## Charles Johnson and Dr. Sean Warnick, Computer Science Department

### Introduction

In complicated interconnected systems the consequences resulting from shocks (the effects resulting from causes) are difficult to anticipate. 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 mathematical relations. For systems that may be modeled linearly (a common modeling assumption) this balance may be struck by building a network (a system that you could graphically represent with something that looks like a brainstorm) and placing linear systems (more mathematical models) on the links of the graph. Measurable elements of interest in a system (which we refer to as observables) become nodes (circles in the brainstorm diagram) and the interrelationships of these observable objects are compacted into the link weights (the lines on the brainstorm diagram). We represent these graphs (brainstorm like diagrams) with an adjacency matrix with rational complex valued functions as its entries (a separate mathematical model). This model is referred to as the Dynamical Structure Function (DSF). At its inception the DSF was used to reconstruct dynamical networks, especially biological networks. DSFs have a certain structural ambiguity.

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 has defined a measure for vulnerability of such systems. My work provides theory to complement some of the network security and analysis attributes of the DSF. I compare two notions of relationship between nodes on DSFs: 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 (an important mathematical result) to develop destabilizing attacks on these networks. 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. My work demonstrates some cases when these distinct notions are equivalent.

### Methodology

My approach was mathematical in nature. I carefully defined net path and net effect. I then used these definitions and other accepted mathematical facts to prove theorems. From these results I proved corollaries.

### Results

I proved two theorems and two corollaries that express how individual net effects may be computed in terms of net paths.

### Discussion

The net effect is a more complete notion of cause and effect in the mathematical modes which we explore. However, computing it is very difficult. On the other hand, the net path is simpler to compute but it currently has fewer applications as it is a less complete notion of effect in networks. These results need further refinement to change the order of magnitude of these computations, however, connecting these notions is a contribution to the budding DSF theory.

### Conclusion

The hidden systems from one observable to another may be represented by a single transfer function (a basic mathematical model). I leveraged this result and its flexibility in representing systems. My project aimed to clearly present and motivate the net path and net effect in linear dynamical networks. I hope that these conditions will facilitate an understanding of how variables of interest affect each other in these networks. I repeat the overview from my 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.