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Register a new userModel-free hidden geometry of complex networks
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The fundamental idea of embedding a network in a metric space is rooted in the principle of proximity preservation. Nodes are mapped into points of the space with pairwise distance that reflects their proximity in the network. Popular methods employed in network embedding either rely on implicit approximations of the principle of proximity preservation or implement it by enforcing the geometry of the embedding space, thus hindering geometric properties that networks may spontaneously exhibit. Here, we take advantage of a model-free embedding method explicitly devised for preserving pairwise proximity, and characterize the geometry emerging from the mapping of several networks, both real and synthetic. We show that the learned embedding has simple and intuitive interpretations: the distance of a node from the geometric center is representative for its closeness centrality, and the relative positions of nodes reflect the community structure of the network. Proximity can be preserved in relatively low-dimensional embedding spaces, and the hidden geometry displays optimal performance in guiding greedy navigation regardless of the specific network topology. We finally show that the mapping provides a natural description of contagion processes on networks, with complex spatiotemporal patterns represented by waves propagating from the geometric center to the periphery. The findings deepen our understanding of the model-free hidden geometry of complex networks.
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Many complex networks exhibit vulnerability to spreading of epidemics, and such vulnerability relates to the viral strain as well as to the network characteristics. For instance, the structure of the network plays an important role in spreading of epidemics. Additionally, properties of previous epidemic models require prior knowledge of the complex network structure, which means the models are limited to only well-known network structures. In this paper, we propose a new epidemiological SIR model based on the continuous time Markov chain, which is generalized to any type of network. The new model is capable of evaluating the states of every individual in the network. Through mathematical analysis, we prove an epidemic threshold exists below which an epidemic does not propagate in the network. We also show that the new epidemic threshold is inversely proportional to the spectral radius of the network. In particular, we employ the new epidemic model as a novel measure to assess the vulnerability of networks to the spread of epidemics. The new measure considers all possible effective infection rates that an epidemic might possess. Next, we apply the measure to correlated networks to evaluate the vulnerability of disassortative and assortative scalefree networks. Ultimately, we verify the accuracy of the theoretical epidemic threshold through extensive numerical simulations. Within the set of tested networks, the numerical results show that disassortative scale-free networks are more vulnerable to spreading of epidemics than assortative scale-free networks.
The ability to control a complex network towards a desired behavior relies on our understanding of the complex nature of these social and technological networks. The existence of numerous control schemes in a network promotes us to wonder: what is the underlying relationship of all possible input nodes? Here we introduce input graph, a simple geometry that reveals the complex relationship between all control schemes and input nodes. We prove that the node adjacent to an input node in the input graph will appear in another control scheme, and the connected nodes in input graph have the same type in control, which they are either all possible input nodes or not. Furthermore, we find that the giant components emerge in the input graphs of many real networks, which provides a clear topological explanation of bifurcation phenomenon emerging in dense networks and promotes us to design an efficient method to alter the node type in control. The findings provide an insight into control principles of complex networks and offer a general mechanism to design a suitable control scheme for different purposes.
We investigate the effects of long-range social interactions in flocking dynamics by studying the dynamics of a scalar model of collective motion embedded in a complex network representing a pattern of social interactions, as observed in several social species. In this scalar model we find a phenomenology analogous to that observed in the classic Vicsek model: In networks with low heterogeneity, a phase transition separates an ordered from a disordered phase. At high levels of heterogeneity, instead, the transition is suppressed and the system is always ordered. This observation is backed up analytically by the solution of a modified scalar model within an heterogeneous mean-field approximation. Our work extends the understanding of the effects of social interactions in flocking dynamics and opens the path to the analytical study of more complex topologies of social ties.
The majority-vote model with noise is one of the simplest nonequilibrium statistical model that has been extensively studied in the context of complex networks. However, the relationship between the critical noise where the order-disorder phase transition takes place and the topology of the underlying networks is still lacking. In the paper, we use the heterogeneous mean-field theory to derive the rate equation for governing the models dynamics that can analytically determine the critical noise $f_c$ in the limit of infinite network size $Nrightarrow infty$. The result shows that $f_c$ depends on the ratio of ${leftlangle k rightrangle }$ to ${leftlangle k^{3/2} rightrangle }$, where ${leftlangle k rightrangle }$ and ${leftlangle k^{3/2} rightrangle }$ are the average degree and the $3/2$ order moment of degree distribution, respectively. Furthermore, we consider the finite size effect where the stochastic fluctuation should be involved. To the end, we derive the Langevin equation and obtain the potential of the corresponding Fokker-Planck equation. This allows us to calculate the effective critical noise $f_c(N)$ at which the susceptibility is maximal in finite size networks. We find that the $f_c-f_c(N)$ decays with $N$ in a power-law way and vanishes for $Nrightarrow infty$. All the theoretical results are confirmed by performing the extensive Monte Carlo simulations in random $k$-regular networks, Erdos-Renyi random networks and scale-free networks.
Routing information through networks is a universal phenomenon in both natural and manmade complex systems. When each node has full knowledge of the global network connectivity, finding short communication paths is merely a matter of distributed computation. However, in many real networks nodes communicate efficiently even without such global intelligence. Here we show that the peculiar structural characteristics of many complex networks support efficient communication without global knowledge. We also describe a general mechanism that explains this connection between network structure and function. This mechanism relies on the presence of a metric space hidden behind an observable network. Our findings suggest that real networks in nature have underlying metric spaces that remain undiscovered. Their discovery would have practical applications ranging from routing in the Internet and searching social networks, to studying information flows in neural, gene regulatory networks, or signaling pathways.
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