2,193 research outputs found
Approximating the Spectrum of a Graph
The spectrum of a network or graph with adjacency matrix ,
consists of the eigenvalues of the normalized Laplacian . This set of eigenvalues encapsulates many aspects of the structure
of the graph, including the extent to which the graph posses community
structures at multiple scales. We study the problem of approximating the
spectrum , of in the regime where the graph is too
large to explicitly calculate the spectrum. We present a sublinear time
algorithm that, given the ability to query a random node in the graph and
select a random neighbor of a given node, computes a succinct representation of
an approximation , such that . Our algorithm has query complexity and running time ,
independent of the size of the graph, . We demonstrate the practical
viability of our algorithm on 15 different real-world graphs from the Stanford
Large Network Dataset Collection, including social networks, academic
collaboration graphs, and road networks. For the smallest of these graphs, we
are able to validate the accuracy of our algorithm by explicitly calculating
the true spectrum; for the larger graphs, such a calculation is computationally
prohibitive.
In addition we study the implications of our algorithm to property testing in
the bounded degree graph model
On the Permanence of Vertices in Network Communities
Despite the prevalence of community detection algorithms, relatively less
work has been done on understanding whether a network is indeed modular and how
resilient the community structure is under perturbations. To address this
issue, we propose a new vertex-based metric called "permanence", that can
quantitatively give an estimate of the community-like structure of the network.
The central idea of permanence is based on the observation that the strength
of membership of a vertex to a community depends upon the following two
factors: (i) the distribution of external connectivity of the vertex to
individual communities and not the total external connectivity, and (ii) the
strength of its internal connectivity and not just the total internal edges.
In this paper, we demonstrate that compared to other metrics, permanence
provides (i) a more accurate estimate of a derived community structure to the
ground-truth community and (ii) is more sensitive to perturbations in the
network. As a by-product of this study, we have also developed a community
detection algorithm based on maximizing permanence. For a modular network
structure, the results of our algorithm match well with ground-truth
communities.Comment: 10 pages, 5 figures, 8 tables, Accepted in 20th ACM SIGKDD Conference
on Knowledge Discovery and Data Minin
The stability of a graph partition: A dynamics-based framework for community detection
Recent years have seen a surge of interest in the analysis of complex
networks, facilitated by the availability of relational data and the
increasingly powerful computational resources that can be employed for their
analysis. Naturally, the study of real-world systems leads to highly complex
networks and a current challenge is to extract intelligible, simplified
descriptions from the network in terms of relevant subgraphs, which can provide
insight into the structure and function of the overall system.
Sparked by seminal work by Newman and Girvan, an interesting line of research
has been devoted to investigating modular community structure in networks,
revitalising the classic problem of graph partitioning.
However, modular or community structure in networks has notoriously evaded
rigorous definition. The most accepted notion of community is perhaps that of a
group of elements which exhibit a stronger level of interaction within
themselves than with the elements outside the community. This concept has
resulted in a plethora of computational methods and heuristics for community
detection. Nevertheless a firm theoretical understanding of most of these
methods, in terms of how they operate and what they are supposed to detect, is
still lacking to date.
Here, we will develop a dynamical perspective towards community detection
enabling us to define a measure named the stability of a graph partition. It
will be shown that a number of previously ad-hoc defined heuristics for
community detection can be seen as particular cases of our method providing us
with a dynamic reinterpretation of those measures. Our dynamics-based approach
thus serves as a unifying framework to gain a deeper understanding of different
aspects and problems associated with community detection and allows us to
propose new dynamically-inspired criteria for community structure.Comment: 3 figures; published as book chapte
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