Network Analyses

Network analyses use graph-theoretic approaches to identifying/quantifying/describing/classifying networks of any sort. For example, a network analysis could be used to identify potential bottlenecks in a city design when large numbers of people have to evacuate, or the robustness of a distributed computer network to disruption in the event that one or more redundant servers goes offline.

In our lab, we apply these analytic approaches to networks that describe functional (primarily) or anatomical (potentially) connectivity. Under certain assumptions about what functional role various network nodes play, we can test hypotheses about how the connections between these nodes relates to cognitive processing. Most of our work here will be done in MATLAB and using networks that are represented as adjacency matrices. An adjacency matrix is an NxN square matrix that has entries that indicate whether there is a connection (and potentially how strong it is) between any two of the N nodes in the network. Using the Louvain 2008 parcellation scheme, for example, we break the cortical surface up into 1000 regions. Each region is a node in the network, and so the adjacency matrix for this network is a 1000 x 1000 square matrix. What follows assumes that you have obtained an adjacency matrix, one way or another, and wish to carry out different types of network analyses on it.

Hierarchical Clustering

Sarah Muldoon recommended starting off with a hierarchical clustering of the adjacency matrix in order to visualize the structure of the matrix and get a sense of the sort of structure that exists within the data. The MATLAB help page for hierarchical clustering has some background information, but here are the highlights.

The first thing to note is that these steps are too computationally intensive for the adjacency matrices generated for the larger networks produced using the Lausanne 2008 parcellation (Lausanne 500 and 250 are too large). The myaparc_60 network seems to work smoothly; I'll try the 125 next.

Obtain Linkage

The first step of a hierarchical cluster is to obtain some kind of distance metric between each node in the graph. This information already exists in the form of the weighted adjacency matrices (either correlations or trained network weights). We pick up here at the linkage step. MATLAB has a linkage function that expects a vector of pairwise distances between all nodes, as would be generated by the pdist function. If you read the documentation for pdist, you will find that it outputs a vector for all pairs ({2,1}, {3,1}, ... , {m, 1}, {3, 2}, ..., {m, 2}, ... , {m, m-1}), where m is the number of nodes in the network. In relation to the adjacency matrices, these values would represent the lower (or upper) triangle of the matrix.

Filter Weights

The lower triangle of the adjacency matrix can be obtained in MATLAB using the tril function. An mxm (i.e., square) matrix has k diagonals, numbered 0 to (-1)*m-1. The main diagonal, k=0, will be the diagonal containing all ones (in a correlation matrix), which are meaningless in this context. We will want to get all the diagonals where k<0 (if you are getting the upper triangle using the triu function, use k>0).

mask=tril(ones(size(M)),-1); %identify the elements in the lower triangle, excluding diagonal k=0
lowert=M(find(mask)); %use mask to obtain the values.

The above code returns the values from the lower triangle of the adjacency matrix M as a vector in the same order that pdist generates.

Run linkage

There are several methods for running the linkage function, but common choices are single, complete, and average.

Z=linkage(lowert,'average');

Note that the solutions provided by each of these methods can yield drastically different results. For example, while I was working through this workflow for the first time, I used the single linkage distance calculation method. The sorted correlation matrix displayed at the end didn't seem to have any well-defined clusters of any size (leading me to assume that I had done something wrong). Since there doesn't seem to be any "correct" method, and the purpose of this exercise is to identify some sort of underlying structure to the weights, you might try different methods until you find something that makes that structure apparent.

Generate the Dendrogram

Using the dendogram function, you'll need to force it to provide the entire leaf structure, specifying p, the maximum number of leaf nodes (if not specified, the default behavior is to display up to 30 leaf nodes):

[H, T, outperm] = dendrogram(Z, size(M,1));

Looking at the dendrogram can be used to inform the number of clusters you might look for in a data set (e.g., if something like k-means clustering was subsequently employed).

Sort and Visualize Adjacency Matrices

Use the outperm vector generated by the dendrogram as a sorting index for the adjacency matrix to reveal the network clustering structure.

sorted_M=M(outperm,outperm); 
figure(2);
imagesc(sorted_M);

Stochastic Clustering

A powerful and widely-used clustering algorithm is the Louvain greedy clustering algorithm. It's stochastic (i.e., contains a random element to it) because it's computationally intractable to completely search the entire space of possible clusterings for anything other than small "toy" networks.

The Louvain algorithm is described here, and has been incorporated in a collection of MATLAB network analysis scripts archived here and here but also stored in our ubfs Scripts/Matlab folder.

Create a Modularity Matrix

The first thing to note is that the genlouvain function doesn't work directly on the adjacency matrix, but instead on a modularity matrix, which is derived from your adjacency matrix. The number of partitions that are detected is influenced in part by a parameter, gamma, that is often omitted when this modularity matrix is calculated. When omitted, it defaults to 1, however larger gammas permit a greater number of final partitions. The following code sample is pulled from the genlouvain wiki site. The MATLAB function, full, transforms a sparse matrix into a full matrix. Sparse matrices are a more compact way of storing large data sets when there are a large number of empty values. I'm not clear whether the code works properly if the adjacency matrix, A, starts off as a full matrix, so instead I will begin by transforming the full matrix M (e.g., a 1000 x 1000 network) into a sparse matrix data type and proceed from there.

A=sparse(M);
gamma = 1.1; %gamma=1 tends to produce 3-community solutions. Higher values allow more communities.
k = full(sum(A));
twom = sum(k); %twom ("2M") is a normalization parameter 
B = full(A - gamma*k'*k/twom); %B is the modularity matrix

Detect Communities Using genlouvain

At this point, you now have B, the modularity matrix for the adjacency matrix M. Running the genlouvain function on B will stochastically generate a partitioning of your network that maximizes modularity (within its search space).

[S,Q] = genlouvain(B);
Q = Q/twom

Because this algorithm is stochastic, repeated applications of the genlouvain function are likely to generate somewhat different partitionings. Fortunately, the code runs rather quickly on relatively small networks, and so we can run the function repeatedly and store each partitioning.

maxiterations=10000; %how many partitioning attempts do you want?
S=repmat(NaN, maxiterations, length(k)); %preallocate storage for each partitioning solution
Q=repmat(NaN, maxiterations, 1); %preallocate storage for the modularity of each partitioning solution
for i=1:maxiterations
 [s, q]=genlouvain(B);
 q=q/twom;
 S(i,:)=s(:);
 Q(i)=q;
end

Determine the Consensus Best Partitioning

With maxiterations different partitionings of your adjacency matrix into different communities (stored in matrix S in the previous step), we can ascertain the consensus community assignment (i.e., the modal partitioning) for each node in the network. The commdetect toolbox has a function, consensus_iterative for that task.

[S2 Q2 X_new3 qpc] = consensus_iterative(S);

S2 is a matrix of size(S). mode(S2) can be used to list the representative partition for each node in the adjacency matrix, M. Each row of S2 should be the same, though I'm not 100% positive that this is an absolute guarantee, which is why I suggested using mode rather than mean or just selecting one of the maxiteration rows of S2. The value gpc is the quality of the consensus, where smaller values are better (it might be the case that gpc>0 is associated with S2 matrices with rows that differ).