The Robustness of Interdependent Directed Networks With Intra-layer Angular Correlations

The robustness of interdependent networks is a frontier topic in current network science. A line of studies has so far been investigated in the perspective of correlated structures on robustness, such as degree correlations and geometric correlations in interdependent networks, in-out degree correlations in interdependent directed networks, and so on. Advances in network geometry point that hyperbolic properties are also hidden in directed structures, but few studies link those features to the dynamical process in interdependent directed networks. In this paper, we discuss the impact of intra-layer angular correlations on robustness from the perspective of embedding interdependent directed networks into hyperbolic space. We find that the robustness declines as increasing intra-layer angular correlations under targeted attacks. Interdependent directed networks without intra-layer angular correlations are always robust than those with intra-layer angular correlations. Moreover, empirical networks also support our findings: the significant intra-layer angular correlations are hidden in real interdependent directed networks and contribute to the prediction of robustness. Our work sheds light that the impact of intra-layer angular correlations should be attention, although in-out degree correlations play a positive role in robustness. In particular, it provides an early warning indicator by which the system decoded the intrinsic rules for designing efficient and robust interacting directed networks.

Diagonal Degree Correlations vs. Epidemic Threshold in Scale-Free Networks

We prove that the presence of a diagonal assortative degree correlation, even if small, has the effect of dramatically lowering the epidemic threshold of large scale-free networks. The correlation matrix considered is P h | k = 1 − r P h k U + r δ h k , where P U is uncorrelated and r (the Newman assortativity coefficient) can be very small. The effect is uniform in the scale exponent γ if the network size is measured by the largest degree n . We also prove that it is possible to construct, via the Porto–Weber method, correlation matrices which have the same k n n as the P h | k above, but very different elements and spectra, and thus lead to different epidemic diffusion and threshold. Moreover, we study a subset of the admissible transformations of the form P h | k ⟶ P h | k + Φ h , k with Φ h , k depending on a parameter which leaves k n n invariant. Such transformations affect in general the epidemic threshold. We find, however, that this does not happen when they act between networks with constant k n n , i.e., networks in which the average neighbor degree is independent from the degree itself (a wider class than that of strictly uncorrelated networks).

Comparison of Simulations with a Mean-Field Approach vs. Synthetic Correlated Networks

It is well known that dynamical processes on complex networks are influenced by the degree correlations. A common way to take these into account in a mean-field approach is to consider the function knn(k) (average nearest neighbors degree). We re-examine the standard choices of knn for scale-free networks and a new family of functions which is independent from the simple ansatz knn∝kα but still displays a remarkable scale invariance. A rewiring procedure is then used to explicitely construct synthetic networks using the full correlation P(h∣k) from which knn is derived. We consistently find that the knn functions of concrete synthetic networks deviate from ideal assortativity or disassortativity at large k. The consequences of this deviation on a diffusion process (the network Bass diffusion and its peak time) are numerically computed and discussed for some low-dimensional samples. Finally, we check that although the knn functions of the new family have an asymptotic behavior for large networks different from previous estimates, they satisfy the general criterium for the absence of an epidemic threshold.

Networks with degree–degree correlations are special cases of the edge-coloured random graph

In complex networks, the degrees of adjacent nodes may often appear dependent—which presents a modelling challenge. We present a working framework for studying networks with an arbitrary joint distribution for the degrees of adjacent nodes by showing that such networks are a special case of edge-coloured random graphs. We use this mapping to study bond percolation in networks with assortative mixing and show that, unlike in networks with independent degrees, the sizes of connected components may feature unexpected sensitivity to perturbations in the degree distribution. The results also indicate that degree–degree dependencies may feature a vanishing percolation threshold even when the second moment of the degree distribution is finite. These results may be used to design artificial networks that efficiently withstand link failures and indicate the possibility of super spreading in networks without clearly distinct hubs.

A generalized configuration model with degree correlations and its percolation analysis

In this paper we present a generalization of the classical configuration model. Like the classical configuration model, the generalized configuration model allows users to specify an arbitrary degree distribution. In our generalized configuration model, we partition the stubs in the configuration model into b blocks of equal sizes and choose a permutation function h for these blocks. In each block, we randomly designate a number proportional to q of stubs as type 1 stubs, where q is a parameter in the range [0,1]. Other stubs are designated as type 2 stubs. To construct a network, randomly select an unconnected stub. Suppose that this stub is in block i. If it is a type 1 stub, connect this stub to a randomly selected unconnected type 1 stub in block h(i). If it is a type 2 stub, connect it to a randomly selected unconnected type 2 stub. We repeat this process until all stubs are connected. Under an assumption, we derive a closed form for the joint degree distribution of two random neighboring vertices in the constructed graph. Based on this joint degree distribution, we show that the Pearson degree correlation function is linear in q for any fixed b. By properly choosing h, we show that our construction algorithm can create assortative networks as well as disassortative networks. We present a percolation analysis of this model. We verify our results by extensive computer simulations.

Degree correlations in scale-free random graph models

We 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.