1 Introduction
The ability to quantify how the human brain is interconnected in vivo has opened the door to a number of possible analyses. In nearly all of these, brain parcellation plays a crucial role. Variations in parcellation significantly impact connectome reproducibility, derived graphtheoretical measures, and the relevance of connectome measures with respect to biological questions of interest [16]. A natural approach is then to use individual densely sampled connectomes to drive the parcellation directly, leading to a more compact, connectivityaware set of brain regions and resulting graph, as done in e.g. [10]. A comprehensive review of parcellation methods and their effects on the derived connectome quality is given in [17]. Because individual connectivity data is at once very informative and highly redundant, there is a great flexibility in how parcels can be derived from dense, highly resolute graphs. It is possible for example to derive (1) a unified populationbased atlas, (2) individuallevel parcellations with crosssubject label mapping, or (3) individual parcellations with no intersubject label correspondence. While the first approach is appealing for its simplicity and ease of interpretation, the second and third may enable the researcher to reveal some individual aspect of the connectome that is lost in the aggregate atlas.
In this work, we attempt to bridge these three approaches by first constructing maximally flexible hierarchical parcellations, and then finding a unifying set of labels and parcels to maximize individual agreement. We use the a continuous representation of a brain connectivity [8] as our initial dense connectome representation. Continuous connectivity is a parcellationfree representation of tractographybased, or “structural” connectomes that is based on the Poisson point process. Once individual parcellations are computed, we obtain a groupwise parcellation using partition ensemble algorithm. We access quality of the resulting parcellations in three ways. (1) We use the continuous connectome framework to compare parcellationapproximate and exact edge distribution functions. (2) We compare perfomance of the resulting graphs on a gender classification task. (3) We also show that without any explicit knowledge of brain geometry and based solely on graph connectivity we obtain comparatively symmetric parcellations.
2 Methods
2.1 Continuous Connectome
The continuous connectome model (ConCon) treats each tract as an observation of an inhomogeneous symmetric Poisson point process with the intensity function given by
(1) 
where denote union of two disjoint toplogically spherical brain hemispheres, representing cortical white matter boundaries. In practice, ConCon uses cortical mesh vertices as nodes of connectivity graph. From such a representation, a “discrete” connectivity graph could be computed from any particular cortical parcellation . We follow definitions from [8] and call a parcellation of if such that , and is the number of parcels (ROIs). Edges between regions and can then be computed by integration of the intensity function:
(2) 
Due to properties of the Poisson Process, is the expectation of the number of observed tracts between and . In the context of connectomics, this is the expected edge strength.
2.2 Graph Clustering
Once we obtain all individual continuous connectomes, we partition each independently into a set of disjoint communities. For graph clustering we use the Louvain modularity algorithm [1], as it has shown good results in multiple neuroimaging studies [9], [12], [5], [7]. This algorithm consist of two steps. The first step combines locally connected nodes into communities, while the second step builds new meta graph. The nodes of the metagraph are communities from the previous step, and the edges are defined as the sum of all intercommunity connections of the new nodes. The algorithm in [1] cycles over these steps iteratively, converging when further node clustering leads to no increase in modularity.
We follow the hierarchical brain concept [7], repeating the clustering procedure iteratively. After the initial parcellation, we further cluster each individual parcel as an independent graph. In this work, we repeat the process three times. For each (’th) continuous connectome this procedure yields a threelevel hierarchically embedded partition: , (see Figure 1).
2.3 Consensus clustering
In order to obtain a unified parcellation for all subjects, we use consensus clustering. The concept was developed for aggregating multiple partitions of the same data into a single partition. We define the average partition over all individual partitions as:
(3) 
where is used to denoted desirable average partition, is a number of averaged partitions, is a distance measure between two partitions and we want to minimize average distance from to all given partitions
. All partitions are represented by a vector of length
, where is a number of clustered objects (vertices of a graph in our case). It contains values from up to , where is a number of clusters (parcels). This task is generally NP complete [14], but there are many approximate algorithms. We use two approaches: Clusterbased Similarity Partitioning Algorithm (cspa) [11] and greedy algorithm from [2].CSPA defines a similarity between data points based on cooccurrence in a same cluster across different partitions, and then partitions a graph induced by this similarity. Specifically, given multiple partitions of a data points . One can define similarity between points as follow:
(4) 
Here is Kroneker delta. Thus is just number of partitions in which points and were in the same cluster. Next we build a graph, with nodes correspond to data points and edge between node and is equal to . We the partition this graph into communities using some clustering algorithm and the resulting partition is our clustering consensus partition.
The authors of [2] propose a greedy approach (Hard Ensemble  HE). Given multiple partitions it combines them iteratively, first it finds average of , next average of and and so on. As a measure of distance the authors take the average square distance between membership functions:
(5) 
Exclusively for this definition we use another way to encode object’s memberships: is a matrix of size (number of objects times number of clusters)
(6) 
In Equation 5 and are rows of memberships matrices and respectively. They correspond to membership vector of the object. Since we are looking for disjoint clusters, only a single element of such row vector is equal to . This representation is defined up to any column permutation of matrix , thus the optimization procedure is done subject to all possible column permutations.
2.4 Comparison metrics
Once we find individual partitions and combine them into an average partition, we want to access their quality. We use two different approaches.
First, we compare representation strength of different parcellations by measuring distance between original and its piecewise approximation given by:
(7) 
where and
. Natural way to compare two statistical distributions is to measure distance between their probability density functions, we will use KullbackLeibler divergence
[4]. For two probability distributions with densities
and the KL divergence is:(8) 
It takes values close to 0 if two distributions are equal almost everywhere. Similar but symmetrized version of KL divergence is JensenShannon divergence [6]. Again for two probability distributions with densities and it is given by:
(9) 
where .
Second, we compare performance of different parcellations on a gender classification task. We use Logistic Regression model with (small)
regularization on a vectors of edge weights (the upper triangle of adjacency matrix excluding diagonal). Classification perfomance is measured in terms of ROC AUC score, which is typical for binary classification tasks.Finally, in order to quantify goodness of consensus clustering and access hemisphere symmetry we use Adjusted Mutual Information [15]. It measures similarity between two partitions, with value 1 corresponds to identical partitions and values close to zero for partitions that are very different.
Using AMI we access ensemble goodness (how good clustering ensemble algorithm combines multiple partitions) using modified 3:
(10) 
We compute parcellation symmetry by comparing hemisphere parcels (labels):
(11) 
3 Experiments
3.1 Data description
We use construct continuous connetocmes of 400 subjects from the Human Connectome Project S900 release [13] following [8]. We use an icosahedral spehrical sampling, at a resolution of mesh vertices per hemisphere. We used Dipy’s implementation of constrained spherical deconvolution (CSD) to perform probabilistic tractography. Prior to clustering, we exclude all mesh vertices that were labeled by FreeSurfer as corpus callosum or cerebellum.
3.2 Experimental pipeline
Our experiments are summarized as follows:

For each subject we reconstruct its Continuous Connectome.

For each Continuous Connectome we iteratively run Louvain clustering algorithm, as described above. Subgraphs of having less then 1 percent of original graph vertices were not divided.

Next we aggregate individual subject partitions and obtain consensus clustering. Aggregation was done over 400 HCP subjects. Further, after finding the optimal parcellation, we obtain two parcellations based on two disjoint sets of 200 HCP subjects in order to compute reproducibility.

We aggregate partitions of the same level (IIIIII) using CSPA and HE.

We compare obtained partitions between themselves and with FreeSurfer’s DesikanKilliani parcellation using KullbackLeibler and JensenShannon divergence. We compute goodness of an ensemble and parcellation symmetry using AMI.

We compare performance of simplified connectomes on a binary classification task using Logistic Regression with penalty. Classification results are measured in terms of ROC AUC score, with averaging over crossvalidation folds.
cspa  cspa  cspa  HE  HE  HE  DKT  
KL  
JS  
Gender Classification  
Hemisphere symmetry  
Ensemble goodness  
Number of ROIs  5  7  8  7  30  83  68 
All results are rounded to 2 significant digits. Where it possible results are reported with standard deviation. Best result in each row is colored. KL, JS divergences,
lower is better; binary Gender Classification was measured in terms of ROC AUC score, higher  better; Ensemble goodness and Hemisphere symmetry were measured using AMI, Ensemble goodness is an average AMI between consensus partition and all individual partitions, higher  better.3.3 Results
Table 1 represent all comparison results. First we can see that CSPA algorithm failed to find good clustering ensemble which result in poor classification performance and high KL and JS divergences. Greedy algorithm performed on on the other hand outperforms standard Desikan atlas accross all comparison metrics (except number of parcels, 68 versus 83). Surprisingly, greedy ensemble of second level partition () performs comparatively with Desikan , despite having twice as lower number of parcels (30 versus 68).
Another interesting property that we get automatically is parcellation symmetry. Our clustering algorithm known nothing about brain topology (all information was contained in graph connectivity), still reconstruct parcellations which are highly symmetrical. For standard Desikan atlas hemisphere symmetry is , and for our best parcellation this value even higher (), and still remains quite high for second level partition ().
Finally we check if our best ensemble parcellation, which combines individual partitions is stable. We split subjects into groups of subjects and independently combine their partitions. We compare resulting parcellations: and between themselves and with original (which is an ensemble of all subjects) again using Adjusted Mutual Information. Both and shows AMI value greater than ( and respectively) when compare with , they also highly similar between themselves.
4 Conclusion
We have presented an approach for generating unified connectivitybased human brain atlases bases on consensus clustering. The method is based on finding a pseudo average over the set of individual partitions. Our approach outperforms standard a anatomical parcellation on several important metrics, including agreement with dense connectomes, improved relevance to biological data, and even improved symmetry. Because our approach is entirely data driven an requires no agreement between individual parcellation labels, it combines both the flexibility of individual parcellations and the interpretability of simple unified atlases.
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