Network measures such as the average shortest path length, betweenness centrality, closeness centrality, and eccentricity of the hub-genes and stress response genes present in each GRN are calculated. Fold-change analysis is used to determine whether the hub-genes and stress response genes are upregulated or downregulated in spaceflight microgravity when compared to ground control.

#### 2.6.2. Graphs and Networks

Let $G=\left(V,E\right)$ be a graph, where $V$ is the set of vertices and $E$ is the set of edges. Let ${p}_{k}$ be the degree distribution and $\langle k\rangle $ be the average degree of the graph $G$.

Outdegree: The outdegree distribution of a given node (gene) in the network determines whether the node targets (or directs) other nodes [

28]. We have calculated the outdegree distributions of hub-genes and stress response genes in the GRN to determine whether the hub genes are regulating other genes.

Average shortest path length: The average shortest path in network topology is defined as the average minimum distance that a node can take to reach all other possible nodes (targets) in the network [

29]. The average shortest path length is calculated using Equation (3) [

28]:

Here,

V is the total number of nodes in the network, and

d(

s,

t) is the minimum distance from the node (

s) to the node (

t). ”

n” denotes the total number of nodes. The average shortest path length is meant to be small for the networks to be small world. Here, one can calculate the average shortest path length of all nodes to evaluate the efficiency of a network or determine the individual average shortest path length of each node. We have measured the average shortest path length of each node to determine the efficiency of each hub-gene and stress response gene in the GRN. We define the average shortest path length for any given node

s as:

Betweenness centrality: Betweenness centrality is a measure that shows how frequently a given node appears on the shortest paths of other nodes [

30]. It acts as a bridge for other nodes to be connected by providing the shortest path. The higher betweenness centrality means that the given node appears very frequently in the shortest paths of a greater number of other nodes. Betweenness centrality can be calculated using Equation (5) [

30]:

The total number of shortest paths from the node (s) to node (t) is represented by ${\sigma}_{st}.$ The total number of paths that pass through the node (v) is represented by ${\sigma}_{st}\left(\vartheta \right)$. The GRN’s are directed networks. Therefore, betweenness values are normalized by {1/((n−1)(n$-$2))}, where n is the number of nodes in the network.

Closeness centrality: The average shortest path length of a given node and all other nodes in the network are measured by Closeness Centrality [

31]. The node that has a higher closeness centrality is the most central node that is connected to the maximum number of nodes in the network.

Closeness centrality is calculated using Equation (6) [

31]. Here,

n is the total number of shortest paths going through a given node, y is the given node, and

x is the node that passes through the node

y.

Eccentricity: Eccentricity is a measure of the maximum distance by which a node can be connected to another node [

32]. Eccentricity shows how one node is indirectly connected to other nodes in the network in the path to its target node. A higher eccentricity implies that the node has the greatest influence on the network compared to other nodes.

Clustering Coefficient: Clustering coefficient measures how the adjacent vertices connect to each other. Given a vertex

${v}_{i}$ we define the clustering coefficient as

where

${k}_{i}$ is the degree of vertex

${v}_{i}$ and

${L}_{i}$ is the number of edges between the adjacent vertices of

${v}_{i}$ [

33]. The average clustering coefficient is defined as follows:

The average clustering coefficient can be interpreted as the probability that two adjacent vertices of a randomly selected vertex are connected to each other [

34].

Assortativity coefficient: The assortativity coefficient measures the Pearson correlation coefficient between pairs of adjacent vertices. The assortativity coefficient

$r$ is given by

where

${q}_{k}=\frac{\left|V\right|{p}_{k}}{\langle k\rangle}$ is the distribution of the remaining degree and

${e}_{jk}$ is the probability of finding vertices with degrees

$j$ and

$k$ as the two ends of a randomly selected edge [

35].

HITS algorithm for detecting hubs and authority genes: Our GRN (similar to other complex networks) follow preferential attachment models, which are scale-free with a degree distribution that follows an exponential law. Unlike the random graph model, these networks have nodes with large degrees, called hubs. Classically, with 0–1 (nonconnection-connection) networks, just the degree distribution is used in the identification of such hubs. A much more sophisticated algorithm is proposed by Kleinberg [

36] called the Hypertext induced topic search (HITS) algorithm. Originally it was meant for the networks such as the Internet. We use it now to study our GRN (see also [

37,

38]). Most of these applications use PageRank to reveal localized information about the graph based on some form of external data. We apply this algorithm in our setting for the weighted and directed networks for the transcription factors-target gene networks and co-expression networks.

In the weighted GRN setting, the traditional simplistic method of detecting hub genes would not yield meaningful information. Our approach uses, iteratively the weighted HITS algorithm in a novel way as follows. At the $k$th iteration, let ${\mathit{h}}_{k}$ (respectively ${a}_{k}$) be whose $i$th entry $\mathit{h}\left(i\right)$ (respectively $a\left(i\right)$) be the hub weight (authority weight) assigned to node $i$. One initially assigns uniform distribution on the nodes. Let ${\mathit{h}}_{k}:={\sum}_{u}{\mathit{a}}_{k-1}\left(j\right)$, the sum being over authority nodes $j$ pointed to by $i$. And similarly, the authority node weights are computed by ${\mathit{a}}_{k}\u2254{\sum}_{j}{\mathit{h}}_{k-1}$. Then, we normalize so that the sum of the weights equal to $1$, with normalization factors ${\psi}_{k}$ (respectively ${\varphi}_{k}$). In matrix notation: ${\mathit{h}}_{k}={\psi}_{k}{\varphi}_{k-1}A{A}^{T}{\mathit{a}}_{k-2}$, and ${\mathit{a}}_{k}={\varphi}_{k}{\psi}_{k-1}{A}^{T}A{\mathit{a}}_{k-2}$. These iterations converge to the dominant eigenvector of the real symmetric matrices $A{A}^{T}$ (respectively ${A}^{T}A$). These give us asymptotically hub and authority weights. In this setup, we have assumed entries in A to be 0 or 1. If there are weights on the edges such as correlation or signal strength ${w}_{ij}$, then they are introduced in the sums. Since adjacency is defined for undirected graphs as well, this algorithm will return hubs and authorities weights for such graphs, as well.

We can describe the algorithms as pseudocode as follows, where in place of the 1-norm (sum of the absolute values), we can use any norm (Algorithm 1):

**Algorithm 1** pseudocode |

HITS(A): **#** A :=(The adjacency matrix of the weighted network N = (V,E)) |

$\mathrm{Local}\mathrm{Variables}:n=\left|V\right|;$$e=\left|E\right|$ | |

$\mathit{h}$; | **#**$\mathrm{hub}\mathrm{rank}\mathrm{real}\mathrm{vector}(\mathrm{in}{\mathbb{R}}^{n})$ |

$\mathit{a}$; | **#**$\mathrm{authority}\mathrm{rank}\mathrm{real}\mathrm{vector}(\mathrm{in}{\mathbb{R}}^{n})$’ |

**m**; | **#** the number of HITS iterations. |

${\mathit{h}}_{\mathbf{1}}=\frac{\mathbf{1}}{\mathit{n}};$ | **#** all entries in h are **1** |

${\mathit{a}}_{\mathbf{1}}=\frac{\mathbf{1}}{\mathit{n}};$ | **#** all entries in the vector are 1. |

${\mathit{\psi}}_{\mathbf{1}}=\mathbf{1}$; | |

${\mathit{\varphi}}_{\mathbf{1}}=\mathbf{1}$; | |

**while**$\mathit{k}\le \mathit{m}$**do**; | |

**begin**; | |

${\mathit{h}}_{k}=A{\mathit{a}}_{k-1}$; | |

${\mathit{a}}_{k}={A}^{T}{\mathit{h}}_{k-1};$ | |

${h}_{k}\u2254\raisebox{1ex}{${h}_{k}$}\!\left/ \!\raisebox{-1ex}{$norm\left({h}_{k}\right)$}\right.;$ | # norm(v) = square-root of sum of squares |

${a}_{k}:=\raisebox{1ex}{${a}_{k}$}\!\left/ \!\raisebox{-1ex}{$norm\left({a}_{k}\right)$}\right.$; | |

**end** | |

We have also used different norms in the computations of the normalization factors. The standard norm (square-root of the sum of squares, is proposed by Kleinberg [

36]). If the eigenvalues of

$A{A}^{T}\left(\mathrm{the}\mathrm{same}\mathrm{as}\mathrm{those}\mathrm{of}{A}^{T}A\right)$ are separated (i.e., the multiplicity of the dominant eigenvalues is 1), then the iterations in the converge in the limit to the corresponding dominant eigenvector of

$A{A}^{T}.$ In the matrix notation, this is the famous QR decomposition method or the unsymmetric eigenvalue problem (see 7.3.1 of [

39]). We used a SAGE implementation package [

40].

Since our GRN graph is sparse but highly connected, it converges rapidly with a large number of iterations, yielding hubs and authority genes in this very complex network. These iterations give us asymptotically hub and authority weights. We have implemented a version of this algorithm in the SAGE software and then applied it to our networks to find hub genes and authority genes. Our algorithm gives weighted-hub genes and weighted-authority (target) genes. In complex networks, the HITS algorithm has very high complexity and cannot be applied successfully. The weighted HITS algorithm has yielded important information about biomolecular networks [

41,

42,

43,

44,

45,

46,

47,

48,

49]. For network topology, we refer to [

34], and for the latest on the origin of biomolecular networks, topological, combinatorial, and spectral methods, we refer to [

18].

Small World Phenomenon: Biomolecular networks have features that are not captured by the Erdos and Renyi random graph model. As we have seen, random graphs have a low clustering coefficient, and they do not account for the formation of hubs. To rectify some of these shortcomings, the

small world model, popularly known as the

six degrees of separation model was introduced as the next level of complexity for a probabilistic model with features that are closer to real world networks [

33].

In this model, the graph

$G$ of

$N$ nodes is constructed as a ring lattice, in which, (i) first,

**wire**: that is, connect every node to

$K/2$ neighbors on each side and (ii) second,

**rewire**: that is, for every edge connecting a particular node, with probability

$p$ reconnect it to a randomly selected node. The average number of such edges is

$pNK/2$. The first step of the algorithm produces local clustering, while the second dramatically reduces the distance in the network. Unlike random graphs, the clustering coefficient of this network

$C=3\left(K-2\right)/4\left(K-1\right)$ is independent of the system size. Thus, the small world network model displays the small world property and the clustering of real networks, however, it does not capture the emergence of hubby nodes (e.g., p53 in biomolecular networks) (part of one of the eight open problems that we formulate in

Section 4 in [

18]).

Scale-free Network Models: Most biomolecular networks are hypothesized to have a degree distribution, described as

scale-free. In a scale free network, the number of nodes

${n}_{k}$ of degree

$k$ is proportional to a power of the degree, namely, the degree distribution of the nodes follows a

power-law
where

$\beta >1$ is a coefficient characteristic of the network. Unlike in random networks, where the degree of all nodes is centered around a single value—with the probability of finding nodes with much larger (or smaller) degree decaying exponentially, in scale-free networks, there are nodes of large degree with relatively higher probability (fat tail). In other words, since the power low distribution decreases much more slowly than exponentially, for large

$k$ (heavy or fat tails), scale-free networks support nodes with the extremely high number of connections called “hubs.” Power law distribution has been observed in many large networks, such as the Internet, the phone-call maps, and other collaboration networks [

34]. A caveat to these reports is that inappropriate statistical techniques have often been used to infer power law distributions, and alternative heavy tailed distributions may fit the data better. However, the power law is a useful approximation that allows mechanisms of network growth to be explored, such as preferential attachment, discussed next, while the examination of alternative heavy tailed distributions is set as an open problem.

Preferential Attachment: The original model of preferential attachment was proposed by Barabási–Albert [

34]. The scheme consists of a local

growth rule that leads to a global consequence, namely a power law distribution. The network grows through the addition of new nodes linking to nodes already present in the system. There is a higher probability to preferentially link to a node with a large number of connections. Thus, this rule gives more preferences to those vertices that have larger degrees. For this reason, it is often referred to as the “rich-get-richer” or “Matthew” effect. This can be formulated as a game-theoretic problem originating from information asymmetry and associated Nash equilibrium, discussed in the Open Problems.

With an initial graph ${G}_{0}$ and a fixed probability parameter $p$, the preferential attachment random graph model $G\left(p,{G}_{0}\right)$ can be described as follows: at each step the graph ${G}_{t}$ is formed by modifying the earlier graph ${G}_{t-1}$ in two steps—with probability $p$ take a vertex-step; otherwise, take an edge-step:

Vertex step: Add a new vertex $v$ and an edge $\left\{u,v\right\}$ from $v$ to $u$ by randomly and independently choosing $u$ proportional its degree;

Edge step: Add a new edge $\left\{r,s\right\}$ by independently choosing vertices $r$ and $s$ with probability proportional to their degrees.

That is, at each step, we add a vertex with probability $p$, while for sure, we add an additional edge. If we denote by ${n}_{t}$ and ${e}_{t}$ the number of vertices and edges respectively at step $t$, then ${e}_{t}=t+1$ and ${n}_{t}=1+{\sum}_{i=1}^{t}{z}_{i}$, where ${z}_{i}$’s are Bernoulli random variables with the probability of success $=p$. Hence the expected value of nodes is $\langle {n}_{t}\rangle =1+pt$.

It can be shown that exponentially (as

$t$ asymptotically approaches infinity) this process leads to a scale-free network. The degree distribution of

$G\left(p\right)$ satisfies a power law with the parameter for exponent being

$\beta =2+\frac{p}{2-p}$. Scale-free networks also exhibit hierarchicity. The local clustering coefficient is proportional to a power of the node degree

where

$\alpha $ is called the hierarchy coefficient.

This distribution implies that the low-degree nodes belong to very dense sub-graphs and those sub-graphs are connected to each other through hubs. In other words, it means that the level of clustering is much larger than that in random networks.

Consequently, many of the network properties in a scale-free network are determined by local structures as observed in a relatively small number of highly connected nodes (hubs). A consequence of this scale-free network property is its extreme robustness to failure, which is also displayed by biomolecular networks and their modular structures. Such networks are highly tolerant of random failures (perturbations); however, they remain extremely sensitive to targeted attacks.