Interconnecting Networks: The Role of Connector Links

Abstract

Recently, some studies have started to show how global structural properties or dynamical processes such as synchronization, robustness, cooperation, transport or epidemic spreading change dramatically when considering a network of networks, as opposed to networks in isolation. In this chapter we examine the effects that the particular way in which networks get connected exerts on each of the individual networks. We describe how choosing the adequate connector links between networks may promote or hinder different structural and dynamical properties of a particular network. We show that different connecting strategies have consequences on the distribution of network centrality, population dynamics or spreading processes. The importance of designing adequate connection strategies is illustrated with examples of social and biological systems. Finally, we discuss how this new approach can be translated to other dynamical processes, such as synchronization in an ensemble of networks.

Appendix

Networks A and B, of N _A and N _B nodes and L _A and L _B links respectively, form the initially disconnected network AB of $N_{A} + N_{B}$ nodes and $L_{A} + L_{B}$ links. We connect them through L connector links to create a new interconnected network T of $N_{T} = N_{A} + N_{B}$ nodes and $L_{T} = L_{A} + L_{B} + L$ links. For convenience, the nodes of network A are numbered from i = 1 to N _A and the nodes of network B from $i = N_{A} + 1$ to $N_{T} = N_{A} + N_{B}$. The adjacency matrix G _AB of the disconnected network consists of two diagonal blocks corresponding to G _A and G _B. The relation between the transition matrix M _AB, also formed by two blocks, and G _AB, depends on the peculiarities of the process. Note that the eigenvectors of M _A and M _B are related to those of M _AB as follows: Let us call $\mathbf{x}_{A,i}$ ($i = 1,\ldots,N_{A}$) and $\mathbf{x}_{B,j}$ (j = 1, …, N _B ) the eigenvectors associated to the eigenvalues $\lambda _{A,i}$ and $\lambda _{B,j}$ of matrices M _A and M _B respectively. Note that the N _A eigenvectors $\mathbf{x}_{A,i}$ are of length N _A , the N _B eigenvectors $\mathbf{x}_{B,j}$ are of length N _B , and the eigenvectors of M _AB are of length N _T . The first i = 1, …, N _A eigenvectors of M _AB verify $(\mathbf{u}_{AB,i})_{k} = (\mathbf{x}_{A,i})_{k}$ for k ≤ N _A and $(\mathbf{u}_{AB,i})_{k} = 0$ for k > N _A . Therefore, $\lambda _{AB,i} =\lambda _{A,i}$ for i = 1, …, N _A . The eigenvectors $i = N_{A} + 1,\ldots,N_{T}$ of M _AB verify $(\mathbf{u}_{AB,i})_{k} = 0$ for k ≤ N _A and $(\mathbf{u}_{AB,i})_{k} = (\mathbf{x}_{B,i})_{k-N_{A}}$ for k > N _A . Therefore, $\lambda _{AB,i} =\lambda _{B,i-N_{A}}$ for $i = N_{A} + 1,\ldots,N_{T}$. For simplicity in the following calculations, due to their evident relation with the eigenvectors of M _A, we denote eigenvectors $\mathbf{u}_{AB,i}$ for i = 1, …, N _A as $\mathbf{u}_{A,i}$. Analogously, we denote $\mathbf{u}_{AB,i+N_{A}}$ for i = 1, …, N _B as $\mathbf{u}_{B,i}$.

Considering the addition of interlinks as represented by the symmetric matrix P (with non-zero entries in the off-diagonal blocks of elements (i, j) with i ≤ N _A and j > N _A and i > N _A and j ≤ N _A ) to be a small perturbation of parameter ε, and Taylor-expanding the largest eigenvalue of M _T and its associated eigenvector around those of M _AB , we obtain

$$\displaystyle{ \mathbf{M_{T}}\mathbf{u}_{T,1} =\lambda _{T,1}\mathbf{u}_{T,1} }$$

(4.19)

where

$$\displaystyle\begin{array}{rcl} \mathbf{M_{T}} = \mathbf{M_{AB}} +\epsilon \mathbf{P},& &{}\end{array}$$

(4.20)

$$\displaystyle\begin{array}{rcl} \mathbf{u}_{T_{1}} =\mathbf{ u}_{A,1} +\epsilon \mathbf{ v}_{1} +\epsilon ^{2}\mathbf{v}_{ 2} + o(\epsilon ^{3}),& &{}\end{array}$$

(4.21)

$$\displaystyle\begin{array}{rcl} \lambda _{T,1} =\lambda _{A,1} +\epsilon t_{1} +\epsilon ^{2}t_{ 2} + o(\epsilon ^{3}).& &{}\end{array}$$

(4.22)

Taking into account that (i) $\vert \mathbf{u}_{T,1}\vert = 1 \Rightarrow \mathbf{ u}_{A,1} \cdot \mathbf{ v}_{1} = 0$ and $\mathbf{u}_{A,1} \cdot \mathbf{ v}_{2} = 0$, and (ii) $\mathbf{u}_{A,1}\mathbf{P}\mathbf{u}_{A,1} = 0$ because $(\mathbf{u}_{A,1})_{i} = 0$ for i > N _A , we include Eqs. (4.20–4.22) in Eq. (4.19), premultiply by $\mathbf{u}_{A,1}$ and equate the terms of the same order in ε. Considering that point (i) above, in its turn, implies that $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ can be expressed as linear combinations of the other eigenvectors of M _T , which are orthogonal to $\mathbf{u}_{A,1}$, and therefore $\mathbf{u}_{A,1} \cdot \mathbf{M_{AB}}\mathbf{v}_{1} = 0$ and $\mathbf{u}_{A,1} \cdot \mathbf{M_{AB}}\mathbf{v}_{2} = 0$, we obtain to first order in ε

$$\displaystyle\begin{array}{rcl} \mathbf{u}_{A,1} \cdot (\mathbf{M_{AB}}\mathbf{v}_{1} + \mathbf{P}\mathbf{u}_{A,1}) =\mathbf{ u}_{A,1} \cdot (\lambda _{A,1}\mathbf{v}_{1} + t_{1}\mathbf{u}_{A,1})& &{}\end{array}$$

(4.23)

$$\displaystyle\begin{array}{rcl} \Rightarrow t_{1} = 0& &{}\end{array}$$

(4.24)

$$\displaystyle\begin{array}{rcl} \Rightarrow (\mathbf{M_{AB}} -\lambda _{A,1})\mathbf{v}_{1} = -\mathbf{P}\mathbf{u}_{A,1}\,,& &{}\end{array}$$

(4.25)

and for order ε ²

$$\displaystyle{ \mathbf{u}_{A,1} \cdot (\mathbf{M_{AB}}\mathbf{v}_{2} + \mathbf{P}\mathbf{v}_{1}) =\mathbf{ u}_{A,1} \cdot (\lambda _{A,1}\mathbf{v}_{2} + t_{2}\mathbf{u}_{A,1}) \Rightarrow t_{2} =\mathbf{ u}_{A,1}\mathbf{P}\mathbf{v}_{1}. }$$

(4.26)

The vector $\mathbf{v}_{1}$ can be numerically obtained solving Eq. (4.25). However, it can also be analytically expressed as

$$\displaystyle{ \mathbf{v}_{1} =\sum _{ k=1}^{N_{T} }c_{k}\mathbf{u}_{AB,k} =\sum _{ k=1}^{N_{A} }c_{k}\mathbf{u}_{A,k} +\sum _{ k=N_{A}+1}^{N_{T} }c_{k}\mathbf{u}_{B,k-N_{A}}. }$$

(4.27)

We know c ₁ = 0 because $\mathbf{u}_{A,1} \cdot \mathbf{ v}_{1} = 0$. Including Eq. (4.27) in Eq. (4.25), and multiplying both sides by $\mathbf{u}_{AB,k}$ from the left, we obtain c _k  = 0 for 1 < k ≤ N _A (because $\mathbf{u}_{A,k}\mathbf{P}\mathbf{u}_{A,1} = 0\,\forall k$) and $c_{k} = \frac{\mathbf{u}_{A,1}\mathbf{P}\mathbf{u}_{B,k-N_{A}}} {\lambda _{A,1}-\lambda _{B,k-N_{A}}}$ for k > N _A . All this yields

$$\displaystyle{ \mathbf{v}_{1} =\sum _{ k=1}^{N_{B} }\frac{\mathbf{u}_{A,1}\mathbf{P}\mathbf{u}_{B,k}} {\lambda _{A,1} -\lambda _{B,k}} \mathbf{u}_{B,k}\,, }$$

(4.28)

and including Eqs. (4.28) and (4.26) in Eq. (4.22), we finally obtain

$$\displaystyle{ \mathbf{u}_{T,1} =\mathbf{ u}_{A,1} +\epsilon \sum _{ k=1}^{N_{B} }\frac{\mathbf{u}_{A,1}\mathbf{P}\mathbf{u}_{B,k}} {\lambda _{A,1} -\lambda _{B,k}} \mathbf{u}_{B,k} + o(\epsilon ^{2})\,, }$$

(4.29)