Spatial Effects on Complex Networks
Final Report Abstract
Our research activities on spatial effects in constrained networks consist of three parts. In part 1 we considered single networks embedded in 1- and 2-dimensional lattices, in part 2 we considered interacting networks with spatial constraints, and in part 3 we studied climate networks and used them for forecasting El Nino events. In part 1 we focused on the most relevant cases of spatially embedded networks, (i) Erdos-Renyi networks where the number k of links per node are distributed by a Poissonian, and (ii) scale free networks where k is distributed algebraically. In both cases, the distribution of the link lengths l follows a power law, P (l) ∼ l^−δ . We studied the structural and dynamical properties as well as the percolation properties of these systems and found that all properties depend crucial on the distribution exponent δ and the embedding dimension de : For δ below de, the spatial constraints are irrelevant and the network behaves as an unconstrained network, with an infinite dimension. For δ above 2de , the network behaves like a regular lattice with dimension de. At intermediate values of δ, for de < δ < 2de, we find new scaling behavior, in particular in Erdos-Renyi networks the dimension of the network becomes finite and decreases monotonically with increasing δ, until it reaches the value of the embedding dimension at δ = 2de. In part 2 we studied spatially embedded networks with a new realistic mechanism of dependency links. We tested how embedded networks are affected by this new type of dependency links. A dependency link is defined as follows: If two nodes (in the same network or different coupled networks) are connected by a dependency link and one of the nodes fails the other one also fails instantaneously. Indeed, many real world complex systems such as critical infrastructure networks are embedded in space and their components depend on one another to function. Surprisingly, we find that in embedded systems, in contrast to nonembedded systems, there is no critical dependency and any small fraction of interdependent nodes leads to an abrupt collapse. We show analytically that this extreme vulnerability of very weakly dependency is a consequence of the critical exponent describing the percolation transition of a single lattice network without dependencies. Furthermore we find that localized attacks are significantly more damaging to the spatially embedded network compared to random attacks. In part 3 we studied a climate network in the Pacific. The climate network is embedded in de = 2. We considered atmospheric temperature records at grid points in the Pacific area and studied the teleconnections between the El Nino basin and the rest of the Pacific. The strength of the teleconnection between 2 grid points characterizes the strength of the link between them. We studied the time evolution of the mean link strengths, between 1950 and present. We found that a large-scale cooperativity between the El Nino basin and the rest of the Pacific characterized by an enhanced mean link strength, builds up in the calendar year before an El-Nino event. On this basis, we could develop an efficient 12-months forecasting scheme, this way achieving some doubling of the early-warning period. In our original paper, we used high-quality temperature data between 1948 and 2011. Compared to the conventional algorithms, our model yields a very high hit rate of 0.76 and a very low false alarm rate of 0.05. After publishing our algorithm, we correctly predicted the absence of El-Nino events in 2012 and 2013. In September 2013, we predicted correctly the start of an El-Nino event in fall 2014.
Publications
- Dimension of spatially embedded networks. Nature Physics 7, 481 (2011)
D. Li, K. Kosmidis, A. Bunde and S. Havlin
- Percolation of spatially constraint networks. EPL 93, 68004 (2011)
D. Li, G. Li, K. Kosmidis, H. E. Stanley, A. Bunde and S. Havlin
- Cascading Failures in Interdependent Lattice Networks: The Critical Role of the Length of Dependency Links . Phys. Rev. Lett. 108, 228702 (2012)
W. Li, A. Bashan, S.V. Buldyrev, H.E. Stanley, S. Havlin
- Diffusion, annihilation, and chemical reactions in complex networks with spatial constraint. Phys. Rev. E 86, 046103 (2012)
T. Emmerich, A. Bunde and S. Havlin
(See online at https://doi.org/10.1103/PhysRevE.86.046103) - Complex networks embedded in space: Dimension and scaling relations between mass, topological distance, and Euclidean distance. Phys. Rev. E 87, 032802 (2013)
T. Emmerich, A. Bunde and S. Havlin
(See online at https://doi.org/10.1103/PhysRevE.87.032802) - Improved El Nio forecasting by cooperativity detection. PNAS 110, 11742 (2013)
J. Ludescher, A. Gozolchiani, M.I. Bogachev, A. Bunde, S. Havlin and H.J. Schellnhuber
(See online at https://doi.org/10.1073/pnas.1309353110) - The extreme vulnerability of interdependent spatially embedded networks. Nature Physics 9, 667 (2013)
A. Bashan, Y. Berezin, S.V. Buldyrev, and S. Havlin
- Structural and functional properties of spatially embedded scale-free networks. Phys. Rev. E 89, 062806 (2014)
T. Emmerich, A. Bunde and S. Havlin
(See online at https://doi.org/10.1103/PhysRevE.89.062806) - Very early warning of next El Nino. PNAS 111, 2064 (2014)
J. Ludescher, A. Gozolchiani, M.I. Bogachev, A. Bunde, S. Havlin and H.J. Schellnhuber
(See online at https://doi.org/10.1073/pnas.1323058111) - Localized attacks on spatially embedded networks with dependencies. Scientific Reports 5, 8934 (2015)
Y. Berezin, A. Bashan, M.M. Danziger, D. Li, and S. Havlin
(See online at https://doi.org/10.1038/srep08934)
