Proof. By construction, the first part of the algorithm produces the network with the most assortative structure possible, and therefore the network with the smallest crosstalk sensitivity. If this network is not connected, we need to carry out a rewiring that will have minimum impact to the overall sensitivity. It suffices to prove that the algorithm works when the graph has two components, since this procedure can easily be generalized for an arbitrary number of components. Suppose that the two compo- nentsFandShave degree sequencedF = [d1, d2, . . . , dc] anddS = [dc+1, dc+2, . . . , dN], respectively. In order to make the graph connected, we need to apply the transforma- tion T2 of Figure 4.9, to rewire one edge in F and one edge inS so that they connect vertices in the different components. The difference of the overall crosstalk sensitivity is given in equation (4.98), where the four vertices have degrees
dc−1 ≤dc≤dc+1 ≤dc+2. (4.103) The difference will be
∆c= (f(dc+1)−f(dc))(f(dc+2)−f(dc−1)), (4.104) which is the smallest positive difference while connecting the two components. Re- peating this procedure for all the components of the graph, we end up with a connected graph, which has the minimum additional overall sensitivity to crosstalk.
affinities. We showed that in this case, crosstalk sensitivity only depends on the vertex degree distribution, regardless of the exact interconnection topology of the network.
The sensitivity to geometric crosstalk depends on the network topology, with the assortative network structure being the one that minimizes crosstalk, where vertices with large degrees form cliques with other vertices of large degrees. Finally, we have presented an algorithm which constructs a connected network that minimizes crosstalk when there are constraints regarding its degree distribution. The methods and algorithms described here can easily be generalized to take into account the spe- cific details of the network function, by assigning weights to each crosstalk interaction, depending on how much it affects the system’s normal operation.
Chapter 5
Noise Propagation in Biological and Chemical Reaction Networks
In this chapter, we describe how noise propagates through a network. Using stochas- tic calculus and dynamical systems theory, we study the network topologies that accentuate or alleviate the effect of random variance in the network for both directed and undirected graphs. Given a linear tree network, the variance in the output is a convex function of the poles of the individual nodes. Cycles create correlations which in turn increase the variance in the output. Feedforward and feedback have a limited effect on noise propagation when the respective cycles are sufficiently long.
Crosstalk between the elements of different pathways helps reduce the output noise, but makes the network slower. Next, we study the differences between disturbances in the inputs and disturbances in the network parameters, and how they propagate to the outputs. Separating internal from external dynamics is important in order to understand fluctuations in various systems [4]. We find that these fluctuations have very different behaviors, and as a result, it is easy to analyze their origin in networked systems. Finally, we show how noise can affect the steady state of the system in chemical reaction networks with reactions of two or more reactants, each of which may be affected by independent or correlated sources.
5.1 Introduction and Overview
Noise is ubiquitous in nature, and virtually all signals carry some amount of random noise. In addition, even the simplest systems can be represented as a set of smaller entities interconnected with each other. There have been numerous studies on how noise affects specific functions (e.g. [54, 59] and references therein), but few of them have looked at how noise propagates in general networks, or the impact of network structure on the robustness of each system to noise. Although there is evidence that noise may degrade system performance, it is sometimes necessary for specific functions [23]. Robustness to noise and disturbances in general is something that biological and engineered systems depend upon for their proper function [39, 60]. It becomes even more important if we take into account that molecule concentrations in the cell might be carrying more than one signal through multiplexing [21, 61]. In this case, noise may have the potential of disrupting the network function in more than one way.
We present a new method to quantify the noise propagation in a system, and the vulnerability of each of its subsystems. We use results from graph theory and control systems theory to quantify noise propagation in networks, and use them to evaluate various network structures in terms of how well they filter out noise. We study how crosstalk can help suppress noise, when the noise sources are independent or correlated. We show that perturbations that depend on the state of the system (for example, feedback loops that are prone to noise or noisy degradation rates) have a fundamentally different effect on the system output, compared to noise in the inputs. Finally, we study noise propagation in chemical reaction networks where all reactants may introduce noise, and analytically find that noise correlations may affect the expected behavior of such systems.