a Quantified Network Portrait of a Population

(1)

Solution of (1) is to choose the aforementioned eigenvectors of L. The centroid based co-regularization approach as proposed by Kumar et al. [12] formulates the problem of finding a common architecture among multiple subjects, by minimizing the disagreement between subject specific subnetworks and the common subnetworks. Similar to the formulation of spectral clustering, we have the following minimization problem:

$\mathop{\min}\nolimits_{{{U_{1}} , \ldots ,U_{N} , U_{c} }} \sum\nolimits_{s = 1}^{N} {tr} \left( {U_{s}^{T} L_{s} U_{s} } \right) - \sum\nolimits_{s = 1}^{N} {\lambda_{s} tr} (U_{s} U_{s}^{T} U_{c} U_{c}^{T} ),$

(2)

where the centroid eigenvector matrix U _cencodes the common set of subnetworks. Eigenvector matrices U _scorrespond to individual subnetworks of subjects $(s = 1 \ldots N)$ . $\lambda_{s}$ ‘s are the weights of each regularization term. Once this problem is solved for U _c, k-means clustering is applied on U _cto get the common k subnetworks. The solution is obtained by a two-step iterative scheme after initializing U _c: (a) solve for U _sby fixing U _cand (b) solve for U _cby fixing U _s. This is repeated until convergence is achieved for U _c. Given U _c, U _sare determined by solving

$\mathop{\min} \nolimits_{{{U_{1}} , \ldots ,U_{N} }} \sum\nolimits_{s = 1}^{N} {tr} \left( {U_{s}^{T} (L_{s} - \lambda_{s} U_{c} U_{c}^{T} )U_{s} } \right).$

(3)

This is equivalent to calculating eigenvectors corresponding to the smallest k eigenvalues of the modified Laplacian $\tilde{L}_{s} = L_{s} - \lambda_{s} U_{c} U_{c}^{T}$ . Then by fixing U _s, U _cis determined by solving

$\mathop{\max}\nolimits_{{{U_{c}} }} \sum\nolimits_{s = 1}^{N} {\lambda_{s} tr(U_{s} U_{s}^{T} U_{c} U_{c}^{T} ),}$

(4)

that is again equivalent to finding the eigenvectors corresponding to the largest k eigenvalues of $\sum\nolimits_{s = 1}^{N} {\lambda_{s} U_{s} U_{s}^{T} }$ .

2.2 Reconstructive Projection onto Subnetworks

Subnetwork detection determines membership of each anatomical region to a specific subnetwork. Based on these memberships, a connectivity matrix of a subject can be decomposed into blocks (after reordering rows and columns), each corresponding to connections in a subnetwork or between two subnetworks. This is illustrated in Fig. 1. The block structure of a connectivity matrix, as illustrated in Fig. 1, naturally defines a generative model for the multiple subject case: for each block we define a common basis (M _ij) that includes connections in a subnetwork or between two subnetworks, and subject specific coefficients ( $\alpha_{ij}^{s}$ ).

Fig. 1.

Decomposition of a connectivity matrix into blocks corresponding to connections in a single subnetwork ( $B_{1} , B_{2} ,B_{3} , B_{4}$ ) or between two subnetworks ( $B_{1 - 2} , B_{1 - 3} , \ldots$ ). Each matrix M _ijincludes zeros everywhere except at the region corresponding to the encoded block. Note that in case of a single subject, this decomposition defines an exact reconstruction of the original connectivity matrix.

Then the connectivity matrix A ^sof a subject s is assumed to be generated as

$A^{s} = \alpha_{11}^{s} M_{11} + \alpha_{12}^{s} M_{12} + \ldots + \alpha_{22}^{s} M_{22} + \ldots + \alpha_{kk}^{s} M_{kk} ,$

(5)

where M _ij(corresponding to a block in Fig. 1) defines a common basis of connections in the subnetwork i (if i = j) or between subnetworks i and j (if $i \ne j$ ), including zeros everywhere except at the region corresponding to the connections encoded by M _ij. The coefficients $\alpha_{ij}^{s}$ are subject specific weights. The estimation of each M _ijcan be done independently since they do not share any connection. Thus, each basis component and corresponding coefficients are determined by solving

$$$ { \hbox{min} }_{{_{{_{m,p} }} }} \;f\left( {m,p} \right) = \left\| {X - mp^{T} } \right\|_{F}^{2} ,\quad {\text{subject to}}\quad m > 0, p > 0 $$” src=”/wp-content/uploads/2016/09/A339424_1_En_51_Chapter_Equ6.gif”></DIV></DIV> <DIV class=EquationNumber>(6)</DIV></DIV>where the sth column of matrix X has the elements of matrix A s in the block corresponding to M ij . m is a vector including only non-zero elements of M ij . p is a vector including the coefficient <IMG alt=$ as its s^th element. This is solved independently for each M _ijby a projected gradient descent algorithm for nonnegative matrix factorization [13]. The solution is found iteratively by updating the current estimate of parameters $\theta^{t} \equiv (m^{t} ,p^{t} )$ as

$\theta^{t + 1} = P[\theta^{t} - \beta^{t} {\nabla }_{\theta } f(\theta^{t} )],$

(7)

$P\left[ x \right]\underline{\underline{\text{def}}} \left\{ {\begin{array}{*{20}l} x \hfill & { if \; l < x < u,} \hfill \\ u \hfill & {if \;x \ge u,} \hfill \\ l \hfill & {if\; x \le l,} \hfill \\ \end{array} } \right.$

(8)

${\nabla }_{\theta } f\underline{\underline{\text{def}}} \left( {\left( {mp^{T} - X} \right)p, m^{T} (mp^{T} - X)} \right).$

(9)

The step size parameter $\beta^{t}$ is selected so that the following inequality is satisfied.

$f\left( {\theta^{t + 1} } \right) - f\left( {\theta^{t} } \right) \le \sigma {\nabla }_{\theta } f\left( {\theta^{t} } \right)^{T} (\theta^{t + 1} - \theta^{t} ),$

(10)

where $\sigma$ is any value between $$ 0 - 1. $$

The projection function $P\left[ x \right]$ projects the value of x into the range defined by the lower and upper bounds $$ l,u $$

2.3 Population Studies

The coefficients of generative model (5), $\alpha_{ij}^{s}$ , are subject-specific and encode the overall strength of connections in a subnetwork or between subnetworks. These coefficients that provide a comprehensive low dimensional representation of each subject facilitate population studies. Similar to edge-wise comparison of groups [5], we can compare two groups, such as controls vs. Patients, on a subnetwork-wise basis i.e. we can identify which group has higher/lower expression of a subnetwork or connections of an inter-subnetwork communication. This approach also increases statistical power by lowering the dimensionality of the comparison. Instead of comparing each edge individually, we divide the connectome into subnetworks and compare only coefficients that depict the overall communication pattern of these subnetworks, reducing the problem of multiple comparisons.

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