128,682 research outputs found
Modularity of regular and treelike graphs
Clustering algorithms for large networks typically use modularity values to
test which partitions of the vertex set better represent structure in the data.
The modularity of a graph is the maximum modularity of a partition. We consider
the modularity of two kinds of graphs.
For -regular graphs with a given number of vertices, we investigate the
minimum possible modularity, the typical modularity, and the maximum possible
modularity. In particular, we see that for random cubic graphs the modularity
is usually in the interval , and for random -regular graphs
with large it usually is of order . These results help to
establish baselines for statistical tests on regular graphs.
The modularity of cycles and low degree trees is known to be close to 1: we
extend these results to `treelike' graphs, where the product of treewidth and
maximum degree is much less than the number of edges. This yields for example
the (deterministic) lower bound mentioned above on the modularity of
random cubic graphs.Comment: 25 page
Modularity bounds for clusters located by leading eigenvectors of the normalized modularity matrix
Nodal theorems for generalized modularity matrices ensure that the cluster
located by the positive entries of the leading eigenvector of various
modularity matrices induces a connected subgraph. In this paper we obtain lower
bounds for the modularity of that set of nodes showing that, under certain
conditions, the nodal domains induced by eigenvectors corresponding to highly
positive eigenvalues of the normalized modularity matrix have indeed positive
modularity, that is they can be recognized as modules inside the network.
Moreover we establish Cheeger-type inequalities for the cut-modularity of the
graph, providing a theoretical support to the common understanding that highly
positive eigenvalues of modularity matrices are related with the possibility of
subdividing a network into communities
Development of modularity in the neural activity of children's brains
We study how modularity of the human brain changes as children develop into
adults. Theory suggests that modularity can enhance the response function of a
networked system subject to changing external stimuli. Thus, greater cognitive
performance might be achieved for more modular neural activity, and modularity
might likely increase as children develop. The value of modularity calculated
from fMRI data is observed to increase during childhood development and peak in
young adulthood. Head motion is deconvolved from the fMRI data, and it is shown
that the dependence of modularity on age is independent of the magnitude of
head motion. A model is presented to illustrate how modularity can provide
greater cognitive performance at short times, i.e.\ task switching. A fitness
function is extracted from the model. Quasispecies theory is used to predict
how the average modularity evolves with age, illustrating the increase of
modularity during development from children to adults that arises from
selection for rapid cognitive function in young adults. Experiments exploring
the effect of modularity on cognitive performance are suggested. Modularity may
be a potential biomarker for injury, rehabilitation, or disease.Comment: 29 pages, 11 figure
Community detection in directed acyclic graphs
Some temporal networks, most notably citation networks, are naturally
represented as directed acyclic graphs (DAGs). To detect communities in DAGs,
we propose a modularity for DAGs by defining an appropriate null model (i.e.,
randomized network) respecting the order of nodes. We implement a spectral
method to approximately maximize the proposed modularity measure and test the
method on citation networks and other DAGs. We find that the attained values of
the modularity for DAGs are similar for partitions that we obtain by maximizing
the proposed modularity (designed for DAGs), the modularity for undirected
networks and that for general directed networks. In other words, if we neglect
the order imposed on nodes (and the direction of links) in a given DAG and
maximize the conventional modularity measure, the obtained partition is close
to the optimal one in the sense of the modularity for DAGs.Comment: 2 figures, 7 table
Extension of Modularity Density for Overlapping Community Structure
Modularity is widely used to effectively measure the strength of the disjoint
community structure found by community detection algorithms. Although several
overlapping extensions of modularity were proposed to measure the quality of
overlapping community structure, there is lack of systematic comparison of
different extensions. To fill this gap, we overview overlapping extensions of
modularity to select the best. In addition, we extend the Modularity Density
metric to enable its usage for overlapping communities. The experimental
results on four real networks using overlapping extensions of modularity,
overlapping modularity density, and six other community quality metrics show
that the best results are obtained when the product of the belonging
coefficients of two nodes is used as the belonging function. Moreover, our
experiments indicate that overlapping modularity density is a better measure of
the quality of overlapping community structure than other metrics considered.Comment: 8 pages in Advances in Social Networks Analysis and Mining (ASONAM),
2014 IEEE/ACM International Conference o
Modularity from Fluctuations in Random Graphs and Complex Networks
The mechanisms by which modularity emerges in complex networks are not well
understood but recent reports have suggested that modularity may arise from
evolutionary selection. We show that finding the modularity of a network is
analogous to finding the ground-state energy of a spin system. Moreover, we
demonstrate that, due to fluctuations, stochastic network models give rise to
modular networks. Specifically, we show both numerically and analytically that
random graphs and scale-free networks have modularity. We argue that this fact
must be taken into consideration to define statistically-significant modularity
in complex networks.Comment: 4 page
Ground state energy of -state Potts model: the minimum modularity
A wide range of interacting systems can be described by complex networks. A
common feature of such networks is that they consist of several communities or
modules, the degree of which may quantified as the \emph{modularity}. However,
even a random uncorrelated network, which has no obvious modular structure, has
a finite modularity due to the quenched disorder. For this reason, the
modularity of a given network is meaningful only when it is compared with that
of a randomized network with the same degree distribution. In this context, it
is important to calculate the modularity of a random uncorrelated network with
an arbitrary degree distribution. The modularity of a random network has been
calculated [Phys. Rev. E \textbf{76}, 015102 (2007)]; however, this was limited
to the case whereby the network was assumed to have only two communities, and
it is evident that the modularity should be calculated in general with communities. Here, we calculate the modularity for communities by
evaluating the ground state energy of the -state Potts Hamiltonian, based on
replica symmetric solutions assuming that the mean degree is large. We found
that the modularity is proportional to regardless of and that only the coefficient depends on . In
particular, when the degree distribution follows a power law, the modularity is
proportional to . Our analytical results are
confirmed by comparison with numerical simulations. Therefore, our results can
be used as reference values for real-world networks.Comment: 14 pages, 4 figure
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