Multiple phase transitions in networks of directed networks

Robustness of Interdependent Networks with Weak Dependency Based on Bond Percolation

Real-world systems interact with one another via dependency connectivities. Dependency connectivities make systems less robust because failures may spread iteratively among systems via dependency links. Most previous studies have assumed that two nodes connected by a dependency link are strongly dependent on each other; that is, if one node fails, its dependent partner would also immediately fail. However, in many real scenarios, nodes from different networks may be weakly dependent, and links may fail instead of nodes. How interdependent networks with weak dependency react to link failures remains unknown. In this paper, we build a model of fully interdependent networks with weak dependency and define a parameter α in order to describe the node-coupling strength. If a node fails, its dependent partner has a probability of failing of 1−α. Then, we develop an analytical tool for analyzing the robustness of interdependent networks with weak dependency under link failures, with which we can accurately predict the system robustness when 1−p fractions of links are randomly removed. We find that as the node coupling strength increases, interdependent networks show a discontinuous phase transition when α<αc and a continuous phase transition when α>αc. Compared to site percolation with nodes being attacked, the crossover points αc are larger in the bond percolation with links being attacked. This finding can give us some suggestions for designing and protecting systems in which link failures can happen.

Group percolation in interdependent networks with reinforcement network layer

In many real-world interdependent network systems, nodes often work together to form groups, which can enhance robustness to resist risks. However, previous group percolation models are always of a first-order phase transition, regardless of the group size distribution. This motivates us to investigate a generalized model for group percolation in interdependent networks with a reinforcement network layer to eliminate collapse. Some backup devices that are equipped for a density ρ of reinforced nodes constitute the reinforcement network layer. For each group, we assume that at least one node of the group can function in one network and a node in another network depends on the group to function. We find that increasing the density ρ of reinforcement nodes and the size S of the dependency group can significantly enhance the robustness of interdependent networks. Importantly, we find the existence of a hybrid phase transition behavior and propose a method for calculating the shift point of percolation types. The most interesting finding is the exact universal solution to the minimal density _ρ of reinforced nodes (or the minimum group size _S) to prevent abrupt collapse for Erdős-Rényi, scale-free, and regular random interdependent networks. Furthermore, we present the validity of the analytic solutions for a triple point _ρ (or _S ), the corresponding phase transition point _p , and second-order phase transition points _p in interdependent networks. These findings might yield a broad perspective for designing more resilient interdependent infrastructure networks.

Identification of Biomarker in Brain-specific Gene Regulatory Network Using Structural Controllability Analysis

Brain tumor research has been stapled for human health while brain network research is crucial for us to understand brain activity. Here the structural controllability theory is applied to study three human brain-specific gene regulatory networks, including forebrain gene regulatory network, hindbrain gene regulatory network and neuron associated cells cancer related gene regulatory network, whose nodes are neural genes and the edges represent the gene expression regulation among the genes. The nodes are classified into two classes: critical nodes and ordinary nodes, based on the change of the number of driver nodes upon its removal. Eight topological properties (out-degree DO, in-degree DI, degree D, betweenness B, closeness CA, in-closeness CI, out-closeness CO and clustering coefficient CC) are calculated in this paper and the results prove that the critical genes have higher score of topological properties than the ordinary genes. Then two bioinformatic analysis are used to explore the biologic significance of the critical genes. On the one hand, the enrichment scores in several kinds of gene databases are calculated and reveal that the critical nodes are richer in essential genes, cancer genes and the neuron related disease genes than the ordinary nodes, which indicates that the critical nodes may be the biomarker in brain-specific gene regulatory network. On the other hand, GO analysis and KEGG pathway analysis are applied on them and the results show that the critical genes mainly take part in 14 KEGG pathways that are transcriptional misregulation in cancer, pathways in cancer and so on, which indicates that the critical genes are related to the brain tumor. Finally, by deleting the edges or routines in the network, the robustness analysis of node classification is realized, and the robustness of node classification is proved. The comparison of neuron associated cells cancer related GRN (Gene Regulatory Network) and normal brain-specific GRNs (including forebrain and hindbrain GRN) shows that the neuron-related cell cancer-related gene regulatory network is more robust than other types.

Robustness and lethality in multilayer biological molecular networks

Robustness is a prominent feature of most biological systems. Most previous related studies have been focused on homogeneous molecular networks. Here we propose a comprehensive framework for understanding how the interactions between genes, proteins and metabolites contribute to the determinants of robustness in a heterogeneous biological network. We integrate heterogeneous sources of data to construct a multilayer interaction network composed of a gene regulatory layer, a protein-protein interaction layer, and a metabolic layer. We design a simulated perturbation process to characterize the contribution of each gene to the overall system's robustness, and find that influential genes are enriched in essential and cancer genes. We show that the proposed mechanism predicts a higher vulnerability of the metabolic layer to perturbations applied to genes associated with metabolic diseases. Furthermore, we find that the real network is comparably or more robust than expected in multiple random realizations. Finally, we analytically derive the expected robustness of multilayer biological networks starting from the degree distributions within and between layers. These results provide insights into the non-trivial dynamics occurring in the cell after a genetic perturbation is applied, confirming the importance of including the coupling between different layers of interaction in models of complex biological systems.

How motifs condition critical thresholds for tipping cascades in complex networks: Linking micro- to macro-scales

In this study, we investigate how specific micro-interaction structures (motifs) affect the occurrence of tipping cascades on networks of stylized tipping elements. We compare the properties of cascades in Erdős-Rényi networks and an exemplary moisture recycling network of the Amazon rainforest. Within these networks, decisive small-scale motifs are the feed forward loop, the secondary feed forward loop, the zero loop, and the neighboring loop. Of all motifs, the feed forward loop motif stands out in tipping cascades since it decreases the critical coupling strength necessary to initiate a cascade more than the other motifs. We find that for this motif, the reduction of critical coupling strength is 11% less than the critical coupling of a pair of tipping elements. For highly connected networks, our analysis reveals that coupled feed forward loops coincide with a strong 90% decrease in the critical coupling strength. For the highly clustered moisture recycling network in the Amazon, we observe regions of a very high motif occurrence for each of the four investigated motifs, suggesting that these regions are more vulnerable. The occurrence of motifs is found to be one order of magnitude higher than in a random Erdős-Rényi network. This emphasizes the importance of local interaction structures for the emergence of global cascades and the stability of the network as a whole.