Multilayered Network Model

Kennedy et al. developed a model to determine minimum cost cut-sets across multiple layers of networks. [73] This model finds the minimum cost (or maximum

determining an overall cut set as driven by objective function. Before this formula-

tion is introduced and developed, some additional variable definitions are explained.

First, let I be a node or arc(s) with common interdependencies across k net- works. That is to say, I has common elements in all layers of the k networks of

interest, or in some subset of the layers. In addition, let Wi be the set of all effects

options, w, which can be applied to the elements in I. The option w may affect all

the elements in I, or it may affect a subset of I.

Associated with each option against a particular interdependent element is the actual effect. For a given I and wi, let δik be the change (effect) on node i of

network k given the selection of wi. Define δijk to be the change (effect) on arc (i, j)

of network k given the selection of wi. Of course, the effect may be zero in some or

all networks. In addition, affecting a node could also affect a number of arcs.

δ(ijk),w =

 



1, if arc(i, j) of network k is affected by option wi ∈ W

0, otherwise

(5.2)

For a given I, it is assumed ywi = 1 if option wi is selected and zero otherwise.

ywi =

 1, if wi ∈ WI chosen

0, otherwise. (5.3)

In a targeting model, it can also be assumed that one would not wish to

double strike a target, or w could be the level of strike(s) required (at least in initial planning). Therefore, at most, one of the common attack options w is selected,

leading to the constraint:

X

w∈WI

where C is the set of all commonalities I. Finally, Cw, represents the relative cost

of cutting the interdependent arcs associated with using option w. This leads to the following model: minX k∈K X (i,j)∈Ak cijkνijk+ X I∈C X w∈WI Cwyw s.t. πik− πjk + νijk+ δijkwyw ≥ 0 πtk− πsk ≥ 1 X I∈C X w∈WI yw ≤ 1 π, ν, y ∈ {0, 1} (5.5)

where cijk is the flow capacity along arc (i, j) of network k, νijk is the dual variable

associated with the capacity constraint of arc (i, j) of network k, πik is the dual

variable associated with the conservation of flow for node i of network k, sk is the

source node for network k, tk is the sink node for network k.

In order to extend this minimum cost cut-set model to an interdiction model, it

is first necessary to convert the notation to be consistent with interdiction literature. The variable names are therefore renamed as demonstrated in Table 5.1.

Table 5.1: Variable Naming Substitutions

Layered Notation Interdiction Notation

Flow capacity cijk uijk

Capacity constraint dual νijk βijk

Node dual πik αik

Selection of interdependency yw γw

Interdependent cost _Cw rw

minX k∈K X (i,j)∈Ak uijkβijk+ X I∈C X w∈WI rwγw s.t. αik − αjk + βijk+ δijkwγw ≥ 0 αtk− αsk ≥ 1 X I∈C X γ∈WI γw ≤ 1 α, β, γ ∈ {0, 1} (5.6)

where uijk is the flow capacity along arc (i, j) of network k, βijk is the dual variable

associated with the capacity constraint of arc (i, j) of network k, αik is the dual

variable associated with the conservation of flow for node i of network k, sk is the

source node for network k, tk is the sink node for network k.

5.3 Multilayered Network Interdiction

A quick comparison reveals that the traditional network interdiction model

(formulation (5.1)) and the layered network cut-set model (formulation 5.6) are sim- ilar. In order to facilitate the extension of interdiction to multiple layers, it is helpful

to have a conceptual understanding of how the maximum flow network interdiction model (formulation 5.1) works. This model sets nodes on either side of the cut, and

it sets γi,j = 1 or βi,j = 1 for forward arcs across the cut, to satisfy the main ”dual”

equation.

In other words, the cuts are identified by setting the αi and αj values. The

model evaluates the capacity of the cut with the βij variables. However, the at-

tacker/interdictor can avoid paying for some of the capacity (which would normally

allow flow through) by interdicting arc (i, j) via γi,j = 1. That is, the γi,j variables

behave like the βi,j variables, except that the γi,j do not have “costs” associated with

them. Unfortunately for the attacker/interdictor, only a limited supply of γ’s are available to be set to one (because of the attacker resource constraint).

The layered network cut-set model (formulation 5.6) works in a similar manner.

As a first step in converting this formulation to an interdiction model, an interdiction variable is added to the dual constraints for each edge in the network. Of course,

these interdictions do not count against the capacity or “costs,” but are subject to the resource constraint of the attacker. In addition, the “cost” of selection of an

effect γw is moved from the objective function to the resource constraint. In network

interdiction, the primary objective is not to minimize (interdiction) cost. Instead,

interdiction cost is converted into a constraint which is limited by the availability of the resource, R. In addition, in the layered network formulation (5.6), it was

assumed that a target would not be attacked multiple times. Since minimizing cost is no longer a primary concern (so long as the resource constraint is not violated),

this restriction can be dropped, if desired. Of course, in circumstances where it is important not to strike a target multiple times (for whatever reason), this restriction

should be retained. Otherwise, other constraints make the constraint limiting attacks to single strikes redundant and unnecessary.

The commonality variables remain largely unchanged. Selection of a common

effect works much like the selection of an interdiction variable. In each case, selection of the variable stops flow through the edge, and associated “costs” are limited by

the resource constraint, not the objective function. The commonality variable type accounts for interdiction across multiple networks with a single cost. Therefore, the

model determines which elements across the layers of networks should be interdicted to maximize disruption across the system of networks.

With the discussed modifications, the single level layered network interdiction

minX k∈K X (i,j)∈Ak uijkβijk s.t. αik − αjk+ βijk+ γij+ X w δijkwγw ≥ 0 αtk − αsk ≥ 1 X (i,j)∈A rijγij + X w∈WI rwγw ≤ R αik ∈ {0, 1} γw ∈ {0, 1} βijk ∈ {0, 1} (5.7)

When considering a system of layered networks as a holistic system, it is impor- tant to use commensurate units. Traditionally, networks are considered in isolation,

partly because each network usually serves a specific purpose. For example, con- sider infrastructure layers as an example. Water, energy, and telecommunications

all have different types of flows across their networks. Although these networks are connected, the material that is flowing does not cross networks (water never uses

electrical lines for transport). Instead, interdependencies are created through other means as discussed in Section 2.2.8.1. For example, the water infrastructure requires

electricity to power its pumps, SCADA systems, and so forth. In addition, water lines and electrical lines may cross the same bridge creating a geographic dependency.

Therefore, when considering the system of networks as a whole, the units must

be scaled and/or normalized. For example, in considering a multi-modal system of transportation networks, a commensurate unit that could be used across all layers

would be tonnage moved per unit of time. Another common unit used across multiple layers is cost/dollars. An additional option would be to use the approach by Wallace

et al. and use a binary variable to represent connectivity of a critical network system without regard to units of physical flow. As discussed in (2.4.0.5), this allows the

networks to retain their non-commensurate units, but still captures their interdepen- dencies. Either approach (using commensurate units or binary variables) could be

used with the developments in the section; but for consistency, commensurate units

are assumed.

5.4 Benders’ Partitioning

As with formulations 5.1 and 5.6, Benders’ partitioning is applied to develop a master problem and subproblem (which is similar to the development in [46]).

For fixed interdictions, γijk and yw, the linear relaxation of (5.7) is a dual of

a network flow problem which has an intrinsically integer solution. Therefore, the dual can be taken which results in the following program:

min γ∈Γ,w∈Wmax X k xtsk s.t. X j:(i,j)∈Ak xijk− X j:(j,i)∈Ak xjik = 0

xijk ≤ uijk(1 − γijk)(1 − ywδijkw)

(5.8) where γijk : γ ∈ {0, 1}, P (i,j)∈Ak rijkγijk+ P w

rwyw ≤ R. If all extreme points of the inner

maximization are enumerated, and the solution with the minimum value subject to

γ ∈ Γ and w ∈ W is selected, then the model becomes

min γ∈Γ,w∈Wmaxxl_∈Xx l tsk− X (i,j)∈Ak xl_ijkγijk− X w X (i,j)∈Ak xl_ijkywδijkw _(5.9)

Alternatively, this can now be written as the following subproblem:

maxX k xtsk s.t. X j:(i,j)∈Ak xijk− X j:(j,i)∈Ak xjik = 0

xijk≤ uijk(1 − ˆγijk)(1 − ˆywδijkw)

and the following master problem: min γ∈Γ z s.t. z ≥ xl_tsk− X (i,j)∈Ak xl_ijkγijk− X w X (i,j)∈Ak xl_ijkywδijkw γijk, w ∈ {0, 1} (5.11)

The subproblem is just the summation across pure maximum flow problems,

with fixed interdiction and commonality selection. Therefore, the subproblems can be solved as relaxed linear programs (but will still have integer solutions). The

solution to the subproblem provides a lower bound to the optimal solution to formu- lation (5.7). The master problem is simply the attackers problem with “fixed” flows

through the network. Therefore, it provides an upper bound to the optimal solution of formulation (5.7). As with the other cases discussed, a Benders’ partitioning al-

gorithm would iterate between these upper and lower bounds until they converged to this optimal solution.