Understanding the Mesoscale Structure of the C. elegans Brain
Network
Prof. Edward
of Warwick, Dept. of Statistics, Coventry, UK;
Cambridge, UK;
3 GlaxoSmithKline,
2 University
of Cambridge, Brain Mapping Unit, Dept. of Psychiatry,
Differences in Community Estimation
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Analysis
We apply all 3 methods to the C. elegans neural network, composed of 279 non pharyngeal neurons and
of 2287 undirected edges, and we use the additional
functional and anatomical measures to evaluate the
estimates of its community structure. For the quantitative ground truth measures, we use the Intra Class
Correlation (ICC) to compare the variance explained
by each community estimates. For the categorical
ground truth measures, we use the Adjusted Rand
Index (ARI) to compare the similarity.
References
[1]
Daudin, Picard, Robin A mixture model for random graphs, Statistics and computing, (2008).
[2]
Newman, Detecting community structure in networks,The European Physical Journal B-Condensed
Matter and Complex Systems, vol. 38, (2004).
[3]
Blondel, Guillaume, Lambiotte and Lefebvre Fast unfolding of communities in large networks,
Journal of Statistical Mechanics: Theory and Experiment, vol. 10, (2008).
[4]
Dobson, An introduction to generalised linear models (2001).
[5]
Hubert and Arabie, Comparing partitions, Journal of classification, vol. 2, (1985).
[6]
Varshney, Chen, Paniagua, Hall, and Chklovskii, Structural properties of the Caenorhabditis elegans neuronal network, PLoS computational biology, vol. 7, (2011).
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Figure 4: Spectral.
Network Compression and Degree Distribution with ERMM
Block 1
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Block 2
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Block 9
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Block 8
Block 3
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Dr
aft
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Block 7
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Empirical
Fitted
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Block 6
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Block 4
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Block 5
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Figure 5: ERMM connectivity structure.
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Degrees
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Figure 6: ERMM’s fitted degree distribution.
Qualitative Assessment
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Methods
ERMM
Louvain algorithm
Spectral algorithm
0.4
ICC
The ERMM treats the communities (blocks) and
their mutual connections as mini Erdős-Rényi models, represented in the likelihood with different proportions. For a given number of communities Q,
a variational approach is used to approximate the
likelihood, while the Integrated Classification Likelihood (ICL) is used to compare the optimised likelihoods over different Q. The final result is an estimate
of Q and the partition, visualised as a reorganised adjacency matrix. The deterministic methods like the
Fast Louvain and Spectral algorithms define community as a group of highly connected nodes whose
between group connections are very small. Both
algorithms are devised to maximise the modularity
but use different strategies to find its maximum. For
example, the Fast Louvain algorithm uses a greedy
approach, while the Spectral algorithm uses eigenvalues of the modularity matrix to find the optimal
partition.
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Figure 3: Louvain.
Figure 2: ERMM.
31
Methods
Dr. Thomas E.
1
Nichols
Clinical Unit Cambridge, Addenbrooke’s Hospital, Cambridge, UK.
Introduction
Recently, there has been much
interest in mesoscale structure
of networks such as: their
organisation into communities and core and periphery. However, it is often difficult to disambiguate the relationship between these two
types of mesoscale structure
or, indeed, to summarise the
full network into the relationships between its mesoscale
Figure 1: Nerve constituents. Here, we use
a stochastic blockmodel aptracts C. elegans.
proach Erdős-Rényi Mixture
Model (ERMM)[1] for community estimation and
compare this to the much more widely used deterministic methods such as: Louvain [3] and Spectral
[2] algorithms. We use the Caenorhabditis elegans
(C. elegans) [6] connectome (Fig. 1) as a model system in which biological knowledge about each node
or neuron can be used to validate the functional relevance of the communities obtained.
2,3
T.Bullmore ,
1−CDF
1 University
Dr. Petra
2
E.Vertes ,
0.2
0.3
Methods
0.2
ERMM
ARI
Dragana
1
M.Pavlovic ,
Louvain algorithm
Spectral algorithm
0.1
0.0
0.0
ALL
ALS
AD
BT
BTD
LD
FC
Figure 7: ICC scores for the Anatomical location
(longitudinal) (ALL), Anatomical location (sectional)
(ALS), Anatomical distance (AD), Birth time (BT),
Birth time difference (BTD) and Lineage distance (LD).
GC
Figure 8: ARI scores for Functional Classification (FC) and Ganglion Classification (GC).
Results
The optimal ERMM fit consists of 9 classes, while the fits of Louvain and Spectral algorithms consist of 5 and
4 communities, shown in Fig. 2-4 as the reorganised adjacency matrices. The ERMM finds dense blocks on
the diagonal, but but also a range of off-diagonal patterns. Note how blocks 5&6, with tight inter-connections
and numerous external connections, form a core-periphery structure. Surprisingly, even though blocks 5&6
fit the standard notion of “community" they are not identified by the determinist algorithms. Furthermore,
ERMM fit provides a compressed view of the C. elegans network (see Fig.5) and a faithful approximation of
the degree distribution (Fig. 6).
To score the quality of each fit, we show the ICC scores (Fig. 7) across the known biological features characterising nodes and edges. Here, we see that the ERMM fit scores consistently higher than Spectral and
Louvain algorithms. In Fig. 8, however, ARI is rather low in general, with the Spectral algorithm showing
slightly better similarity with the functional classifications and all methods having similar ARI for ganglion
classification.
Conclusion
We showed that the Erdős- Rényi Mixture Model not only produces more biologically plausible communities
but also that it provides an integrated picture of the full mesoscale structure (including core-periphery) and
that it allows for compression of the network into a set of super-nodes and their connectivities. We expect
these methods to prove useful for the analysis of other types of networks such as human brain functional
connectivity.