NET PATH AND NET EFFECT OF LINEAR NETWORKS WITH

DYNAMICS

Charles A. Johnson

A thesis submitted to the faculty ofBrigham Young University and Oak Ridge National Laboratory

in partial fulfillment of the requirements for

Graduation with Honors

Sean Warnick, ChairPaul Jenkins

Robert Bridges

Department of Mathematics

Brigham Young University

ABSTRACT

NET PATH AND NET EFFECT OF LINEAR NETWORKS WITH

DYNAMICS

Charles A. Johnson

Department of Mathematics, BYU

Graduation with Honors

As the world grows increasingly connected, an understanding of network security grows

ever more relevant. We introduce two notions of relationships between nodes in networks:

net path and net effect. Net effect has several security applications, however, net path can be

easier to compute. We characterize and compare net path and net effect on linear dynamical

networks. We specify cases in which the net path may be used to compute the net effect.

Keywords: Linear Dynamical Systems, Networks, Network Security

ACKNOWLEDGMENTS

I would like to thank the three members of my thesis committee for the contributions

they have made for my undergraduate experience.

Thanks to Dr. Robert Bridges for taking time out of his busy schedule to host me at

Oak Ridge National Laboratory and to review both my proposal and finished thesis. Though

the direction of my thesis has changed significantly from its inception, his perspective and my

experience with Oak Ridge have expanded my view of security and applied mathematics and

opened my eyes to the importance of computer science as our world moves into the future.

Thanks to Dr. Paul Jenkins to being patient with many of the unconventional

circumstances surrounding my honors thesis process.

Additionally, I would like to thank Julie Radle and Vika Filimoeatu for their flexibility

and support of my honors experience and for the hours and energy they dedicate to the

honors program. Their support, along with that of Dr. Warnick, Dr. Rosborough and

Dr. Black-Hults has been crucial in shaping my wife’s and my educational goals and plans.

Additional thanks to Dr. Sean Warnick for being my advisor and mentor over the past two

years and sticking with me as I learned the basics of linear systems theory.

Thanks to my wife for her support and for helping to clarify my exposition of state-

space models, transfer functions and dynamical structure functions.

Table of Contents

List of Figures xi

1 Introduction 1

2 Dynamical Structure Function Theory 3

2.1 State-Space Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 The Dynamical Structure Function . . . . . . . . . . . . . . . . . . . . . . . 5

2.3.1 Observing a Subset of the State . . . . . . . . . . . . . . . . . . . . . 5

2.3.2 The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3.3 Computing the DSF . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 Relation to Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4.1 Transfer Functions and the Reachability of DSFs . . . . . . . . . . . 10

3 Net Path 13

3.1 Reducing DSFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Definition and Interpretation of Net Path . . . . . . . . . . . . . . . . . . . . 14

3.3 Example: Multiple State Removal . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3.1 Example: Net Path of Zero . . . . . . . . . . . . . . . . . . . . . . . 19

4 Net Effect 21

4.1 Example: Three Node Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.2 Applications of the Net Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 23

vii

4.2.1 Minimal Effort Denial of Service Attacks in Cyber-Physical Systems . 24

4.2.2 Net Effect and Target Reachability . . . . . . . . . . . . . . . . . . . 25

5 A Comparison of Net Effect and Net Path 27

5.1 Direct Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.2 Networks where (I` −Q)−11` = (I` −Q)−1`` Q1 . . . . . . . . . . . . . . . . . . . 30

5.2.1 Equivalence of the Net Path and the Net Effect . . . . . . . . . . . . 32

5.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.3.1 Three Node Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.3.2 Three Node Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.3.3 Three Node Chains Leading into Cycle . . . . . . . . . . . . . . . . . 35

6 Conclusion 39

References 41

viii

x

List of Figures

3.1 A DSF, (Q,P ), with four observables. . . . . . . . . . . . . . . . . . . . . . . 14

3.2 (Q,P ) reduced to three observables. . . . . . . . . . . . . . . . . . . . . . . . 15

3.3 (Q,P ) reduced to two observables. . . . . . . . . . . . . . . . . . . . . . . . 15

3.4 A fully connected network. Blue arrows indicate non-zero entries of Q and

black arrows represent non-zero entries of P . . . . . . . . . . . . . . . . . . . 19

3.5 A network after the removal of a single observable. Blue arrows indicate

non-zero entries of Q and black arrows represent non-zero entries of P . We see

where signals cancel each other in a net path relating y1 and y3. Additionally

the effect of U on y3 is nullified. . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.1 The three node chain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.2 The net effect of Y1 on Y3 in the three node chain. . . . . . . . . . . . . . . . 23

5.1 The net path from Y1 to Y3 in the three node chain. . . . . . . . . . . . . . . 33

5.2 The three node cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.3 Two chains from one node leading into a two node cycle. . . . . . . . . . . . 36

xi

xii

Chapter 1

Introduction

In complicated interconnected systems the consequences resulting from shocks (the

effects resulting from causes) are difficult to anticipate [15]. Yet, the vast interconnection of

digital age technology with critical infrastructure systems and other elements of the physical

world exposes human convenience, property and safety to distant shocks. Understanding

models of interconnected and interrelated systems is more relevant than ever.

Modelers of complicated interconnected systems must strike a careful balance between

including relevant system features and compacting non-essential features into simple math-

ematical relations. For systems that may be modeled linearly this balance may be struck

by building a network (a graph) of the system and placing linear systems on the links of

the graph. Measurable elements of interest in a system (which we refer to as observables)

become nodes and the interrelationships of these observable objects are compacted into the

link weights. We represent these graphs with an adjacency matrix with rational complex

valued functions as its entries. This model is referred to as the Dynamical Structure Function

(DSF).

At its inception the DSF was used to reconstruct dynamical networks [8, 11, 14],

especially biological networks [9, 13]. Much like transfer functions, DSFs have a certain

structural ambiguity which is addressed by Yuan et al. [21], Yeung et al. [20], Goncalves et

al. [10] and again by Yuan et al. [23]. The network reconstructing techniques were expanded

upon, improved and sped up [1, 3, 5, 6, 16, 22].

1

From dynamical structure functions one may answer several network security questions

tied to the structure of the system itself. Where could attackers cause the most harm? What

could an attacker with access to one section of the system do to a target on the other side of

the system?

Previous work by Rai et al. [17], Chetty et al. [4] and Woodbury [19] has defined

a measure for vulnerability of such systems. This paper provides theory to complement

some of the network security and analysis attributes of the DSF. We compare two notions

of relationship between nodes on linear dynamical networks: net path and net effect. The

former has a special role in understanding how one can build equivalent but simpler (having

fewer nodes) networks to represent the same system. The latter has been used in conjunction

with the Small Gain Theorem to develop destabilizing attacks on these networks. In one

sense we may think of the net path as being an open-loop measurement of effect in the

linear dynamical network. Observables are checkpoints the limit measurement of effect when

calculating the net path. On the other hand the net effect considers the closed-loop system

between observables in which feedback through observables is taken in to account. We

demonstrate when these distinct notions are equivalent.

In chapter (2) we lay a mathematical foundation for modeling interconnected linear

systems, focusing especially on the dynamical structure function (DSF). In chapter (3) we

define and explore the net path. In chapter (4) we likewise explore the net effect. In chapter

(5) we present our results on the relationship between these two concepts. In chapter (5.3)

we illustrate these results with three examples. In chapter (6) we draw our conclusions.

2

Chapter 2

Dynamical Structure Function Theory

In this section we will introduce several different methods for modeling dynamical

systems. For examples and a formal presentation of linear, time-invariant state-space models

and transfer functions see Hespanha [12].

2.1 State-Space Model

We model a linear time invariant system in the state space by a set of linear differential

equations and a second set of linear equations governing the outputs of the system. The

systems that we consider are linear and time invariant. Thus we consider the general

state-space system below:

x(t) = Ax(t) +Bu(t)

y(t) = Cx(t)

where:

1. A ∈ Rn×n, is the state matrix,

2. B ∈ Rn×m, is the input matrix,

3. C ∈ R`×n, is the output matrix,

4. x(t) ∈ Rn, is the state vector,

5. y(t) ∈ R`, is the vector of observables (also known as outputs or manifest variables),

6. u(t) ∈ Rm, is the vector of inputs,

3

7. and n,m, ` ∈ N with n ≥ m, `.

For clarity’s sake we note that n,m, ` ∈ N with n ≥ m, `. We further note that x(t) denotes

the time derivative of x at time t.

2.2 Transfer Functions

To capture the relationship between u(t) and y(t) in the LTI-system above we may solve

for y(t) in terms of u(t). This results in a loss of information about the structure of the

system above; for example we lose the interconnection (graph) structure of the n elements of

the vector x(t). To start, we make a change of basis on this system by taking the Laplace

transform of the equations above yielding the following system:

sX(s)− x(0) = AX(s) +BU(s)

Y (s) = CX(s).

(2.1)

In this form we may easily solve for Y (s) in terms of U(s) by eliminating the variable X(s).

Assume the initial conditions of the system to be zero so:

X(s) = (sIn − A)−1BU(s) (2.2)

.

Thus:

Y (s) = CX(s)

= C(sIn − A)−1BU(s).

(2.3)

We define G(s) := C(sIn − A)−1B to be the transfer function from U(s) to Y (s). If

we then return the system to the original set of variables in the time domain we have that

y(t) = L−1(G) ∗ u(t)

4

, where ∗ denotes the continuous convolution operator.

2.3 The Dynamical Structure Function

When we consider a state-space model of a linear time invariant system, we may think of

what we observe (the system output y) in two different ways. We may either observe a subset

of the state vector directly or we may observe linear combinations of the states. Observing

a subset of the state vector directly corresponds (up to a reordering of the state vector,

x(t)) to a system where C =

[I` 0`×n−`

]. In other words, up to a reordering of the state

vector, y =

[x1(t) x2(t) . . . x`(t)

]T. When y(t) may be linear combinations of a set of

states, we call the values that we observe (the output of the system) by the more general

term observables. When our observables are linear combinations of the states C is not even

required to be full row rank (we may be measuring redundant information). In that case our

strategy is to rewrite the state-space system in terms of the observables so that it reduces

to the case where C =

[Ik 0k×n−k

], where k is the rank of C. In other words we apply a

transformation to the system to acquire a new system so that the new state vector (x) and

observables (y) satisfy y =

[x1(t) x2(t) . . . xk(t)

]T.

2.3.1 Observing a Subset of the State

In the first case (y =

[x1(t) x2(t) . . . x`(t)

]T), we may solve for the value of the observed

(manifest) states in terms of themselves and of the inputs into the system. By doing so

we preserve information such as the graphical relationship between these manifest states.

However, the hidden states are suppressed in this mapping and their influence is modeled

by a number of single-input-single-output (1× 1) transfer functions (rational functions over

C). We solve for this by initially partitioning the system in terms of manifest states, y, and

hidden states, xh(t) =

[x`+1(t) x`+2(t) . . . xn(t)

]. The states in xh are hidden in the sense

that, from the perspective of the transfer function, the role that xh plays in the relationship

between inputs u and outputs y is totally ambiguous. In other words many different network

5

topologies and number of states of xh could explain the relationship between the manifest

states.

We reorder and partition the states as follows:

y(t)

xh(t)

=

A11 A12

A21 A22

y(t)

xh(t)

+

B1

B2

[u(t)

]. (2.4)

Where:

1. A11 ∈ R`×`,

2. A12 ∈ R`×n−`,

3. A21 ∈ Rn−`×`,

4. A22 ∈ Rn−`×n−`,

5. B1 ∈ R`×m,

6. and B2 ∈ Rn−`×m.

Having defined this representation of the system, we now interpret the case where

C 6=[I 0

].

2.3.2 The General Case

When C 6=[I` 0`×n−`

]we still can rewrite the system in terms of the observables. Doing so

creates an equivalent state-space model of the system built on a composite state vector, x.

There are shared hidden states within a subset of that state. As explained by Chetty and

Warnick [2] such a transformation may be obtained as follows.

First, let k be the rank of C. One then begins by reordering the composite states of

the system so that C =

[C1 C2

]T, where C1 ∈ Rk×n and C2 ∈ R`−k×n so that each row of

C1 is linearly independent and each row of C2 is linearly dependent on rows of C1. We apply

the transformation T to the system:

6

x = Ax+Bu,

y = Cx,

(2.5)

to obtain:˙x = TAT−1x+ TBu,

y = CT−1x,

(2.6)

where T =

[CT

1 E1

]T, so that E1 ∈ Rn×n−k is a basis for the nullspace of C1. This means

that T−1 =

[CT

1 (C1CT1 )−1 E1

]. C2 is in the rowspace of C1 and thus is canceled by the

nullspace of C1:

CT−1 =

C1

C2

[CT1 (C1C

T1 )−1 E1

]=

I 0

C2CT1 (C1C

T1 )−1 0.

We note that only the first m states of our transformed system are measured. At this

point we choose only to measure the observables which we directly observe, as measuring

additional linear combination of these observables would be redundant. Thus we simplify to

have C =

[I 0

]. At this point we may solve for the observables and their interrelationship

and structure in terms of the (now modified) inputs to the system. The technique is identical

to the case above when we observed a subset of the state vector directly. However, in this

case we do not observe the relationship between manifest states of the system but instead

the relationship between observables of the system. Thus we have:

˙y(t)

˙xh(t)

=

C1ACT1 (C1C

T1 )−1 C1AE1

ET1 AC

T1 (C1C

T1 )−1 ET

1 AE1

y(t)

xh(t)

+

CT1 B

E1B

[u(t)

](2.7)

where we use xh ∈ Rn−k to distinguish the redundant observables that will not be manifest

in the final system representation.

7

2.3.3 Computing the DSF

Once our system is in a form of either Equation (2.8) or (2.7) we proceed to compute the

relationship between the observables and the inputs. Without loss of generality we assume

the notation of Equation (2.8). So our system is of the form:

y(t)

xh(t)

=

A11 A12

A21 A22

y(t)

xh(t)

+

B1

B2

[u(t)

],

y(t) =

[I 0

] y(t)

xh(t)

.(2.8)

Much as we did in the previous section, we perform a change of basis on the vector

x(t) =

[y(t) xh(t)

]Tby taking the Laplace transform of the system. We then assume zero

initial conditions and solve out the hidden states, xh(t). We find first that:

sY (s) = A11Y (s) + A12Xh(s) +B1U(s), (2.9)

and

Xh(s) = (sIn−` − A22)−1A21Y (s) + (sIn−` − A22)

−1B2U(s). (2.10)

Substituting (2.10) into (2.9) we have that:

sY (s) = A11Y (s) + A12((sIn−` − A22)−1A21Y (s) + (sIn−` − A22)

−1B2U(s)) +B1U(s)

= (A11 + A12(sIn−` − A22)−1A21)Y (s) + (A12(sIn−` − A22)

−1B2 +B1)U(s).

(2.11)

We define this as follows:

sY (s) = W (s)Y (s) + V (s)U(s)

8

We then subtract Diag(W (s))Y (s) from both sides of the equation and then multiply by

(sI`−Diag(W (s)))−1. (sI`−Diag(W (s)))−1 is invertible as it is both diagonal and the entries

of W (s) are proper and so they do not equal s. So we have:

Y (s) = (sI` −Diag(W (s)))−1(W (s)−Diag(W (s)))Y (s) + (sI` −Diag(W (s)))−1V (s)U(s)

(2.12)

We rewrite (2.12) by performing the substitutions

Q(s) = (sI` −Diag(W (s)))−1(W (s)−Diag(W (s)))

and

P (s) = (sI` −Diag(W (s)))−1V (s)

resulting in the relation:

Y (s) = Q(s)Y (s) + P (s)U(s), (2.13)

where Q(s) is `× ` and P (s) is `×m.

Definition 1. We define the pair, (Q(s), P (s)), to be the dynamical structure function,

or the DSF.

2.4 Relation to Transfer Function

We note that to solve for the observables, Y (s), in terms of the system inputs, U(s), one

simply subtracts Q(s)Y (s) from both sides of the expression and then left multiplies by

(I` −Q(s))−1. This yields

Y (s) = (I` −Q(s))−1P (s)U(s).

9

This relation is equal to the system’s transfer function. We again emphasize that within the

transfer function the interactions between observables are lost. We only have a description of

input-output behavior. In other words, (I` −Q(s))−1P (s) = G(s), the transfer function, for

our system.

Observe that, in the case where P (s) = I`, exactly a single input directly applies to

every manifest variable. Assuming that our original system directly measured a subset of the

system state, we may think that any input into the system is equivalent to directly changing

a corresponding observable. In this paper we consider this a special case of the DSF.

Definition 2. A DSF with direct inputs is a dynamical structure function where P (s) =

I`.

If the original system measured linear combinations of state, then a DSF with direct

inputs only occurs when TB =

[I 0

]. We would interpret this occurrence as a special

alignment of the inputs directly in terms of a linearly independent subset of our observables.

It would likely only occur by design or definition of the inputs to directly influence the

outputs. In such a special case, the nullspace of C equals the left nullspace of B and the

transpose of the linearly independent rows of C are the left inverse of B. In any case, when

P (s) = I` we have that:

G(s) = (I` −Q(s))−1 =

[I` 0

](sI` − A)−1

I`0

.2.4.1 Transfer Functions and the Reachability of DSFs

A natural question to ask of a DSF given by (Q,P ) is: “Given my P and my Q can I drive

Y , the vector of observables, to any desired value using U , the vector of inputs?” We make

an elementary observation that, if for all s ∈ C, G(s) = (I` −Q)−1P is full row rank then

the system of equations represented in matrix form as: Y (s) = G(s)U(s) will have a solution.

Thus, Y may be driven to any set of values by choices of U .

10

Specifically, given the system

Y (s) = Q(s)Y (s) + P (s)U(s),

we have that

Y (s) = (I` −Q(s))−1P (s)U(s) = G(s)U(s).

So, if Y (s) = [Y1(s), Y2(s), . . . , Y`(s)]T and U(s) = [U1(s), U2(s), . . . , Um(s)]T , the

network is reachable if the system of equations

Y1(s) =m∑i=1

G1iUi(s)

Y2(s) =m∑i=1

G2iUi(s)

...

Y`(s) =m∑i=1

G`iUi(s)

(2.14)

has a solution for all s ∈ C.

Thus, given a DSF, we may glean the reachability of the observables in the network

by calculating and checking the rank of the transfer function G.

This is relevant to questions regarding network security. Assume that an attacker

may influence the system via U(s). Then, a reachable system implies that the attacker has

enough influence over the network of observables to drive them to any desired combination of

values. Full reachability implies ability to hijack the network.

11

12

Chapter 3

Net Path

As a preliminary to defining and interpreting the net path, we briefly discuss reducing

DSFs. This will come into play as much of this paper focuses on the net paths in a reduced

DSF.

3.1 Reducing DSFs

Given the DSF Y (s) = Q(s)Y (s) + P (s)U(s), we may remove an observable from the system

to acquire an equivalent (in terms of input-output modeling) system model. We do so by

partitioning Y (s), U(s), Q(s) and P (s) as follows:

Y1Y ′

=

0 T12

B21 H

Y1Y ′

+

P1

P2

[U]

We then solve out the observable, Y1:

Y1 = T12Y′ + P1U

Y ′ = B21Y1 +HY ′ + P2U

= B21(T12Y′ + P1U) +HY ′ + P2U

= (B21T12 +H)Y ′ + (B21P1 + P2)U

(3.1)

From here, to get a representation of this system as a DSF we need to ensure that

the transfer function matrix on Y ′ is hollow. To this end we subtract the diagonal entries of

13

Figure 3.1: A DSF, (Q,P ), with four observables.

B21T12 +H diag(B21T12 +H) from both sides and then apply (I`−1 − diag(B21T12 +H))−1

to obtain

Y ′ =(I`−1 − diag(B21T12 +H))−1(B21T12 +H − diag(B21T12 +H))Y ′+

(I`−1 − diag(B21T12 +H))−1(B21P1 + P2)U.

(3.2)

We see that this forms a new pair of transfer function matrices which act as a new

DSF:

Y ′(s) = Q(s)Y ′(s) + P (s)U(s).

To demonstrate this reducing process we visually reduce a DSF with four observables

down to a DSF with two observables in Figure (3.1), Figure (3.2) and Figure (3.3). In this

case there are no cancellations in net paths and the reduction proceeds as expected (contrast

this with the example illustrated in Figure (3.4) and Figure (3.5)).

3.2 Definition and Interpretation of Net Path

We begin by defining net path for a general DSF. We discuss this definition and interpret the

net path when observables are removed from a DSF. We then apply a DSF reduction technique

as in Section (3.1) to consider the definition of net path on a DSF with two observables.

After this section, we will focus exclusively on the net path on these reduced DSFs. We begin

14

Figure 3.2: (Q,P ) reduced to three observables.

Figure 3.3: (Q,P ) reduced to two observables.

15

with a preliminary definition of net path, which we explore and then overwrite at the end of

the section.

Given a set of observables, Y (s), the net path between two observables, Yi and Yj is

the transfer function on the link Qji, the jith entry of Q.

This definition suggests an interpretation of the Q matrix in a DSF as a collection

of net paths from observables to observables. We interpret the net path when the DSF

represents observables which are a subset of the state and when it represents observables

which are linear combinations of the state.

When our observables are a subset of the system state (when C =

[I` 0

]) we have

that measuring fewer observables corresponds to removing a row of C. Thus the system’s

internal dynamics do not change; however, the DSF representation gains an additional

(possibly shared) hidden variable.

For example, given Q, we may reduce Q to Q by solving out an observable, say Y1.

Every entry in Q that is different from its corresponding entry in Q2:,2: (the ` − 1 × ` − 1

lower right block of Q) after removing Y1 is a link representing a net path that shares the

hidden observable Y1.

Similarly, when the system observables are linear combinations of the state, we have

that such links in Q share the hidden observable Y1. In this case the observables themselves

are composed of multiple states, some of which are shared, and so an interpretation of the

shared hidden states in reduced order equivalent DSFs is much more murky and case specific.

However, it is important to recognize that system states will often be shared between links

and observables in the DSF.

In the example that follows we remove multiple observables to calculate a net path

between two system observables in the absence of all other observables.

Definition 3. The net path from Yi to Yj is the link Qji where all observables except for

Yi and Yj have been removed from Q.

16

3.3 Example: Multiple State Removal

As a specific example, if we wished to extract the net path from Y1 to Y`, we could preform

the above simplification all at once. This process will yield both the SISO transfer function

from Y` to Y1 and from Y1 to Y`, where ` is the number of observables in our DSF. First we

rewrite the system above to isolate Y` and Y1Y1

Y ′

Y`

=

0 T12 t13

B21 H T23

b31 B32 0

Y1

Y ′

Y`

+

P1

P2

P3

[U

]

where

1. T12 ∈ R1×`−2,

2. t13 ∈ R,

3. T23 ∈ R`−2×1,

4. B21 ∈ R`−2×1,

5. b31 ∈ R,

6. B32 ∈ R1×`−2,

7. H ∈ R`−2×`−2,

8. P1 ∈ R1×m,

9. P2 ∈ R`−2×m,

10. and P3 ∈ R1×m.

We then simplify as follows:

Y1 = T12Y′ + t13Y` + P1U

Y ′ = B21Y1 +HY ′ + T23Y` + P2U

= (I`−2 −H)−1B21Y1 + (I`−2 −H)−1T23Y` + (I`−2 −H)−1P2U

(3.3)

17

We note here that the hollow matrix (I`−2−H) is invertible, as H, like Q from which it

was taken, is a strictly proper transfer function matrix. This is an implicit consideration given

by Goncalves and Warnick [11] (note their choice to solve for sY by subtracting diagonal

elements of W (s)).

Thus we have that:

Y1 = T12((I`−2 −H)−1B21Y1 + (I`−2 −H)−1T23Y` + (I`−2 −H)−1P2U) + t13Y` + P1U

= T12(I`−2 −H)−1B21Y1 + (T12(I`−2 −H)−1T23 + t13)Y` + (T12(I`−2 −H)−1P2 + P1)U

= (1− T12(I`−2 −H)−1B21)−1(T12(I`−2 −H)−1T23 + t13)Y`

+ (1− T12(I`−2 −H)−1B21)−1(T12(I`−2 −H)−1P2 + P1)U

(3.4)

The complementary transfer function from Y1 to Y` would be:

Y` = (1−B32(I`−2 −H)−1T23)−1(B32(I`−2 −H)−1B21 + b31)Y1

+ (1−B32(I`−2 −H)−1T23)−1(B32(I`−2 −H)−1P2 + P3)U

(3.5)

We then have the relation:

Y1Y`

=

0 Q1

Q2 0

Y1Y`

+

P1

P2

[U] (3.6)

where:

1. Q1 = (1− T12(I`−2 −H)−1B21)−1(T12(I`−2 −H)−1T23 + t13),

2. Q2 = (1−B32(I`−2 −H)−1T23)−1(B32(I`−2 −H)−1B21 + b31),

3. P1 = (1− T12(I`−2 −H)−1B21)−1(T12(I`−2 −H)−1P2 + P1),

4. and P2 = (1−B32(I`−2 −H)−1T23)−1(B32(I`−2 −H)−1P2 + P3).

This representation of the system highlights the path between these two variables,

while excluding the effect each variable has on itself. In other words, the effect of Y1 on

18

Figure 3.4: A fully connected network. Blue arrows indicate non-zero entries of Q and blackarrows represent non-zero entries of P .

itself is abstracted into the reciprocal effect of Y1 on Y` and Y` on Y1, but it is not directly

represented here. This would be useful to determine if there is any direct effect from a source

variable Y1 to a target Y`. When this value (the net path) is zero, Y` may not be influenced

by Y1. Looking at representations of the system which include more than two observables

may not yield this same insight. We give an example of this below.

3.3.1 Example: Net Path of Zero

We consider an example of a fully-connected network where signal relations are such that

they cancel each other out. To see the original representation of this system, refer to Figure

(3.4).

Q =

0 1

s+21

s2+3s+2

−1s−1 0 1

s−1

1s2−1

1s+1

0

and P =

s+2s2−1

−1s

1s2+s

To explore the relationship between the observables Y1 and Y3, we remove Y2 from the

system using the technique above. This results in the following two state DSF:

19

Figure 3.5: A network after the removal of a single observable. Blue arrows indicate non-zeroentries of Q and black arrows represent non-zero entries of P . We see where signals canceleach other in a net path relating y1 and y3. Additionally the effect of U on y3 is nullified.

Y1Y3

=

0 Q1

0 0

Y1Y3

+

P1

0

U,where Q1 = 2

s2−s−1 and P1 = − 1s3

.

To see this representation, refer to Figure (3.5).

Though this system is fully connected with an input that affects all of the states,

we notice by removing the middle observables that there is no net path from Y1 to Y3.

Furthermore, the input U never reaches the observable Y3 as all the signals sent along those

channels cancel each other out.

20

Chapter 4

Net Effect

Recall from Section 2.2 that the transfer function of a system gives a direct relationship

(with no structural detail) between a selection of system inputs and a selection of system

outputs. Thus, given a DSF of the standard form demonstrated in Equation (2.13), the

transfer function description of the system,

Y (s) = (I −Q(s))−1P (s)U(s),

contains all of the input output relationships of the system where U(s) is the input and Y (s)

is the output.

To understand the net effect of one observable on another, say the ith entry of Y on

the jth, we consider the effect of changing Yi on the value of Yj . We do so by formulating our

system so that we can directly influence Yi and directly (and solely) measure Yj. We do so

by artificially creating a new DSF with direct inputs: (Q, I). We then only send inputs to Yi.

To do this, we set U(s) to be a complex valued rational function times the ith component

projection vector (a vector of all zeros with a one in the ith location). We then observe that

the jth component of Y evolves according to the relation: Y (s)j = (I −Q(s))−1ji U(s)i.

So, if one were to measure the jth observable while sending direct input into the

ith, one would have that the input-output relationship characterized here would be the one

dimensional (single input single output) transfer function (I −Q(s))−1ji .

Definition 4. The SISO transfer function (I −Q(s))−1ji is the net effect of the observable Yi

on Yj.

21

Figure 4.1: The three node chain.

Net effect is a more complete notion of relationships between observables than that of

the net path since it captures the effect of the feedback of Yi with itself on Yj.

This generalizes to the jith entry of the transfer function which is not constrained by

the assumption that we may directly input to a single state. Thinking in these broader terms

of input is not the objective of the net effect. Net effect focuses on the relationship between

the observables themselves, not the inputs and the observables. Here, we use direct inputs to

observables to explain this concept in terms of our current system formulation.

4.1 Example: Three Node Chain

For DSF of the form of Equation (2.13), let Q represent the three node chain shown in Figure

(4.1). Thus

Q =

0 0 0

1s+1

0 0

0 1s+2

0

.We can place a direct input on to the first observable, call it U , and read the output

of the third observable, call it Y , and consider the system from that input as seen by Y , see

Figure (4.2).

Mathematically we calculate that:

22

Figure 4.2: The net effect of Y1 on Y3 in the three node chain.

(I −Q)−1 =

1 0 0

−1s+1

1 0

1(s+1)(s+2)

−1s+2

1

And we note that (I − Q)−113 = 1

(s+1)(s+2). So we have that Y (s) = 1

(s+1)(s+2)U(s).

When we remove the observable Y2 we find the equivalent system representation,

Y1Y3

=

0 Q1

Q2 0

Y1Y3

+

P1

P2

[U] ,is such that Q2 = 1

(s+1)(s+2).

In this case we note that the net path from Y1 to Y3 equals the net effect of Y1 on Y3.

This is not true in general. We will demonstrate the relationship between the net path and

the net effect in the presence and absence of self feedback in section (5) and with examples

(5.3.2) and (5.3.3).

4.2 Applications of the Net Effect

We would like to further motivate and clarify the interpretation of the net effect by referring

to an example of its application. We also highlight the usefulness of the net effect when

exploring an alternative notion of reachability in networks.

23

4.2.1 Minimal Effort Denial of Service Attacks in Cyber-Physical Systems

There are several bodies of work which concern themselves with utilizing the net effect of links

that induce feedback on networks to measure the vulnerability of the networks to crippling

destabilization (our exposition of this measure of vulnerability is presented and defined by Rai

et al. [17]). A primary question that motivated my interest in net effect is the interpretation

of this vulnerability score.

According to the Small Gain Theorem: when one changes the link of a system transfer

function (an entry of (I −Q)−1 in our case) that is in feedback with itself through the rest of

the system, the modified system is guaranteed to be stable unless the product of the infinity

norms of the affected link and the change are greater than 1 [7]. In other words, a cleverly

designed systematic perturbation (∆) to a link (Qij) in feedback with itself through the

system has potential to destabilize the system if and only if

||∆||∞||(I −Q)−1ji ||∞ ≥ 1.

When a system is destabilized by such a perturbation, at least some element of the state

vector will be driven to +−∞. In a physical system a state going to infinity corresponds to

some critical failure in the system. Depending on the system, this failure may imply a denial

of services rendered by the normally functional system.

Thus, if an attacker wished to make a minimal energy perturbation to a link, Qij, of

the system with the intention of creating a denial of service through system destabilization,

his perturbation would only need to be of size 1||(I−Q)−1

ji ||∞. So, for example, given two links in

Q, say Q12 and Q34, it is reasonable to assert that Q12 is more vulnerable to a minimal effort

destabilizing perturbation than Q34 if:

1

||(I −Q)−121 ||∞<

1

||(I −Q)−143 ||∞.

24

Thus the infinity-norm of the net effect (or its reciprocal) may be used to measure

and rank links in systems for potential to seed denial of service attacks.

4.2.2 Net Effect and Target Reachability

As we have already seen in section (2.4.1), the row rank of the transfer function of a Dynamical

Network is an effective test for determining the reachability of the nodes of that network given

defined inputs. However, the net effect can be used to verify a weaker form of reachability in

networks.

Definition 5. Given a network (Q, I`), an output signal Yi(t) is target reachable from the

jth input if ∀r ∈ R there exists an input from the signal Uj(t) to drive Yi(t) to r.

In other words, target reachability from node Yi to node Yj is defined as the existence

of a signal starting at a direct input to Yi and ending at Yj. The existence of such a signal

implies that, by sending signals in Yi it is possible to move Yj to take on any desired value.

Target reachability has previously been defined on state-space models in Vosughi et al. [18].

In terms of network security, the target reachability from node Yi to node Yj answers

the questions: “Could an attacker with access to node Yi effectively hijack Yj? How much

effort would the attacker need to make?”

When the net path from observable Yi to Yj is nonzero then node Yi is target reachable

from node Yj. The norm of the net path from Yi to Yj may be interpreted as the effort

required to perform a unit change in Yj from Yi. The definition of effort is norm specific

and may be tailored to the application. For example, a signal H2-norm of that net effect

could be interpreted as average energy required over all signal frequencies to make a unit

change in Yj from Yi. A signal H∞-norm of that net effect could be interpreted as the largest

possible change in Yj that an ideal unit change in Yi could induce. Future work will capture

this notion more rigorously with a theorem of the form:

Conjecture 1. Given network (Q,P ), node j is target reachable from node i if ||(I`−Q)−1ji ||∗ 6=

0. Moreover:

25

1. If ∗ = 2, then ||(I` −Q)−1ji || is the average signal amplification from node i to node j

over all frequencies of sinusoidal signals.

2. If ∗ =∞, then ||(I` −Q)−1ji || is the maximum gain that a signal from node i can gain

before reaching node j.

26

Chapter 5

A Comparison of Net Effect and Net Path

Given the system described by Equation (2.13) we consider both the net path from

Y1 to Y` and its relationship to the net effect of Y1 on Y`. We will describe a set of results

that allow one to calculate individual entries of the net effect in terms of the net path. This

is advantageous as calculating the net effect directly requires the calculation of (I` −Q)−1,

which simultaneously gives all the net effects of the DSF. With these results we can calculate

individual entries of (I` −Q)−1 without preforming matrix inversion.

5.1 Direct Inputs

Given the setup from example (3.3), specifically

Y (s) = Q(s)Y (s) + P (s)U(s)

implies Y1Y`

=

0 Q1

Q2 0

y1Y`

+

P1

P2

U(s)

where:

1. Q1 = (1− T12(I`−2 −H)−1B21)−1(T12(I`−2 −H)−1T23 + t13),

2. Q2 = (1−B32(I`−2 −H)−1T23)−1(B32(I`−2 −H)−1B21 + b31),

3. P1 = (1− T12(I`−2 −H)−1B21)−1(T12(I`−2 −H)−1P2 + P1),

4. and P2 = (1−B32(I`−2 −H)−1T23)−1(B32(I`−2 −H)−1P2 + P3).

27

we have the following theorem.

Theorem 1. Given (Q, In), a DSF with direct inputs, (In −Q(s))−11` = (1− Q1Q2)−1Q1(1−

B32(I`−2 − H)−1T23)−1 = (1 − Q1Q2)

−1Q1P2 where P2 is calculated from the DSF (Q,P )

where

P =

P1

P2

P3

=

0 . . . 0 0

.... . .

......

0 . . . 0 1

=

[0`×`−1 e`

]

where e` is the `th indicator vector.

Proof. We assume direct inputs into each of the observables; thus we may write Equation

(2.13) as

Y (s) = Q(s)Y (s) + IU(s).

When we set all inputs but the `th to be zero we have

Y (s) = Q(s)Y (s) +

0 . . . 0 0

.... . .

......

0 . . . 0 1

U(s).

From here, as in example (3.3), we calculate the DSF relating Y1 to Y`. As before we gain

the relation of Equation (3.6):

Y1Y`

=

0 Q1

Q2 0

Y1Y`

+

P1

P2

U(s)

where:

1. P1 = (1− T12(I`−2 −H)−1B21)−1(T12(I`−2 −H)−1P2 + P1), and

2. P2 = (1−B32(I`−2 −H)−1T23)−1(B32(I`−2 −H)−1P2 + P3).

28

In this case we note that P1 = 0 and P2U(s) = (1−B32(I`−2−H)−1T23)−1[0 0 . . . 0 1

]U(s) =

(1−B32(I`−2 −H)−1T23)−1U`(s). So we have that:

Y1 = Q1Y` = Q1(Q2Y1 + (1−B32(I`−2 −H)−1T23)−1U`(s)

And, thus, solving for Y1 in terms of U`(s) we have that:

Y1 = (1− Q1Q2)−1Q1(1−B32(I`−2 −H)−1T23)

−1U`(s).

We compare this to the relation established in section (4):

Y1(s) = (I` −Q(s))−11` U`(s)

and thus we conclude that (I` −Q(s))−11` = (I − Q1Q2)−1Q1(I −B32(I`−2 −H)−1T23)

−1

Corollary 1. Given a DSF with direct inputs and no self-loop from Y1 to Y1 we have that

(I` −Q(s))−11` = Q1(1−B32(I`−2 −H)−1T23)−1U`(s) = Q1P2 where P2 is calculated from the

DSF (Q,P ) where

P =

[0`×`−1 e`

].

Proof. Since we have assumed that there is no self-loop from Y1 to Y1 we have that Q1Q2 = 0.

Thus the result from Theorem 1 simplifies to (I`−Q)−1`1 = Q1(I−B32(I`−2−H)−1T23)−1.

These results are helpful as a computational tool for calculating the net effect (which

involves a difficult inversion of a matrix of rational polynomials) using a simple abstraction

of a DSF, the net path. The net path’s expression given here includes taking the inverse of a

matrix of rational functions over C, however, it may be found by successively solving out

observables without performing matrix inversion.

29

5.2 Networks where (I` −Q)−11` = (I` −Q)−1`` Q1

The goal of this section is to show network conditions under which the net effect from Y` to

Y1 may be decomposed in to the product of the net effect of Y` with itself and the net path

from Y` to Y1. This extends to the relationship between any distinct observables in a linear

network without loss of generality.

To compare the net path and the net effect we consider the net path from the `th

observable to the first observable (from Y` to Y1) of system (2.13). This is done by removing

all other observables from the DSF as in the example in section (3). We thus have that the

net path from Y` to Y1 is Q1 of Equation (3.6). Explicitly:

Q1 = (1− T12(I`−2 −H)−1B21)−1(T12(I`−2 −H)−1T23 + t13)

We now apply 2 × 2 block matrix inversion to (I` − Q)−1 to find the net effect:

(I` −Q)−11` . We note that, by two applications of the block 2× 2 matrix inversion formula

using the Schur complement, we have:

(I` −Q)−1 =

1 −Q12

−Q21 I`−1 −Q22

−1

=

1 +Q12∆−1Q21 Q12∆

−1

∆−1Q21 ∆−1

where

∆ =

I`−2 − (H +B21T12) −(T23 +B21t13)

−(b31T12 +B32) 1− b31t13

=

∆11 ∆12

∆21 ∆22

so:

∆−1 =

∆−111 + ∆−111 ∆12δ−1∆21∆

−111 −∆−111 ∆12δ

−1

δ−1∆21∆−111 δ−1

30

where

δ = (1− b31t13)− (b31T12 +B32)(I`−2 − (H +B21T12))−1(T23 +B21t13) = ∆22 −∆12∆

−111 ∆21

and thus

δ−1 =1

∆22 −∆12∆−111 ∆21

.

At this point we note that (I` −Q)−1`` = δ−1.

Theorem 2. In the case where T12 = 0 or B21 = 0 we have that (I` − Q)−11` = δ−1Q1 =

(I` −Q)−1`` Q1.

Proof. We note that

(I` −Q)−11` = (Q12∆−1)1,`−1 =

[T12 t13

]−∆−111 ∆12δ−1

δ−1

= (T12(I`−2 − (H +B21T12))

−1(T23 +B21t13) + t13)δ−1

(5.1)

If T12 = 0 then we have that (I` −Q)−11` = t13δ−1 and we also have that Q1 = t13. On

the other hand, if B21 = 0 then we have that (I` −Q)−11` = (T12(I`−2 −H)−1T23 + t13)δ−1 and

we also have that Q1 = T12(I`−2 −H)−1T23 + t13.

When Y` is not in a feedback loop, this theorem simplifies as follows:

Corollary 2. If T12 = 0 or B21 = 0 and Y` is not in a feedback loop we have that (I`−Q)−11` =

Q1. The net effect of Y` on Y1 equals the net path from Y` to Y1.

Proof. Given our assumption that Y` is not in a feedback loop (thinking of Q as an adjacency

matrix for the observables) we know that, for all k ∈ N, [Qk]`` = 0 = [−Qk]``. Thus,

[(I` −Q)k]`` = 1 for all k ∈ N as adding the identity to −Q will only create self-links of size

one on hollow Q. As this is the case [(I`−Q)−1]`` = 1, since there is only one path from n to

n and it is of size one, thus the inverse path is also of size one from n to n.

31

Theorem 2 describes network conditions under which the net effect from Y` to Y1 is

simply the product of the net effect of Y` with itself and the net path from Y` to Y1. Corollary

2 describes conditions which imply that the net effect of Y` on Y1 equals the net path from Y`

to Y1.

These results explain the equivalence of the upcoming examples (4.1) and (5.3.2). In

each of those two examples we may permute Q to swap the locations of Y1 and Y3 (to consider

the path from Y1 to Y3) and we note that the B21 entry is zero.

However, when we permute Q to change the location of the first and third observables

in example (5.3.3) we note that neither T12 = 0 nor B21 = 0. However, in that case the

identity (I`−Q)−11` = (I`−Q)−1`` Q1 holds; this demonstrates that the above condition (T12 = 0

or B21 = 0), while sufficient, is not necessary.

5.2.1 Equivalence of the Net Path and the Net Effect

Before we follow with our illustrative examples, we propose a necessary and sufficient condition

for the net path to equal the net effect. However, this is not a condition on Q itself as is

Theorem 2, but rather it is a condition on the matrix (I −Q)−1. This perhaps would limit

this result to theoretical applications.

Conjecture 2. The net path from Y` to Y1 equals the net effect of Y` on Y1 if and only if

the net effect of Y1 on Y` is zero, in other words (I` −Q)−1`1 = 0.

This conjecture would follow from the net path being a measure of the total effect of

one observable on another which passes through no other manifest variables. In other words,

observables are checkpoints where we stop measuring effect when calculating the net path.

When the net effect of Y1 on Y` is zero, then there is no path, direct or indirect from Y1 to Y`.

Thus, there would be no manifest variable checkpoints from Y1 to Y`. So, once the system

was cut down to two observables the path from from Y` to Y1 would constitute the entire

effect of Y` on Y1.

32

Figure 5.1: The net path from Y1 to Y3 in the three node chain.

5.3 Examples

For observables Y1, Y2, Y3, we consider a system with the following DSF

Y (s) = Q(s)Y (s) + IU(s),

where Y (s) =

[Y1(s) Y2(s) Y3(s)

]T. In each of the three examples below we use this stock

system and its three observables (Y1, Y2 and Y3) but we allow Q to vary.

5.3.1 Three Node Chain

We take Q to be given by the three node chain in example (4.1). So,

Q =

0 0 0

1s+1

0 0

0 1s+2

0

and

(I −Q)−1 =

1 0 0

−1s+1

1 0

1(s+1)(s+2)

−1s+2

1

33

Figure 5.2: The three node cycle.

Assume we want to consider the relationship between the first and third observables.

To do so we may remove the second observable (Y2), set U(s) = 0 and compute the new DSF,

see Figure (5.1). We make the straightforward substitution:

Y2 =1

s+ 1x1

Y3 =1

s+ 2x2

=1

(s+ 1)(s+ 2)x1 := Q2x1

(5.2)

As before we note that the net path from Y1 to Y3 equals the net effect of Y1 on

Y3. This agrees with Corollary 2 as there is no feedback from Y1 to Y1. So, (I3 − Q)−131 =

1(s+1)(s+2)

= (1) 1(s+1)(s+2)

= (1− 0)−1Q1.

5.3.2 Three Node Cycle

For this example let Q be a three node cycle, see Figure (5.2). Explicitly we have:

Q =

0 0 2

s

−2s+2

0 0

0 1s−2 0

34

As before we study the relationship between the first and third observables. We first

set U(s) = 0 then remove Y2 and consider the resulting system representation:

Y1(s)Y3(s)

=

0 2s

−2s2−4 0

x1(s)x3(s)

=

0 Q1

Q2 0

x1(s)x3(s)

So we have the relation that Y3 = −2

s2−4Y1 = Q2Y1. This is the net path from Y1 to Y3.

Then we compare by placing a direct input on to Y1 (as before call it U(s)) and read

the output of x3 (call it Y (s)). Thus we consider the net effect of Y1 on Y3 or (I3 − Q)−131 .

Firstly:

(I3 −Q)−1 =

s

s+12

(s−2)(s+1)2

s+1

−2s(s+1)(s+2)

ss+1

−4(s+1)(s+2)

−2s(s2−4)(s+1)

2s(s−2)(s+1)

ss+1

So we have an insightful comparison between the net path and the net effect. The

net effect is the product of the net path with the self feedback loop from Y1 to itself. Note:

(I3 −Q)−131 = −2s(s2−4)(s+1)

=(

ss+1

) ( −2s2−4

)= (I3 −Q)−111 Q2.

5.3.3 Three Node Chains Leading into Cycle

For our third example we let Q represent two chains from one node leading into a simple

cycle (see Figure (5.3)) as below:

Q =

0 0 0

1s+1

0 s+2s2−1

1s+2

3s+2

0

We set U(s) = 0 and solve out Y2, obtaining the relation

35

Figure 5.3: Two chains from one node leading into a two node cycle.

Y1Y3

=

0 0

(s+4)(s−1)(s−2)(s+2)2

0

Y1Y3

=

0 0

Q2 0

Y1Y3

.We compare the net path from Y1 to Y3, Q2 = (s+4)(s−1)

(s−2)(s+2)2, to the net effect of Y1 on Y3.

First we note that:

(I3 −Q)−1 =

1 0 0

ss2−4

s2−1s2−4

1s−2

(s+4)(s−1)(s−2)(s+2)2

3(s2−1)(s+2)(s2−4)

s2−1s2−4

So, (I3 −Q)−131 = (s+4)(s−1)

(s−2)(s+2)2= (I3 −Q)−111 Q2 = Q2.

For this example let us also consider both the net path from Y2 to Y3 and the net

effect of Y2 on Y3.

When we set U(s) = 0 and solve out Y1 we obtain:

Y2Y3

=

0 s+2s2−1

3s+2

0

Y2Y3

=

0 Q1

Q2 0

Y2Y3

so our net path from Y2 to Y3 is Q2 = 3

s+2.

36

We note that (I3 − Q)−132 = 3(s2−1)(s+2)(s2−4) =

(s2−1s2−4

) (3

s+2

)= (I3 − Q)−122 Q2. We again

stress, though this fact has been true across these examples, it is not true in the general case.

We have shown in the previous section why it worked out so nicely in the first two cases. We

have shown in this third example that those conditions are sufficient but not necessary.

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Chapter 6

Conclusion

The hidden systems from one observable to another may be represented by a single

transfer function. We have leveraged this result and its flexibility in representing systems. We

hope to have clearly presented and motivated the net path and net effect in linear dynamical

networks. We further hope that these conditions will facilitate an understanding of how

variables of interest affect each other in these networks. We repeat the overview from our

introduction below.

In one sense we may think of the net path as being an open-loop measurement of effect

in the linear dynamical network. Observables are checkpoints the limit measurement of effect

when calculating the net path. On the other hand the net effect considers the closed-loop

system between observables in which feedback through observables is taken in to account.

The relationship between the net path and net effect not only clarifies each other’s respective

roles in thinking about complicated interconnected systems, but also demonstrates when one

can use the more computationally feasible net path to understand the more thorough notion

of net effect.

The goal of these tools is to facilitate and even speed up the computation of entries of

the rational matrix inverse: (I − Q)−1. Through network analysis tools such as these, we

hope to contribute to a greater ability to understand and predict the behavior of complex,

dynamic interconnected systems.

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40

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