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Percolation theory has been widely used to study phase transitions in complex networked systems. It has also successfully explained several macroscopic phenomena across different fields. Yet, the existent theoretical framework for percolation places the focus on the direct interactions among the systems components, while recent empirical observations have shown that indirect interactions are common in many systems like ecological and social networks, among others. Here, we propose a new percolation framework that accounts for indirect interactions, which allows to generalize the current theoretical body and understand the role of the underlying indirect influence of the components of a networked system on its macroscopic behavior. We report a rich phenomenology in which first-order, second-order or hybrid phase transitions are possible depending on whether the links of the substrate network are directed, undirected or a mix, respectively. We also present an analytical framework to characterize the proposed induced percolation, paving the way to further understand network dynamics with indirect interactions.
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Understanding the mechanisms of complex systems is very important. Networked dynamical system, that understanding a system as a group of nodes interacting on a given network according to certain dynamic rules, is a powerful tool for modelling complex systems. However, finding such models according to the time series of behaviors is hard. Conventional methods can work well only on small networks and some types of dynamics. Based on a Bernoulli network generator and a Markov dynamics learner, this paper proposes a unified framework for Automated Interaction network and Dynamics Discovery (AIDD) on various network structures and different types of dynamics. The experiments show that AIDD can be applied on large systems with thousands of nodes. AIDD can not only infer the unknown network structure and states for hidden nodes but also can reconstruct the real gene regulatory network based on the noisy, incomplete, and being disturbed data which is closed to real situations. We further propose a new method to test data-driven models by experiments of control. We optimize a controller on the learned model, and then apply it on both the learned and the ground truth models. The results show that both of them behave similarly under the same control law, which means AIDD models have learned the real network dynamics correctly.
We propose a maximally disassortative (MD) network model which realizes a maximally negative degree-degree correlation, and study its percolation transition to discuss the effect of a strong degree-degree correlation on the percolation critical behaviors. Using the generating function method for bipartite networks, we analytically derive the percolation threshold and the order parameter critical exponent, $beta$. For the MD scale-free networks, whose degree distribution is $P(k) sim k^{-gamma}$, we show that the exponent, $beta$, for the MD networks and corresponding uncorrelated networks are same for $gamma>3$ but are different for $2<gamma<3$. A strong degree-degree correlation significantly affects the percolation critical behavior in heavy-tailed scale-free networks. Our analytical results for the critical exponents are numerically confirmed by a finite-size scaling argument.
Industrial symbiosis involves creating integrated cycles of by-products and waste between networks of industrial actors in order to maximize economic value, while at the same time minimizing environmental strain. In such a network, the global environmental strain is no longer equal to the sum of the environmental strain of the individual actors, but it is dependent on how well the network performs as a whole. The development of methods to understand, manage or optimize such networks remains an open issue. In this paper we put forward a simulation model of by-product flow between industrial actors. The goal is to introduce a method for modelling symbiotic exchanges from a macro perspective. The model takes into account the effect of two main mechanisms on a multi-objective optimization of symbiotic processes. First it allows us to study the effect of geographical properties of the economic system, said differently, where actors are divided in space. Second, it allows us to study the effect of clustering complementary actors together as a function of distance, by means of a spatial correlation between the actors by-products. Our simulations unveil patterns that are relevant for macro-level policy. First, our results show that the geographical properties are an important factor for the macro performance of symbiotic processes. Second, spatial correlations, which can be interpreted as planned clusters such as Eco-industrial parks, can lead to a very effective macro performance, but only if these are strictly implemented. Finally, we provide a proof of concept by comparing the model to real world data from the European Pollutant Release and Transfer Register database using georeferencing of the companies in the dataset. This work opens up research opportunities in interactive data-driven models and platforms to support real-world implementation of industrial symbiosis.
Higher order interactions are increasingly recognised as a fundamental aspect of complex systems ranging from the brain to social contact networks. Hypergraph as well as simplicial complexes capture the higher-order interactions of complex systems and allow to investigate the relation between their higher-order structure and their function. Here we establish a general framework for assessing hypergraph robustness and we characterize the critical properties of simple and higher-order percolation processes. This general framework builds on the formulation of the random multiplex hypergraph ensemble where each layer is characterized by hyperedges of given cardinality. We reveal the relation between higher-order percolation processes in random multiplex hypergraphs, interdependent percolation of multiplex networks and K-core percolation. The structural correlations of the random multiplex hypergraphs are shown to have a significant effect on their percolation properties. The wide range of critical behaviors observed for higher-order percolation processes on multiplex hypergraphs elucidates the mechanisms responsible for the emergence of discontinuous transition and uncovers interesting critical properties which can be applied to the study of epidemic spreading and contagion processes on higher-order networks.
In the last two decades, network science has blossomed and influenced various fields, such as statistical physics, computer science, biology and sociology, from the perspective of the heterogeneous interaction patterns of components composing the complex systems. As a paradigm for random and semi-random connectivity, percolation model plays a key role in the development of network science and its applications. On the one hand, the concepts and analytical methods, such as the emergence of the giant cluster, the finite-size scaling, and the mean-field method, which are intimately related to the percolation theory, are employed to quantify and solve some core problems of networks. On the other hand, the insights into the percolation theory also facilitate the understanding of networked systems, such as robustness, epidemic spreading, vital node identification, and community detection. Meanwhile, network science also brings some new issues to the percolation theory itself, such as percolation of strong heterogeneous systems, topological transition of networks beyond pairwise interactions, and emergence of a giant cluster with mutual connections. So far, the percolation theory has already percolated into the researches of structure analysis and dynamic modeling in network science. Understanding the percolation theory should help the study of many fields in network science, including the still opening questions in the frontiers of networks, such as networks beyond pairwise interactions, temporal networks, and network of networks. The intention of this paper is to offer an overview of these applications, as well as the basic theory of percolation transition on network systems.
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