Editing Open Problems:98
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{{Header | {{Header | ||
+ | |title=Estimating a Graph's Degree Distribution | ||
|source=wola19 | |source=wola19 | ||
|who=C. Seshadhri | |who=C. Seshadhri | ||
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Assume one has access to the graph $G$ via the following (standard) three types of queries: | Assume one has access to the graph $G$ via the following (standard) three types of queries: | ||
− | + | - sampling a u.a.r. vertex | |
− | + | - querying the degree of a given vertex | |
− | + | - sample a u.a.r. neighbor of a given vertex | |
− | and the goal is to obtain the following $(1\pm \varepsilon)$- | + | and the goal is to obtain the following $(1\pm \varepsilon)$-"bicriteria" approximation $\hat{N}$ of the degree distribution: for all $d$, |
\[ | \[ | ||
(1-\varepsilon)N( (1-\varepsilon)d) \leq \hat{N}(d) \leq (1+\varepsilon) N((1+\varepsilon)d)\,. | (1-\varepsilon)N( (1-\varepsilon)d) \leq \hat{N}(d) \leq (1+\varepsilon) N((1+\varepsilon)d)\,. | ||
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And also, slightly less well-defined: | And also, slightly less well-defined: | ||
− | '''Question:''' Can one obtain better upper bounds when relaxing the goal to only learn the ''high-degree'' (tail) part of the distribution? What about testing properties of the degree distribution (e.g., | + | '''Question:''' Can one obtain better upper bounds when relaxing the goal to only learn the ''high-degree'' (tail) part of the distribution? What about testing properties of the degree distribution (e.g., "power-law-ness") in this setting? And what about the first type of queries — can one relax it, or work with a different type of sampling than uniform (for instance, via random walks)? |