Let $\mathcal{V}$ and $\mathcal{U}$ be the point sets of two independent homogeneous Poisson processes on $\mathbb{R}^d$. A graph $\mathcal{G}_\mathcal{V}$ with vertex set $\mathcal{V}$ is constructed by first connecting pairs of points (v, u) with $v\in\mathcal{V}$ and $u\in\mathcal{U}$ independently with probability $g(v-u)$, where g is a non-increasing radial function, and then connecting two points $v_1,v_2\in\mathcal{V}$ if and only if they have a joint neighbor $u\in\mathcal{U}$. This gives rise to a random intersection graph on $\mathbb{R}^d$. Local properties of the graph, including the degree distribution, are investigated and quantified in terms of the intensities of the underlying Poisson processes and the function g. Furthermore, the percolation properties of the graph are characterized and shown to differ depending on whether g has bounded or unbounded support.

We consider linear preferential attachment trees with additive fitness, where fitness is the random initial vertex attractiveness. We show that when the fitnesses are independent and identically distributed and have positive bounded support, the local weak limit can be constructed using a sequence of mixed Poisson point processes. We also provide a rate of convergence for the total variation distance between the r-neighbourhoods of a uniformly chosen vertex in the preferential attachment tree and the root vertex of the local weak limit. The proof uses a Pólya urn representation of the model, for which we give new estimates for the beta and product beta variables in its construction. As applications, we obtain limiting results and convergence rates for the degrees of the uniformly chosen vertex and its ancestors, where the latter are the vertices that are on the path between the uniformly chosen vertex and the initial vertex.

There are many deviation sources in the assembly process of aircraft complex thin-walled structures. To get important factors that affect quality, it is crucial to diagnose the key deviation resources. The deviation transfer between deviation sources and assembly parts has the characteristics of small sample size, nonlinearity, and strong coupling, so it is difficult to diagnose the key deviation sources by constructing assembly dimension chains. Therefore, based on the deviation detection data, transfer entropy and complex network theory are introduced. Integrating the depth-first traversal algorithm with degree centrality theory, a key deviation diagnosis method for complex thin-walled structures is proposed based on weighted transfer entropy and complex networks. The application shows that key deviation sources that affect assembly quality can be accurately identified by the key deviation source diagnosis method based on complex networks and weighted transfer entropy.

We study a general model of recursive trees where vertices are equipped with independent weights and at each time-step a vertex is sampled with probability proportional to its fitness function, which is a function of its weight and degree, and connects to $\ell$ new-coming vertices. Under a certain technical assumption, applying the theory of Crump–Mode–Jagers branching processes, we derive formulas for the limiting distributions of the proportion of vertices with a given degree and weight, and proportion of edges with endpoint having a certain weight. As an application of this theorem, we rigorously prove observations of Bianconi related to the evolving Cayley tree (Phys. Rev. E 66, paper no. 036116, 2002). We also study the process in depth when the technical condition can fail in the particular case when the fitness function is affine, a model we call ‘generalised preferential attachment with fitness’. We show that this model can exhibit condensation, where a positive proportion of edges accumulates around vertices with maximal weight, or, more drastically, can have a degenerate limiting degree distribution, where the entire proportion of edges accumulates around these vertices. Finally, we prove stochastic convergence for the degree distribution under a different assumption of a strong law of large numbers for the partition function associated with the process.

We provide a general purpose result for the coupling of exploration processes of random graphs, both undirected and directed, with their local weak limits when this limit is a marked Galton–Watson process. This class includes in particular the configuration model and the family of inhomogeneous random graphs with rank-1 kernel. Vertices in the graph are allowed to have attributes on a general separable metric space and can potentially influence the construction of the graph itself. The coupling holds for any fixed depth of a breadth-first exploration process.

Data-driven analysis of complex networks has been in the focus of research for decades. An important area of research is to study how well real networks can be described with a small selection of metrics, furthermore how well network models can capture the relations between graph metrics observed in real networks. In this paper, we apply machine-learning techniques to investigate the aforementioned problems. We study 500 real-world networks along with 2000 synthetic networks generated by four frequently used network models with previously calibrated parameters to make the generated graphs as similar to the real networks as possible. This paper unifies several branches of data-driven complex network analysis, such as the study of graph metrics and their pair-wise relationships, network similarity estimation, model calibration, and graph classification. We find that the correlation profiles of the structural measures significantly differ across network domains and the domain can be efficiently determined using a small selection of graph metrics. The structural properties of the network models with fixed parameters are robust enough to perform parameter calibration. The goodness-of-fit of the network models highly depends on the network domain. By solving classification problems, we find that the models lack the capability of generating a graph with a high clustering coefficient and relatively large diameter simultaneously. On the other hand, models are able to capture exactly the degree-distribution-related metrics.

We prove concentration inequality results for geometric graph properties of an instance of the Cooper–Frieze [5] preferential attachment model with edge-steps. More precisely, we investigate a random graph model that at each time $t\in \mathbb{N}$ , with probability p adds a new vertex to the graph (a vertex-step occurs) or with probability $1-p$ an edge connecting two existent vertices is added (an edge-step occurs). We prove concentration results for the global clustering coefficient as well as the clique number. More formally, we prove that the global clustering, with high probability, decays as $t^{-\gamma(p)}$ for a positive function $\gamma$ of p, whereas the clique number of these graphs is, up to subpolynomially small factors, of order $t^{(1-p)/(2-p)}$ .

We study propagation of avalanches in a certain excitable network. The model is a particular case of the one introduced by Larremore et al. (Phys. Rev. E, 2012) and is mathematically equivalent to an endemic variation of the Reed–Frost epidemic model introduced by Longini (Math. Biosci., 1980). Two types of heuristic approximation are frequently used for models of this type in applications: a branching process for avalanches of a small size at the beginning of the process and a deterministic dynamical system once the avalanche spreads to a significant fraction of a large network. In this paper we prove several results concerning the exact relation between the avalanche model and these limits, including rates of convergence and rigorous bounds for common characteristics of the model.

We investigate the impact of misinformation about the contact structure on the ability to predict disease outbreaks. We base our study on 31 empirical temporal networks and tune the frequencies in errors in the node identities or time stamps of contacts. We find that for both these spreading scenarios, the maximal misprediction of both the outbreak size and time to extinction follows an stretched exponential convergence as a function of the error frequency. We furthermore determine the temporal-network structural factors influencing the parameters of this convergence.

Neural mass models are ubiquitous in large-scale brain modelling. At the node level, they are written in terms of a set of ordinary differential equations with a non-linearity that is typically a sigmoidal shape. Using structural data from brain atlases, they may be connected into a network to investigate the emergence of functional dynamic states, such as synchrony. With the simple restriction of the classic sigmoidal non-linearity to a piecewise linear caricature, we show that the famous Wilson–Cowan neural mass model can be explicitly analysed at both the node and network level. The construction of periodic orbits at the node level is achieved by patching together matrix exponential solutions, and stability is determined using Floquet theory. For networks with interactions described by circulant matrices, we show that the stability of the synchronous state can be determined in terms of a low-dimensional Floquet problem parameterised by the eigenvalues of the interaction matrix. Moreover, this network Floquet problem is readily solved using linear algebra to predict the onset of spatio-temporal network patterns arising from a synchronous instability. We further consider the case of a discontinuous choice for the node non-linearity, namely the replacement of the sigmoid by a Heaviside non-linearity. This gives rise to a continuous-time switching network. At the node level, this allows for the existence of unstable sliding periodic orbits, which we explicitly construct. The stability of a periodic orbit is now treated with a modification of Floquet theory to treat the evolution of small perturbations through switching manifolds via the use of saltation matrices. At the network level, the stability analysis of the synchronous state is considerably more challenging. Here, we report on the use of ideas originally developed for the study of Glass networks to treat the stability of periodic network states in neural mass models with discontinuous interactions.

Most people simultaneously belong to several distinct social networks, in which their relations can be different. They have opinions about certain topics, which they share and spread on these networks, and are influenced by the opinions of other persons. In this paper, we build upon this observation to propose a new nodal centrality measure for multiplex networks. Our measure, called Opinion centrality, is based on a stochastic model representing opinion propagation dynamics in such a network. We formulate an optimization problem consisting in maximizing the opinion of the whole network when controlling an external influence able to affect each node individually. We find a mathematical closed form of this problem, and use its solution to derive our centrality measure. According to the opinion centrality, the more a node is worth investing external influence, and the more it is central. We perform an empirical study of the proposed centrality over a toy network, as well as a collection of real-world networks. Our measure is generally negatively correlated with existing multiplex centrality measures, and highlights different types of nodes, accordingly to its definition.

We propose two random network models for complex networks, which are treelike and always grow very fast. One is the uniform model and the other is the preferential attachment model, and both of them depends on a parameter 0<p<1. We first briefly discuss the network sizes, each of which can be corresponding to a supercritical branching process. And then we mainly study the degree distributions of both models. The asymptotic degree distribution of the first one with any parameter 0<p<1 is a geometric distribution with parameter 1/2, whereas that of the second one, which depends on p, can be uniquely determined by a functional equation of its probability generating function.

We introduce NetworKit, an open-source software package for analyzing the structure of large complex networks. Appropriate algorithmic solutions are required to handle increasingly common large graph data sets containing up to billions of connections. We describe the methodology applied to develop scalable solutions to network analysis problems, including techniques like parallelization, heuristics for computationally expensive problems, efficient data structures, and modular software architecture. Our goal for the software is to package results of our algorithm engineering efforts and put them into the hands of domain experts. NetworKit is implemented as a hybrid combining the kernels written in C++ with a Python frontend, enabling integration into the Python ecosystem of tested tools for data analysis and scientific computing. The package provides a wide range of functionality (including common and novel analytics algorithms and graph generators) and does so via a convenient interface. In an experimental comparison with related software, NetworKit shows the best performance on a range of typical analysis tasks.

The central theme of complex systems research is to understand the emergent macroscopic properties of a system from the interplay of its microscopic constituents. The emergence of macroscopic properties is often intimately related to the structure of the microscopic interactions. Here, we present an analytical approach for deriving necessary conditions that an interaction network has to obey in order to support a given type of macroscopic behaviour. The approach is based on a graphical notation, which allows rewriting Jacobi's signature criterion in an interpretable form and which can be applied to many systems of symmetrically coupled units. The derived conditions pertain to structures on all scales, ranging from individual nodes to the interaction network as a whole. For the purpose of illustration, we consider the example of synchronization, specifically the (heterogeneous) Kuramoto model and an adaptive variant. The results complete and extend the previous analysis of Do et al. (2012Phys. Rev. Lett.108, 194102).

We study the behaviour of network diffusions based on the PageRank random walk from a set of seed nodes. These diffusions are known to reveal small, localized clusters (or communities), and also large macro-scale clusters by varying a parameter that has a dual-interpretation as an accuracy bound and as a regularization level. We propose a new method that quickly approximates the result of the diffusion for all values of this parameter. Our method efficiently generates an approximate solution path or regularization path associated with a PageRank diffusion, and it reveals cluster structures at multiple size-scales between small and large. We formally prove a runtime bound on this method that is independent of the size of the network, and we investigate multiple optimizations to our method that can be more practical in some settings. We demonstrate that these methods identify refined clustering structure on a number of real-world networks with up to 2 billion edges.

The master stability function is a powerful tool for determining synchrony in high-dimensional networks of coupled limit cycle oscillators. In part, this approach relies on the analysis of a low-dimensional variational equation around a periodic orbit. For smooth dynamical systems, this orbit is not generically available in closed form. However, many models in physics, engineering and biology admit to non-smooth piece-wise linear caricatures, for which it is possible to construct periodic orbits without recourse to numerical evolution of trajectories. A classic example is the McKean model of an excitable system that has been extensively studied in the mathematical neuroscience community. Understandably, the master stability function cannot be immediately applied to networks of such non-smooth elements. Here, we show how to extend the master stability function to non-smooth planar piece-wise linear systems, and in the process demonstrate that considerable insight into network dynamics can be obtained. In illustration, we highlight an inverse period-doubling route to synchrony, under variation in coupling strength, in globally linearly coupled networks for which the node dynamics is poised near a homoclinic bifurcation. Moreover, for a star graph, we establish a mechanism for achieving so-called remote synchronisation (where the hub oscillator does not synchronise with the rest of the network), even when all the oscillators are identical. We contrast this with node dynamics close to a non-smooth Andronov–Hopf bifurcation and also a saddle node bifurcation of limit cycles, for which no such bifurcation of synchrony occurs.