14,076 research outputs found
Algorithmic Complexity of Power Law Networks
It was experimentally observed that the majority of real-world networks
follow power law degree distribution. The aim of this paper is to study the
algorithmic complexity of such "typical" networks. The contribution of this
work is twofold.
First, we define a deterministic condition for checking whether a graph has a
power law degree distribution and experimentally validate it on real-world
networks. This definition allows us to derive interesting properties of power
law networks. We observe that for exponents of the degree distribution in the
range such networks exhibit double power law phenomenon that was
observed for several real-world networks. Our observation indicates that this
phenomenon could be explained by just pure graph theoretical properties.
The second aim of our work is to give a novel theoretical explanation why
many algorithms run faster on real-world data than what is predicted by
algorithmic worst-case analysis. We show how to exploit the power law degree
distribution to design faster algorithms for a number of classical P-time
problems including transitive closure, maximum matching, determinant, PageRank
and matrix inverse. Moreover, we deal with the problems of counting triangles
and finding maximum clique. Previously, it has been only shown that these
problems can be solved very efficiently on power law graphs when these graphs
are random, e.g., drawn at random from some distribution. However, it is
unclear how to relate such a theoretical analysis to real-world graphs, which
are fixed. Instead of that, we show that the randomness assumption can be
replaced with a simple condition on the degrees of adjacent vertices, which can
be used to obtain similar results. As a result, in some range of power law
exponents, we are able to solve the maximum clique problem in polynomial time,
although in general power law networks the problem is NP-complete
Pruning based Distance Sketches with Provable Guarantees on Random Graphs
Measuring the distances between vertices on graphs is one of the most
fundamental components in network analysis. Since finding shortest paths
requires traversing the graph, it is challenging to obtain distance information
on large graphs very quickly. In this work, we present a preprocessing
algorithm that is able to create landmark based distance sketches efficiently,
with strong theoretical guarantees. When evaluated on a diverse set of social
and information networks, our algorithm significantly improves over existing
approaches by reducing the number of landmarks stored, preprocessing time, or
stretch of the estimated distances.
On Erd\"{o}s-R\'{e}nyi graphs and random power law graphs with degree
distribution exponent , our algorithm outputs an exact distance
data structure with space between and
depending on the value of , where is the number of vertices. We
complement the algorithm with tight lower bounds for Erdos-Renyi graphs and the
case when is close to two.Comment: Full version for the conference paper to appear in The Web
Conference'1
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