Algorithmic Complexity of Power Law Networks

It was experimentally observed that the majority of real-world networks are scale-free and 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 [1, 2] 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. Again, we experimentally validate that many real-world graphs satisfy our property. 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. In contrast to previously done average-case analyses, we believe that this is the first “waterproof” argument that explains why many real-world networks are easier. Moreover, an interesting aspect of this study is the existence of structure oblivious algorithms, i.e., algorithms that run faster on power law networks without explicit knowledge of this fact or explicit knowledge of the parameters of the degree distribution, e.g., algorithms for maximum clique or triangle counting. ∗University of Warsaw, [email protected], Supported by the ERC StG PAAl project no. 259515. †University of Warsaw, [email protected] ‡University of Warsaw, [email protected]. Jakub Łącki is a recipient of the Google Europe Fellowship in Graph Algorithms, and this research is supported in part by this Google Fellowship. §University of Warsaw, [email protected], Supported by the ERC StG PAAl project no. 259515. ar X iv :1 50 7. 02 42 6v 1 [ cs .D S] 9 J ul 2 01 5