Scale-free Random Graphs Research Articles

Barely Supercritical Percolation on Poissonian Scale-free Networks

We study the giant component problem slightly above the critical regime for percolation on Poissonian random graphs in the scale-free regime, where the vertex weights and degrees have a diverging second moment. Critical percolation on scale-free random graphs has been observed to have incredibly subtle features that are markedly different compared to those in random graphs with a converging second moment. In particular, the critical window for percolation depends sensitively on whether we consider single- or multi-edge versions of the Poissonian random graph. In this paper, and together with our companion paper [3], we build a bridge between these two cases. Our results characterize the part of the barely supercritical regime where the size of the giant components are approximately same for the single- and multi-edge settings. The methods for establishing concentration of giant for the single- and multi-edge versions are quite different. While the analysis in the multi-edge case is based on scaling limits of exploration processes, the single-edge setting requires identi fication of a core structure inside certain high-degree vertices that forms the giant component.

The contact process on scale-free geometric random graphs

We study the contact process on a class of geometric random graphs with scale-free degree distribution, defined on a Poisson point process on Rd. This class includes the age-dependent random connection model and the soft Boolean model. In the ultrasmall regime of these random graphs we provide exact asymptotics for the non-extinction probability when the rate of infection spread is small and show for a finite version of these graphs that the extinction time is of exponential order in the size of the graph.

N-strain epidemic model using bond percolation.

In this paper we examine the emergent structures of random networks that have undergone bond percolation an arbitrary, but finite, number of times. We define two types of sequential branching processes: a competitive branching process, in which each iteration performs bond percolation on the residual graph (RG) resulting from previous generations, and a collaborative branching process, where percolation is performed on the giant connected component (GCC) instead. We investigate the behavior of these models, including the expected size of the GCC for a given generation, the critical percolation probability, and other topological properties of the resulting graph structures using the analytically exact method of generating functions. We explore this model for Erdős-Renyi and scale-free random graphs. This model can be interpreted as a seasonal N-strain model of disease spreading.

Large degrees in scale-free inhomogeneous random graphs

We consider a class of scale-free inhomogeneous random graphs, which includes some long-range percolation models. We study the maximum degree in such graphs in a growing observation window and show that its limiting distribution is Frechet. We achieve this by proving convergence of the underlying point process of the degrees to a certain Poisson process. Estimating the index of the power-law tail for the typical degree distribution is an important question in statistics. We prove consistency of the Hill estimator for the inverse of the tail exponent of the typical degree distribution.

Penalising transmission to hubs in scale-free spatial random graphs

Penalising transmission to hubs in scale-free spatial random graphs

Critical Percolation on Scale-Free Random Graphs: New Universality Class for the Configuration Model

In this paper, we study the critical behavior of percolation on a configuration model with degree distribution satisfying an infinite second-moment condition, which includes power-law degrees with exponent $$\tau \in (2,3)$$ . It is well known that, in this regime, many canonical random graph models, such as the configuration model, are robust in the sense that the giant component is not destroyed when the percolation probability stays bounded away from zero. Thus, the critical behavior is observed when the percolation probability tends to zero with the network size, despite of the fact that the average degree remains bounded. In this paper, we initiate the study of critical random graphs in the infinite second-moment regime by identifying the critical window for the configuration model. We prove scaling limits for component sizes and surplus edges, and show that the maximum diameter the critical components is of order $$\log n$$ , which contrasts with the previous universality classes arising in the literature. This introduces a third and novel universality class for the critical behavior of percolation on random networks, that is not covered by the multiplicative coalescent framework due to Aldous and Limic (Electron J Probab 3(3):1–59, 1998). We also prove concentration of the component sizes outside the critical window, and that a unique, complex giant component emerges after the critical window. This completes the picture for the percolation phase transition on the configuration model.

Degree correlations in scale-free random graph models

AbstractWe study the average nearest-neighbour degree a(k) of vertices with degree k. In many real-world networks with power-law degree distribution, a(k) falls off with k, a property ascribed to the constraint that any two vertices are connected by at most one edge. We show that a(k) indeed decays with k in three simple random graph models with power-law degrees: the erased configuration model, the rank-1 inhomogeneous random graph, and the hyperbolic random graph. We find that in the large-network limit for all three null models, a(k) starts to decay beyond $n^{(\tau-2)/(\tau-1)}$ and then settles on a power law $a(k)\sim k^{\tau-3}$, with $\tau$ the degree exponent.

Counting Cliques and Cycles in Scale-Free Inhomogeneous Random Graphs

Scale-free networks contain many small cliques and cycles. We model such networks as inhomogeneous random graphs with regularly varying infinite-variance weights. For these models, the number of cliques and cycles have exact integral expressions amenable to asymptotic analysis. We obtain various asymptotic descriptions for how the average number of cliques and cycles, of any size, grow with the network size. For the cycle asymptotics we invoke the theory of circulant matrices.

Diameter in ultra-small scale-free random graphs.

It is well known that many random graphs with infinite variance degrees are ultra‐small. More precisely, for configuration models and preferential attachment models where the proportion of vertices of degree at least k is approximately k −(τ − 1) with τ ∈ (2,3), typical distances between pairs of vertices in a graph of size n are asymptotic to 2loglogn|log(τ−2)| and 4loglogn|log(τ−2)|, respectively. In this paper, we investigate the behavior of the diameter in such models. We show that the diameter is of order loglogn precisely when the minimal forward degree d fwd of vertices is at least 2. We identify the exact constant, which equals that of the typical distances plus 2/logdfwd. Interestingly, the proof for both models follows identical steps, even though the models are quite different in nature.

Efficiency of complex networks under failures and attacks: A percolation approach

Network efficiency, defined as the average of the reciprocal of the shortest path lengths between each node pair in a network, indicates how efficiently the network propagates information. The change of efficiency caused by failures or attacks can be used to assess the robustness or resilience of networks. However, due to the uncertainties of failures or attacks, the lengths of the shortest paths and in turn efficiency in a disrupted network cannot be easily calculated. The normal practice in the literature is to estimate the efficiency in different scenarios by simulation. In this paper, we propose an analytical way to assess the efficiency of complex networks under failures and attacks using percolation theory. We find that the efficiency of an affected network is exactly the product of global connectivity and local connectivity, where global connectivity refers to the size of the giant component, and local connectivity represents the average number of neighbors with different distances. This approach not only provides a more efficient and systematic way to analyze efficiency, but also reveals the relation between efficiency and connectivity. We discuss the application of our approach to scale-free networks and random graphs.

Spin-glass-like transition in the majority-vote model with anticonformists

Majority-vote model on scale-free networks and random graphs is investigated in which a randomly chosen fraction p of agents (called anticonformists) follows an antiferromagnetic update rule, i.e., they assume, with probability governed by a parameter q (0 < q < 1∕2), the opinion opposite to that of the majority of their neighbors, while the remaining 1 − p fraction of agents (conformists) follows the usual ferromagnetic update rule assuming, with probability governed by the same parameter q, the opinion in accordance with that of the majority of their neighbors. For p = 1 it is shown by Monte Carlo simulations and using the Binder cumulants method that for decreasing q the model undergoes second-order phase transition from a disordered (paramagnetic) state to a spin-glass-like state, characterized by a non-zero value of the spin-glass order parameter measuring the overlap of agents’ opinions in two replicas of the system, and simultaneously by the magnetization close to zero. In the case of the model on scale-free networks the critical value of the parameter q weakly depends on the details of the degree distribution. As p is decreased, the critical value of q falls quickly to zero and only the disordered phase is observed. On the other hand, for p close to zero for decreasing q the usual ferromagnetic transition is observed.

Unimodular lattice triangulations as small-world and scale-free random graphs

Real-world networks, e.g., the social relations or world-wide-web graphs, exhibit both small-world and scale-free behaviour. We interpret lattice triangulations as planar graphs by identifying triangulation vertices with graph nodes and one-dimensional simplices with edges. Since these triangulations are ergodic with respect to a certain Pachner flip, applying different Monte Carlo simulations enables us to calculate average properties of random triangulations, as well as canonical ensemble averages, using an energy functional that is approximately the variance of the degree distribution. All considered triangulations have clustering coefficients comparable with real-world graphs; for the canonical ensemble there are inverse temperatures with small shortest path length independent of system size. Tuning the inverse temperature to a quasi-critical value leads to an indication of scale-free behaviour for degrees . Using triangulations as a random graph model can improve the understanding of real-world networks, especially if the actual distance of the embedded nodes becomes important.

Degree-Degree Dependencies in Random Graphs with Heavy-Tailed Degrees

We investigate degree-degree correlations for scale-free graph sequences. The main conclusion of this paper is that the assortativity coefficient is not the appropriate way to describe degree-dependences in scale-free random graphs. Indeed, we study the infinite volume limit of the assortativity coefficient, and show that this limit is always non-negative when the degrees have finite first but infinite third moment, i.e., when the degree exponent $\gamma + 1$ of the density satisfies $\gamma \in (1,3)$. More generally, our results show that the correlation coefficient is inappropriate to describe dependencies between random variables having infinite variance. We start with a simple model of the sample correlation of random variables $X$ and $Y$, which are linear combinations with non-negative coefficients of the same infinite variance random variables. In this case, the correlation coefficient of $X$ and $Y$ is not defined, and the sample covariance converges to a proper random variable with support that is a subinterval of $(-1,1)$. Further, for any joint distribution $(X,Y)$ with equal marginals being non-negative power-law distributions with infinite variance (as in the case of degree-degree correlations), we show that the limit is non-negative. We next adapt these results to the assortativity in networks as described by the degree-degree correlation coefficient, and show that it is non-negative in the large graph limit when the degree distribution has an infinite third moment. We illustrate these results with several examples where the assortativity behaves in a non-sensible way. We further discuss alternatives for describing assortativity in networks based on rank correlations that are appropriate for infinite variance variables. We support these mathematical results by simulations.

On thermodynamic states of the Ising model on scale-free graphs

There is proposed a model of scale-free random graphs which are locally close to the uncorrelated complex random networks with divergent 〈k2〉 studied in, e.g., Dorogovtsev S.N. et al., Rev. Mod. Phys., 2008, 80, 1275. It is shown that the Ising model on the proposed graphs with interaction intensities of arbitrary signs with probability one is in a paramagnetic state at sufficiently high finite values of the temperature. For the same graphs, the bond percolation model with probability one is in a nonpercolative state for positive values of the percolation probability. These results and their possible extensions are also discussed.

Pheromone Static Routing Strategy for Complex Networks

We adopt the concept of using pheromones to generate a set of static paths that can reach the performance of global dynamic routing strategy [Phys. Rev. E 81 (2010) 016113]. The path generation method consists of two stages. In the first stage, a pheromone is dropped to the nodes by packets forwarded according to the global dynamic routing strategy. In the second stage, pheromone static paths are generated according to the pheromone density. The output paths can greatly improve traffic systems' overall capacity on different network structures, including scale-free networks, small-world networks and random graphs. Because the paths are static, the system needs much less computational resources than the global dynamic routing strategy.

Clustering of random scale-free networks

We derive the finite-size dependence of the clustering coefficient of scale-free random graphs generated by the configuration model with degree distribution exponent 2<γ<3. Degree heterogeneity increases the presence of triangles in the network up to levels that compare to those found in many real networks even for extremely large nets. We also find that for values of γ≈2, clustering is virtually size independent and, at the same time, becomes a de facto non-self-averaging topological property. This implies that a single-instance network is not representative of the ensemble even for very large network sizes.

Identifying Critical Components in Infrastructure Networks Using Network Topology

This paper applies graph theory metrics to network flow models, with the aim of assessing the possibility of using these metrics to identify vulnerable areas within infrastructure systems. To achieve this, a reduced complexity flow model that can be used to simulate flows in infrastructure networks is developed. The reason for developing this model is not to make the analysis easier, but to reduce the physical problem to its most basic level and therefore produce the most general flow model (i.e., applicable to the widest range of infrastructure networks). An initial assessment of the applicability of graph theory metrics to infrastructure networks is made by comparing the distribution of flows, calculated using this model, to the shortest average path length in three of the most recognized classes of network—scale-free networks, small-world networks, and random graph models—and it is demonstrated that for all three classes of network there is a strong correlation. This suggests that at least parts of graph theory may be used to inform one about the behavior of physical networks. The authors further demonstrate the utility of graph theory metrics by using them to improve their predictive skill in identifying vulnerable areas in a specific type of infrastructure system. This is done using a hydraulic model to calculate the flows in a sample water distribution network and then to show that using a combination of graph theory metrics and flow gives superior predictive skill over just one of these measures in isolation.

Degree distribution of a scale-free random graph model

In this paper, we consider the degree distribution of a general random graph with multiple edges and loops from the perspective of probability. Based on the first-passage probability of Markov chains, we give a new and rigorous proof to the existence of the network degree distribution and obtain the precise expression of the degree distribution. The analytical results are in good agreement with numerical simulations.

Treewidth of Erdős–Rényi random graphs, random intersection graphs, and scale-free random graphs

We study conditions under which the treewidth of three different classes of random graphs is linear in the number of vertices. For the Erdős–Rényi random graph G(n,m), our result improves a previous lower bound obtained by Kloks (1994) [22]. For random intersection graphs, our result strengthens a previous observation on the treewidth by Karoński et al. (1999) [19]. For scale-free random graphs based on the Barabási-Albert preferential-attachment model, it is shown that if more than 11 vertices are attached to a new vertex, then the treewidth of the obtained network is linear in the size of the network with high probability.

The clustering coefficient of a scale-free random graph

We consider a random graph process in which, at each time step, a new vertex is added with m out-neighbours, chosen with probabilities proportional to their degree plus a strictly positive constant. We show that the expectation of the clustering coefficient of the graph process is asymptotically proportional to log n n . Bollobás and Riordan have previously shown that when the constant is zero, the same expectation is asymptotically proportional to ( log n ) 2 n .