517 research outputs found
Optimal subgraph structures in scale-free configuration models
Subgraphs reveal information about the geometry and functionalities of
complex networks. For scale-free networks with unbounded degree fluctuations,
we obtain the asymptotics of the number of times a small connected graph
occurs as a subgraph or as an induced subgraph. We obtain these results by
analyzing the configuration model with degree exponent and
introducing a novel class of optimization problems. For any given subgraph, the
unique optimizer describes the degrees of the vertices that together span the
subgraph.
We find that subgraphs typically occur between vertices with specific degree
ranges. In this way, we can count and characterize {\it all} subgraphs. We
refrain from double counting in the case of multi-edges, essentially counting
the subgraphs in the {\it erased} configuration model.Comment: 50 pages, 2 figure
Identifying networks with common organizational principles
Many complex systems can be represented as networks, and the problem of
network comparison is becoming increasingly relevant. There are many techniques
for network comparison, from simply comparing network summary statistics to
sophisticated but computationally costly alignment-based approaches. Yet it
remains challenging to accurately cluster networks that are of a different size
and density, but hypothesized to be structurally similar. In this paper, we
address this problem by introducing a new network comparison methodology that
is aimed at identifying common organizational principles in networks. The
methodology is simple, intuitive and applicable in a wide variety of settings
ranging from the functional classification of proteins to tracking the
evolution of a world trade network.Comment: 26 pages, 7 figure
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