The structure of the network underlying many complex systems, whether artificial or natural, plays a significant role in how these systems operate. As a result, much emphasis has been placed on accurately describing networks using network theoretic metrics. When it comes to generating networks with similar properties, however, the set of available techniques and properties that can be controlled for remains limited. Further, whilst it is becoming clear that some of the metrics currently used to control the generation of such networks are not very prescriptive so that networks could potentially exhibit very different higher-order structure within those constraints, network generating algorithms typically produce fairly contrived networks and lack mechanisms by which to systematically explore the space of network solutions. In this paper, we explore the potential of a multi-objective novelty-biased GA to provide a viable alternative to these algorithms. We believe our results provide the first proof of principle that (i) it is possible to use GAs to generate graphs satisfying set levels of key classical graph theoretic properties and (ii) it is possible to generate diverse solutions within these constraints. The paper is only a preliminary step, however, and we identify key avenues for further development.
The structure of a network plays a key role in the outcome of dynamical processes operating on it. Two prevalent network descriptors are the degree distribution and the global clustering. However, when generating networks with a prescribed degree distribution and global clustering, it has been shown that changes in structural properties other than that controlled for are induced and these changes have been found to alter the outcome of spreading processes on the network. This therefore begs the question of our understanding of the potential diversity of networks sharing a given degree distribution and global clustering. As the space of all possible networks is too large to be systematically explored, a heuristic approach is needed. In our genetic algorithm-based approach, networks are encoded by their subgraph counts from a chosen family of subgraphs. Coverage of the space of possible networks is then maximised by focusing the search through optimising the diversity of counts by the Map-Elite algorithm. We provide preliminary evidence of our approach's ability to sample from the space of possible networks more widely than some state of the art methods.
Methods that generate networks sharing a given degree distribution and global clustering can induce changes in structural properties other than that controlled for. Diversity in structural properties, in turn, can affect the outcomes of dynamical processes operating on those networks. Since exhaustive sampling is not possible, we propose a novel evolutionary framework for mapping this structural diversity. The three main features of this framework are: (a) subgraph-based encoding of networks, (b) exact mutations based on solving systems of Diophantine equations, and (c) heuristic diversity-driven mechanism to drive resolution changes in the MapElite algorithm. We show that our framework can elicit networks with diversity in their higher-order structure and that this diversity affects the behaviour of the complex contagion model. Through a comparison with state of the art clustered network generation methods, we demonstrate that our approach can uncover a comparably diverse range of networks without needing computationally unfeasible mixing times. Further, we suggest that the subgraph-based encoding provides greater confidence in the diversity of higher-order network structure for low numbers of samples and is the basis for explaining our results with complex contagion model. We believe that this framework could be applied to other complex landscapes that cannot be practically mapped via exhaustive sampling.
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