Finite connected components in infinite directed and multiplex networks with arbitrary degree distributions

Phase transition behavior of finite clusters under localized attack

Most previous studies focused on the giant component to explore the structural robustness of complex networks under malicious attacks. As an important failure mode, localized attacks (LA) can excellently describe the local failure diffusion mechanism of many real scenarios. However, the phase transition behavior of finite clusters, as important network components, has not been clearly understood yet under LA. Here, we develop a percolation framework to theoretically and simulatively study the phase transition behavior of functional nodes belonging to the finite clusters of size greater than or equal to s(s=2,3,…) under LA in this paper. The results reveal that random network exhibits second-order phase transition behavior, the critical threshold p_c increases significantly with increasing s, and the network becomes vulnerable. In particular, we find a new general scaling relationship with the critical exponent δ=-2 between the fraction of finite clusters and s. Furthermore, we apply the theoretical framework to some real networks and predict the phase transition behavior of finite clusters in real networks after they face LA. The framework and results presented in this paper are helpful to promote the design of more critical infrastructures and inspire new insights into studying phase transition behaviors for finite clusters in the network.

Percolation in networks with local homeostatic plasticity

Percolation is a process that impairs network connectedness by deactivating links or nodes. This process features a phase transition that resembles paradigmatic critical transitions in epidemic spreading, biological networks, traffic and transportation systems. Some biological systems, such as networks of neural cells, actively respond to percolation-like damage, which enables these structures to maintain their function after degradation and aging. Here we study percolation in networks that actively respond to link damage by adopting a mechanism resembling synaptic scaling in neurons. We explain critical transitions in such active networks and show that these structures are more resilient to damage as they are able to maintain a stronger connectedness and ability to spread information. Moreover, we uncover the role of local rescaling strategies in biological networks and indicate a possibility of designing smart infrastructures with improved robustness to perturbations.

Effect of volume growth on the percolation threshold in random directed acyclic graphs with a given degree distribution

In every network, a distance between a pair of nodes can be defined as the length of the shortest path connecting these nodes, and therefore one may speak of a ball, its volume, and how it grows as a function of the radius. Spatial networks tend to feature peculiar volume scaling functions, as well as other topological features, including clustering, degree-degree correlation, clique complexes, and heterogeneity. Here we investigate a nongeometric random graph with a given degree distribution and an additional constraint on the volume scaling function. We show that such structures fall into the category of m-colored random graphs and study the percolation transition by using this theory. We prove that for a given degree distribution the percolation threshold for weakly connected components is not affected by the volume growth function. Additionally, we show that the size of the giant component and the cyclomatic number are not affected by volume scaling. These findings may explain the surprisingly good performance of network models that neglect volume scaling. Even though this paper focuses on the implications of the volume growth, the model is generic and might lead to insights in the field of random directed acyclic graphs and their applications.

Enhancing the robustness of a multiplex network leads to multiple discontinuous percolation transitions

Determining design principles that boost the robustness of interdependent networks is a fundamental question of engineering, economics, and biology. It is known that maximizing the degree correlation between replicas of the same node leads to optimal robustness. Here we show that increased robustness might also come at the expense of introducing multiple phase transitions. These results reveal yet another possible source of fragility of multiplex networks that has to be taken into the account during network optimization and design.

Dynamic Networks that Drive the Process of Irreversible Step-Growth Polymerization

Many research fields, reaching from social networks and epidemiology to biology and physics, have experienced great advance from recent developments in random graphs and network theory. In this paper we propose a generic model of step-growth polymerisation as a promising application of the percolation on a directed random graph. This polymerisation process is used to manufacture a broad range of polymeric materials, including: polyesters, polyurethanes, polyamides, and many others. We link features of step-growth polymerisation to the properties of the directed configuration model. In this way, we obtain new analytical expressions describing the polymeric microstructure and compare them to data from experiments and computer simulations. The molecular weight distribution is related to the sizes of connected components, gelation to the emergence of the giant component, and the molecular gyration radii to the Wiener index of these components. A model on this level of generality is instrumental in accelerating the design of new materials and optimizing their properties, as well as it provides a vital link between network science and experimentally observable physics of polymers.

Bond percolation in coloured and multiplex networks

Percolation in complex networks is a process that mimics network degradation and a tool that reveals peculiarities of the network structure. During the course of percolation, the emergent properties of networks undergo non-trivial transformations, which include a phase transition in the connectivity, and in some special cases, multiple phase transitions. Such global transformations are caused by only subtle changes in the degree distribution, which locally describe the network. Here we establish a generic analytic theory that describes how structure and sizes of all connected components in the network are affected by simple and colour-dependent bond percolations. This theory predicts locations of the phase transitions, existence of wide critical regimes that do not vanish in the thermodynamic limit, and a phenomenon of colour switching in small components. These results may be used to design percolation-like processes, optimise network response to percolation, and detect subtle signals preceding network collapse.

Percolation is a tool used to investigate a network’s response as random links are removed. Here the author presents a generic analytic theory to describe how percolation properties are affected in coloured networks, where the colour can represent a network feature such as multiplexity or the belonging to a community.