Network cloning unfolds the effect of clustering on dynamical processes

Abstract

We introduce network L-cloning, a technique for creating ensembles of random networks from any given real-world or artificial network. Each member of the ensemble is an L-cloned network constructed from L copies of the original network. The degree distribution of an L-cloned network and, more importantly, the degree-degree correlation between and beyond nearest neighbors are identical to those of the original network. The density of triangles in an L-cloned network, and hence its clustering coefficient, is reduced by a factor of L compared to those of the original network. Furthermore, the density of loops of any fixed length approaches zero for sufficiently large values of L. Other variants of L-cloning allow us to keep intact the short loops of certain lengths. As an application, we employ these network cloning methods to investigate the effect of short loops on dynamical processes running on networks and to inspect the accuracy of corresponding tree-based theories. We demonstrate that dynamics on L-cloned networks (with sufficiently large L) are accurately described by the so-called adjacency tree-based theories, examples of which include the message passing technique, some pair approximation methods, and the belief propagation algorithm used respectively to study bond percolation, SI epidemics, and the Ising model.

Beyond the locally treelike approximation for percolation on real networks

“…An approach based on the same rationale was also used by Hamilton and Pryadko to study site percolation in isolated networks [18], and by Radicchi in the analysis of bond and site percolation models in interdependent networks [19]. These methods still suffer from a fundamental limitation: they are based on the locally tree-like approximation [6][7][8], and as such they are potentially not reliable for networks with nonnegligible density of triangles, or short loops in general [20,21].In this paper, we make a step forward, by generalizing the approach developed in [16,18] to clustered networks. Through a systematic analysis of about one hundred realworld networks as well as clustered synthetic ones, we demonstrate that our framework provides excellent pre-diction of the whole phase diagram for the site percolation model.…”

Message-Passing Methods for Complex Contagions

“…The corresponding values for the Facebook networks are also very close, indicating that the configuration-model theory is very accurate for these networks (as also found in [13,43]). For the other networks, there is a considerable difference between θ config and θ crit , indicating that the configuration-model result is inaccurate on these networks (although it is also known that the message-passing approach, being based on a tree-like assumption of independence of messages [44], is inaccurate for spatial networks [35,45] such as the power-grid example in this table). applying configuration-model theory (which uses only the degree distribution of a network).…”

Smeared phase transitions in percolation on real complex networks

“…The MPA takes the entire network structure as an input, but then ignores loops when solving the percolation process. Because of this approximation, the MPA is effectively solving percolation not on the true network, but on an infinite network where there exist an infinite number of copies of every node [19]. In practice, this means that any modular structure is mapped to a core-periphery structure.…”