Abstract
We investigate the asymptotic number of induced subgraphs in power-law uniform random graphs. We show that these induced subgraphs appear typically on vertices with specific degrees, which are found by solving an optimization problem. Furthermore, we show that this optimization problem allows to design a linear-time, randomized algorithm that distinguishes between uniform random graphs and random graph models that create graphs with approximately a desired degree sequence: the power-law rank-1 inhomogeneous random graph. This algorithm uses the fact that some specific induced subgraphs appear significantly more often in uniform random graphs than in rank-1 inhomogeneous random graphs.
Highlights
Many networks were found to have a degree distribution that is well approximated by a power-law distribution with exponent τ ∈ (2, 3)
In the case of power-law degrees with exponent τ ∈ (2, 3) the probability of the configuration model resulting in a simple random graph vanishes, so that the configuration model cannot be used as a method to analyze power-law uniform random graphs [11]
We are interested in N (H ), the induced subgraph count of H, the number of subgraphs of URG(n)(d) that are isomorphic to H
Summary
Many networks were found to have a degree distribution that is well approximated by a power-law distribution with exponent τ ∈ (2, 3). These power-law real-world networks are often modeled by random graphs: randomized mathematical models that create networks. The configuration model creates random multigraphs with a specified degree sequence, i.e., graphs where multiple edges and self-loops can be present. When conditioning on the event that the configuration model results in a simple graph, it is distributed as a uniform random graph. If the probability of the event that the configuration model results in a simple graph is sufficiently large, it is possible to translate results from the configuration model to the uniform random graph
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