+1). By Lemma 9 we know that the

largest remaining component in a cycle network after the disruption by a network

disruptor with a disruption budget of D, will be (n-D-1)/D for a network using a

linking budget of B=(n-1). Solving this inequality, it can be shown that for any

D<B/2, the post-disruption network in a cycle network will always be larger than the

post-disruption budget in a minimally connected network.

This of course does not imply that the cycle network is the best possible network for

the network designer to build. What it does show, however, is that it is never good to

build a network including all nodes, if you have a limited linking budget. It also

implies, though, that there is a trade off between building a larger and a stronger

network. Here the trade off only deals with one single node, however, this result hints

that better networks may exist, where even less nodes are used, but that are more

highly connected and therefore stronger. However, this leads to multiple different pre-

disruption networks that are all equally good. Since these cases are hard to

characterize, we will only hint at what such networks can possibly look like by means

of an example.

We have already shown in the case of the low cost links, that to make his network

maximally robust against any attack, the network designer would have to build an

optimally connected symmetric r-regular network with r=D+1. However, due to the

limited linking budget this would lead to leaving out already a considerable amount of

nodes. For a disruption budget of 2nD , this would then lead to building a

connected, symmetric 3-regular network. Thus knowing the linking budget of B=(n-

23

1), and that the network designer needs exactly 3/2*n links to make a network 3-

regular, leads to the conclusion that the network designer is forced to leave 1/3*(n-1)

nodes out of the connected network to build a symmetric 3-regular network.

Now taking as an example a network with n=25 nodes and a linking budget of B=24

links.

Figure 7: 25 node networks

The 3-regular network in graphic a, only uses 2/3*B nodes, as has been calculated

above. In network b, there is exactly one node between any two highly connected

nodes. Thus we know that the post-disruption network will consist of 3 nodes less

than the pre-disruption network. In network c, there are 3 nodes between any two

highly connected ones. Therefore we know that any post disruption network will

consist of 5 nodes less than the pre-disruption network. However, the second factor

24

that we need to take into account is the size of the pre-disruption network. Network c,

uses 22 nodes, network b uses 20 nodes and network a uses only 16 nodes. Therefore,

although less damage in terms of disconnected nodes can be done to network a, as

compared to b and c, the post-disruption network might still turn out to be smaller

than in the larger networks. In this particular example, we can see, that networks b

and c have the largest post-disruption network, since they contains a connected

component of 17 nodes, as compared to 14 in network a.

Even in this small example with a limited number of nodes only, we can see that there

is a definite trade-off between including more nodes in the pre-disruption network and

making a stronger network. It can be seen that the 3-regular network has a smaller

post-disruption network than the larger and less strong networks b and c. However,

when comparing this to the cycle network, we have shown in Lemma 10 above that

the largest remaining component will only consist of 11 nodes, which is even smaller

than the post-disruption network in the 3-regular network. Thus, a cycle network

seems to be a too weak pre-disruption network.

Now turning back to the link deletion case, we will see that here a full

characterization of the pre-disruption network is easier to achieve.

4.3.3 Attacks on the links of a network with a disruption budget of 1lD

Turning to link deletion, we have shown in the benchmark case, that if there is no

threat of an attack on the links of the network, all minimally connected networks are

equally efficient. However, now taking into account that there is a network disruptor

with a disruption budget >0, not all minimally connected networks are equally

efficient any longer. We will show in the following that the star network is the only

efficient network in case the network designer faces a network disruptor with a

disruption budget of .

lD

1lD

Lemma 10: Any minimally connected network contains at least two end-nodes.

Proof: The proof is organized in two simple steps. We will first show by

contradiction that any minimally connected network does not contain a cycle. This

result will then be used to show that there are at least two end-nodes within any

minimally connected network.

Suppose not: Any cycle uses the same number of links as it contains nodes, since all

nodes need to be connected of exactly degree 2)( gi . Any cycle including x nodes,

25

then needs to use x links. Then there are n-x nodes remaining that are not connected to

the network yet, but only n-x-1 links. Therefore, no minimally connected network can

include a cycle. Knowing that no cycle is contained in any minimally connected

network, there have to be at least 2 end-nodes in the network, since without a cycle at

least two nodes need to be connected of degree ( ) 1i g , which are end-nodes by

definition.

The highest number of end-nodes a minimally connected network can have is (n-1).

Such a network is a star network.

Lemma 11: The maximal damage a network disruptor with a disruption budget of

=1 can cause in a star network is to disconnect 1 node. lD

Proof: By definition, a star network consists of one central node and (n-1) end-

players. Therefore any node except for the central node is connected only of degree 1

( ( ) 1i g ). Thus, any link that will be deleted will only be able to disconnect one

node from the network. Let the disruptor delete link 1ijg . Then there is still a

path1 1 2 1

... 1k k kii i i i i i jg g g g

. Exactly the same argument applies to any link

in the star. So the maximal disruption the network disruptor can cause is separating

one node.

Since by Lemma 10 we know that in any minimally connected network, there are at

least 2 end-nodes, we know that no matter what network the network designer with a

linking budget of B=(n-1) builds, at least one node can always be separated from the

network by a network disruptor with a disruption budget of 1lD . Thus there is no

network that is strictly better than the star network.

4.3.4 Attacks on the links of a network with a disruption budget of lD

Now if we generalize the argument, we see that no matter how large the disruption

budget is, as long as there is no defense budget of the network designer, the star

network is the most robust network topology against attack.

Lemma 12: The maximal damage a network disruptor with a disruption budget of

can cause in a star network is to disconnect D nodes.

lD

Proof: By Lemma 11 we know that with a disruption budget of 1, only 1 node can be

separated from the network. The same argument applies to any disruption budget.

With each link that the network disruptor can delete, he can only separate one node

26

from the network and therefore, with a disruption budget of D, he can separate exactly

D nodes.

These results then lead us to conclude that any minimally connected network that is

not the star, will have at least two nodes that are not end-nodes.

Lemma 13: Any minimally connected network that is not the star contains at least

two nodes that are connected of a degree larger than one.

Proof: In a star network, by definition, there is one central node i and ( 1n ) end-

players j 1 2 1, ,..., nj j j j . Therefore there are ( 1n ) nodes connected of degree

one and one node connected of degree ( 1n ) . In any minimally connected network

that is not a strict star, there will be no direct link from i to 1j , so that 1 1, , 0i j j ig g .

For node 1j to still be connected to the network, it must be connected to any

node 2 ,..., n 1j j j . Since all of these nodes are also connected to i, there will be at

least one node except for i that is connected of a degree larger than one, so in total at

least two nodes that are connected of a degree larger than 2.

The fact that the star is the only minimally connected network that consists of (n-1)

nodes that are end-players is important as it shows that the star is the only efficient

minimally connected network in case of an impending attack, since in all other cases

the damage that can be done by a network disruptor is greater.

Proposition 4: The star network is the only efficient minimally connected network, if

an attack by a network disruptor with any disruption budget of , lD 1D is

faced. For a disruption budget of D> -1, it does not matter with which minimally

connected network we start as only 1 link will remain.

Proof: By Lemma 13, we know that the network disruptor with a disruption budget of

1 will be able to disconnect at least one node from any minimally connected network.

By Lemma 11, we know that in a star the maximal damage that can be caused is

disconnecting exactly one node. Therefore the star is efficient. The star is the only

efficient network, since by Lemma 12 we know that in any other network, at least one

more node is connected of a degree larger than one, and therefore more than one node

can be separated from the network by deleting one link. So there is always the

possibility to disconnect at least one more node from any other minimally connected

network than from the star.

27

So while all minimally connected networks are equally efficient if there is no threat of

an attack, the star is the only efficient network in case of an impending attack, if the

network designer has no means to defend the network. Also while there is a trade-off

between larger and stronger networks for the case of node deletion, no such trade-off

takes place in the case of link deletion. Therefore, the most robust network topology is

quite different for the two cases. In the link deletion case, it is always optimal to

include all nodes in the pre-disruption component, whereas in the node deletion case,

the network designer is better off leaving out a number of nodes to build a smaller but

stronger pre-disruption network. In general it seems that nodes are harder to protect in

a network then links, not only because in the node deletion case nodes will be

disrupted by definition in the node disruption case but also because all links attached

to a node are disrupted once the node is disrupted. Therefore it is much harder to keep

nodes safe from disruption than to keep links safe.

5. Conclusion

We have shown that when other considerations except for prices, quality or quantity

of a product, enter competition amongst firms the structure of the within firm network

becomes a decisive factor. As soon as a competitor tries to lure away players from a

competing firm or disrupts communication channels between players in a firm

network, the possibility of survival of that firm largely depends on the within firm

structure. We have shown that structure becomes increasingly important if links

between nodes are very costly. Then the topology of the within firm network does not

only depend on the size of the disruption budget that the network disruptor has but

also on the network designer’s expectation on whether the attack is directed on the

nodes or links of his network. If it is directed on the links, he will include all nodes in

the pre-disruption network and arrange then in form of a star network, thus a

hierarchical network with one key player. If the attack, however, is directed on the

nodes of the network, the network designer faces a trade-off between the size of the

pre-disruption network and the strength of that network, since it is always better to not

include all nodes within the network. In both cases it is clear that forming clusters or

local cliques is not a good mechanism, since these clusters are likely to become

targets of attack.

Of course this work is exploratory work and there is still a lot of room for further

research. The first step being to research how exactly networks should be structured in

28

the case of node deletion, what happens once the network designer has any positive

defense budget and in what relation the size of the network stands to the post-

disruption network. For this it will be necessary to deal more with the mathematical

discipline of graph theory, which, if used properly can be enormously helpful.

However, since they have a different goal-setting some of their results need to be

reworked to fit into the economics literature.

29

30

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