9 research outputs found
On the limiting behavior of parameter-dependent network centrality measures
We consider a broad class of walk-based, parameterized node centrality
measures for network analysis. These measures are expressed in terms of
functions of the adjacency matrix and generalize various well-known centrality
indices, including Katz and subgraph centrality. We show that the parameter can
be "tuned" to interpolate between degree and eigenvector centrality, which
appear as limiting cases. Our analysis helps explain certain correlations often
observed between the rankings obtained using different centrality measures, and
provides some guidance for the tuning of parameters. We also highlight the
roles played by the spectral gap of the adjacency matrix and by the number of
triangles in the network. Our analysis covers both undirected and directed
networks, including weighted ones. A brief discussion of PageRank is also
given.Comment: First 22 pages are the paper, pages 22-38 are the supplementary
material
Using Triangles to Improve Community Detection in Directed Networks
In a graph, a community may be loosely defined as a group of nodes that are
more closely connected to one another than to the rest of the graph. While
there are a variety of metrics that can be used to specify the quality of a
given community, one common theme is that flows tend to stay within
communities. Hence, we expect cycles to play an important role in community
detection. For undirected graphs, the importance of triangles -- an undirected
3-cycle -- has been known for a long time and can be used to improve community
detection. In directed graphs, the situation is more nuanced. The smallest
cycle is simply two nodes with a reciprocal connection, and using information
about reciprocation has proven to improve community detection. Our new idea is
based on the four types of directed triangles that contain cycles. To identify
communities in directed networks, then, we propose an undirected edge-weighting
scheme based on the type of the directed triangles in which edges are involved.
We also propose a new metric on quality of the communities that is based on the
number of 3-cycles that are split across communities. To demonstrate the impact
of our new weighting, we use the standard METIS graph partitioning tool to
determine communities and show experimentally that the resulting communities
result in fewer 3-cycles being cut. The magnitude of the effect varies between
a 10 and 50% reduction, and we also find evidence that this weighting scheme
improves a task where plausible ground-truth communities are known.Comment: 10 pages, 3 figure
An Ensemble Framework for Detecting Community Changes in Dynamic Networks
Dynamic networks, especially those representing social networks, undergo
constant evolution of their community structure over time. Nodes can migrate
between different communities, communities can split into multiple new
communities, communities can merge together, etc. In order to represent dynamic
networks with evolving communities it is essential to use a dynamic model
rather than a static one. Here we use a dynamic stochastic block model where
the underlying block model is different at different times. In order to
represent the structural changes expressed by this dynamic model the network
will be split into discrete time segments and a clustering algorithm will
assign block memberships for each segment. In this paper we show that using an
ensemble of clustering assignments accommodates for the variance in scalable
clustering algorithms and produces superior results in terms of
pairwise-precision and pairwise-recall. We also demonstrate that the dynamic
clustering produced by the ensemble can be visualized as a flowchart which
encapsulates the community evolution succinctly.Comment: 6 pages, under submission to HPEC Graph Challeng
Ranking hubs and authorities using matrix functions
The notions of subgraph centrality and communicability, based on the
exponential of the adjacency matrix of the underlying graph, have been
effectively used in the analysis of undirected networks. In this paper we
propose an extension of these measures to directed networks, and we apply them
to the problem of ranking hubs and authorities. The extension is achieved by
bipartization, i.e., the directed network is mapped onto a bipartite undirected
network with twice as many nodes in order to obtain a network with a symmetric
adjacency matrix. We explicitly determine the exponential of this adjacency
matrix in terms of the adjacency matrix of the original, directed network, and
we give an interpretation of centrality and communicability in this new
context, leading to a technique for ranking hubs and authorities. The matrix
exponential method for computing hubs and authorities is compared to the well
known HITS algorithm, both on small artificial examples and on more realistic
real-world networks. A few other ranking algorithms are also discussed and
compared with our technique. The use of Gaussian quadrature rules for
calculating hub and authority scores is discussed.Comment: 28 pages, 6 figure
Local Rewiring Algorithms to Increase Clustering and Grow a Small World
Many real-world networks have high clustering among vertices: vertices that
share neighbors are often also directly connected to each other. A network's
clustering can be a useful indicator of its connectedness and community
structure. Algorithms for generating networks with high clustering have been
developed, but typically rely on adding or removing edges and nodes, sometimes
from a completely empty network. Here, we introduce algorithms that create a
highly clustered network by starting with an existing network and rearranging
edges, without adding or removing them; these algorithms can preserve other
network properties even as the clustering increases. They rely on local
rewiring rules, in which a single edge changes one of its vertices in a way
that is guaranteed to increase clustering. This greedy step can be applied
iteratively to transform a random network into a form with much higher
clustering. Additionally, the algorithms presented grow a network's clustering
faster than they increase its path length, meaning that network enters a regime
of comparatively high clustering and low path length: a small world. These
algorithms may be a basis for how real-world networks rearrange themselves
organically to achieve or maintain high clustering and small-world structure.Comment: 20 pages, 13 figure