DOI•
Network processes on clique-networks with high average degree: the limited effect of higher-order structure
30 Apr 2021-Vol. 2, Iss: 4, pp 045011
TL;DR: In this paper, the effect of local clique structures on network processes is investigated and it is shown that when the average degree of a vertex is large, the influence of the deviations from the locally tree-like structure is small.
Abstract: In this paper, we investigate the effect of local structures on network processes. We investigate a random graph model that incorporates local clique structures to deviate from the locally tree-like behavior of most standard random graph models. For the process of bond percolation, we derive analytical approximations for large outbreaks and the critical percolation value. Interestingly, these derivations show that when the average degree of a vertex is large, the influence of the deviations from the locally tree-like structure is small. Our simulations show that this insensitivity to local clique structures often already kicks in for networks with average degrees as low as 6. Furthermore, we show that the different behavior of bond percolation on clustered networks compared to tree-like networks that was found in previous works can be almost completely attributed to differences in degree sequences rather than differences in clustering structures. We finally show that these results also extend to completely different types of dynamics, by deriving similar conclusions and simulations for the Kuramoto model on the same types of clustered and non-clustered networks.
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TL;DR: This paper defines a joint-degree correlation function for vertices in the giant component of clustered configuration model networks which are composed of clique subgraphs and uses this model to investigate the organization among nearest-neighbor sub graphs for random graphs as a function of subgraph topology as well as clustering.
Abstract: Correlations among the degrees of vertices in random graphs often occur when clustering is present. In this paper we define a joint-degree correlation function for vertices in the giant component of clustered configuration model networks which are composed of clique subgraphs. We use this model to investigate, in detail, the organization among nearest-neighbor subgraphs for random graphs as a function of subgraph topology as well as clustering. We find an expression for the average joint degree of a neighbor in the giant component at the critical point for these networks. Finally, we introduce a novel edge-disjoint clique decomposition algorithm and investigate the correlations between the subgraphs of empirical networks.
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References
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TL;DR: A dynamical model of epidemic spreading on complex networks in which there are explicit correlations among the node's connectivities finds an epidemic threshold inversely proportional to the largest eigenvalue of the connectivity matrix that gives the average number of links.
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TL;DR: It is shown how standard random-graph models can be generalized to incorporate clustering and give exact solutions for various properties of the resulting networks, including sizes of network components, size of the giant component if there is one, position of the phase transition at which the giant components forms, and position ofThe phase transition for percolation on the network.
Abstract: We offer a solution to a long-standing problem in the theory of networks, the creation of a plausible, solvable model of a network that displays clustering or transitivity---the propensity for two neighbors of a network node also to be neighbors of one another. We show how standard random-graph models can be generalized to incorporate clustering and give exact solutions for various properties of the resulting networks, including sizes of network components, size of the giant component if there is one, position of the phase transition at which the giant component forms, and position of the phase transition for percolation on the network.
476 citations