The structure of real-world multilayer infrastructure systems usually exhibits anisotropy due to constraints of the embedding space. For example, geographical features like mountains, rivers and shores influence the architecture of critical infrastru
cture networks. Moreover, such spatial networks are often non-homogeneous but rather have a modular structure with dense connections within communities and sparse connections between neighboring communities. When the networks of the different layers are interdependent, local failures and attacks may propagate throughout the system. Here we study the robustness of spatial interdependent networks which are both anisotropic and heterogeneous. We also evaluate the effect of localized attacks having different geometrical shapes. We find that anisotropic networks are more robust against localized attacks and that anisotropic attacks, surprisingly, even on isotropic structures, are more effective than isotropic attacks.
In the growth of bacterial colonies, a great variety of complex patterns are observed in experiments, depending on external conditions and the bacterial species. Typically, existing models employ systems of reaction-diffusion equations or consist of
growth processes based on rules, and are limited to a discrete lattice. In contrast, the two-dimensional model proposed here is an off-lattice simulation, where bacteria are modelled as rigid circles and nutrients are point-like, Brownian particles. Varying the nutrient diffusion and concentration, we simulate a wide range of morphologies compatible with experimental observations, from round and compact to extremely branched patterns. A scaling relationship is found between the number of cells in the interface and the total number of cells, with two characteristic regimes. These regimes correspond to the compact and branched patterns, which are exhibited for sufficiently small and large colonies, respectively. In addition, we characterise the screening effect observed in the structures by analysing the multifractal properties of the growth probability.
Interdependencies are ubiquitous throughout the world. Every real-world system interacts with and is dependent on other systems, and this interdependency affects their performance. In particular, interdependencies among networks make them vulnerable
to failure cascades, the effects of which are often catastrophic. Failure propagation fragments network components, disconnects them, and may cause complete systemic failure. We propose a strategy of avoiding or at least mitigating the complete destruction of a system of interdependent networks experiencing a failure cascade. Starting with a fraction $1-p$ of failing nodes in one network, we reconnect with a probability $gamma$ every isolated component to a functional giant component (GC), the largest connected cluster. We find that as $gamma$ increases the resilience of the system to cascading failure also increases. We also find that our strategy is more effective when it is applied in a network of low average degree. We solve the problem theoretically using percolation theory, and we find that the solution agrees with simulation results.
We introduce a contrarian opinion (CO) model in which a fraction p of contrarians within a group holds a strong opinion opposite to the opinion held by the rest of the group. At the initial stage, stable clusters of two opinions, A and B exist. Then
we introduce contrarians which hold a strong B opinion into the opinion A group. Through their interactions, the contrarians are able to decrease the size of the largest A opinion cluster, and even destroy it. We see this kind of method in operation, e.g when companies send free new products to potential customers in order to convince them to adopt the product and influence others. We study the CO model, using two different strategies, on both ER and scale-free networks. In strategy I, the contrarians are positioned at random. In strategy II, the contrarians are chosen to be the highest degrees nodes. We find that for both strategies the size of the largest A cluster decreases to zero as p increases as in a phase transition. At a critical threshold value p_c the system undergoes a second-order phase transition that belongs to the same universality class of mean field percolation. We find that even for an ER type model, where the degrees of the nodes are not so distinct, strategy II is significantly more effctive in reducing the size of the largest A opinion cluster and, at very small values of p, the largest A opinion cluster is destroyed.
In a network, we define shell $ell$ as the set of nodes at distance $ell$ with respect to a given node and define $r_ell$ as the fraction of nodes outside shell $ell$. In a transport process, information or disease usually diffuses from a random node
and reach nodes shell after shell. Thus, understanding the shell structure is crucial for the study of the transport property of networks. For a randomly connected network with given degree distribution, we derive analytically the degree distribution and average degree of the nodes residing outside shell $ell$ as a function of $r_ell$. Further, we find that $r_ell$ follows an iterative functional form $r_ell=phi(r_{ell-1})$, where $phi$ is expressed in terms of the generating function of the original degree distribution of the network. Our results can explain the power-law distribution of the number of nodes $B_ell$ found in shells with $ell$ larger than the network diameter $d$, which is the average distance between all pairs of nodes. For real world networks the theoretical prediction of $r_ell$ deviates from the empirical $r_ell$. We introduce a network correlation function $c(r_ell)equiv r_{ell+1}/phi(r_ell)$ to characterize the correlations in the network, where $r_{ell+1}$ is the empirical value and $phi(r_ell)$ is the theoretical prediction. $c(r_ell)=1$ indicates perfect agreement between empirical results and theory. We apply $c(r_ell)$ to several model and real world networks. We find that the networks fall into two distinct classes: (i) a class of {it poorly-connected} networks with $c(r_ell)>1$, which have larger average distances compared with randomly connected networks with the same degree distributions; and (ii) a class of {it well-connected} networks with $c(r_ell)<1$.
We study the transition between the strong and weak disorder regimes in the scaling properties of the average optimal path $ell_{rm opt}$ in a disordered ErdH{o}s-Renyi (ER) random network and scale-free (SF) network. Each link $i$ is associated with
a weight $tau_iequivexp(a r_i)$, where $r_i$ is a random number taken from a uniform distribution between 0 and 1 and the parameter $a$ controls the strength of the disorder. We find that for any finite $a$, there is a crossover network size $N^*(a)$ at which the transition occurs. For $N ll N^*(a)$ the scaling behavior of $ell_{rm opt}$ is in the strong disorder regime, with $ell_{rm opt} sim N^{1/3}$ for ER networks and for SF networks with $lambda ge 4$, and $ell_{rm opt} sim N^{(lambda-3)/(lambda-1)}$ for SF networks with $3 < lambda < 4$. For $N gg N^*(a)$ the scaling behavior is in the weak disorder regime, with $ell_{rm opt}simln N$ for ER networks and SF networks with $lambda > 3$. In order to study the transition we propose a measure which indicates how close or far the disordered network is from the limit of strong disorder. We propose a scaling ansatz for this measure and demonstrate its validity. We proceed to derive the scaling relation between $N^*(a)$ and $a$. We find that $N^*(a)sim a^3$ for ER networks and for SF networks with $lambdage 4$, and $N^*(a)sim a^{(lambda-1)/(lambda-3)}$ for SF networks with $3 < lambda < 4$.
