Parallel dynamics and computational complexity of network growth models

Abstract

The parallel computational complexity or depth of growing network models is investigated. The networks considered are generated by preferential attachment rules where the probability of attaching a new node to an existing node is given by a power, α of the connectivity of the existing node. Algorithms for generating growing networks very quickly in parallel are described and studied. The sublinear and superlinear cases require distinct algorithms. As a result, there is a discontinuous transition in the parallel complexity of sampling these networks corresponding to the discontinuous structural transition at α = 1, where the networks become scale free. For α > 1 networks can be generated in constant time while for 0 ≤ α < 1 logarithmic parallel time is required. The results show that these networks have little depth and embody very little history dependence despite being defined by sequential growth rules.

Characterization of complex networks: A survey of measurements

“…Some recent proposals are briefly presented below. Machta and Machta [160] proposed the use of the computational complexity of a parallel algorithm [161] for the generation of a network as a complexity measurement of the network model. If there is a known parallel algorithm for the generation of the network of order O(f (N )), with f (x) a given function, then the complexity of the network model is defined as O(f (N )).…”

“…If there is a known parallel algorithm for the generation of the network of order O(f (N )), with f (x) a given function, then the complexity of the network model is defined as O(f (N )). For example, Barabási-Albert networks can be generated in O(log log N ) parallel steps [160].…”

Generation of Realistic Social Network Datasets For Testing of Analysis and Simulation Tools

“…Use of this algorithm becomes impractical for networks over 2000 agents, where generation of the graph took approximately 10000 seconds, or a little under 3 hours. While the computational complexity of this algorithm is very high, it can be executed by a parallel machine in near linear time [26].…”

Distributed-memory parallel algorithms for generating massive scale-free networks using preferential attachment model

“…Several other studies were done on the evolving and growth model. Machta and Machta [22] described how an evolving network can be generated in parallel. Dorogovtsev et al [10] proposed a model that can generate graphs with fat-tailed degree distributions.…”