Scaling up real networks by geometric branching growth

Abstract

Real networks often grow through the sequential addition of new nodes that connect to older ones in the graph. However, many real systems evolve through the branching of fundamental units, whether those be scientific fields, countries, or species. Here, we provide empirical evidence for self-similar growth of network structure in the evolution of real systems—the journal-citation network and the world trade web—and present the geometric branching growth model, which predicts this evolution and explains the symmetries observed. The model produces multiscale unfolding of a network in a sequence of scaled-up replicas preserving network features, including clustering and community structure, at all scales. Practical applications in real instances include the tuning of network size for best response to external influence and finite-size scaling to assess critical behavior under random link failures.

Spatial search by continuous-time quantum walks on renormalized Internet networks

“…In Ref. [30], it was also found that dynamical processes running on the replicas exhibit the same behaviour as in the original network, which enables not only to study dynamics on smaller substrates (with reduced computational costs), but also to analyze the dependence of dynamical processes on the system size even for real-world systems, for which one typically only has access to a single, fixed-size instance [31].…”

“…While synthetic models allow for the generation of networks with an arbitrary number of nodes, with real networks one is limited to the observed ones. In order to obtain replicas of the network with different sizes, in this paper we employ a recently proposed technique based on the geometric renormalization group [30,31]. The method takes the original complex network and uses geometric scaling to renormalize it while preserving the overall structure of the network.…”

“…Note that, while here we focus on the Internet as a paradigmatic example, the approach is general, as demonstrated in [30]. Remarakably, it has been shown that the same replicas can be used to study the inverse process, namely scaling up of real networks, through a process called geometric branching growth [31]. Specifically, in our case, the optimality and scaling analysis on replicas allow us to infer the CTQW behaviour on the original Internet network, and to predict its properties as the Internet grows.…”

The Shortest Path to Network Geometry

“…However, many real systems evolve in a self-similar way that preserves their topology throughout the growth process over a long time span that is better explained by branching of fundamental units -whether those be scientific fields or countries (Zheng et al [2021]). The Geometric Branching Growth model predicts this evolution and explains the symmetries observed (Zheng et al [2021]). The model produces multiscale unfolding of a network in a sequence of scaled-up replicas preserving network features, including clustering and community structure.…”

Dimension matters when modeling network communities in hyperbolic spaces

“…In this framework, each node exists on a surface where a radial dimension encodes its popularity, or how likely it is to have many neighbors, and an angular dimension encodes the similarity between nodes, where similar nodes are more likely to be related [5]. The successes of hyperbolic network geometry cover a wide range of practical applications, like predicting economic patterns across time [6], making sense of the resilience of the Internet [1] or modeling information flow in the brain [2,7], to name a few. Furthermore, hyperbolic space is the only known metric space on which maximum-entropy random graphs can reproduce real network properties like clustering, sparsity, and heterogeneous degree distributions all at once [4].…”