Spatial network surrogates for disentangling complex system structure from spatial embedding of nodes

ACKNOWLEDGMENTS MW and RVD have been supported by the German Federal Ministry for Education and Research (BMBF) via the Young Investigators Group CoSy-CC2 (grant no. 01LN1306A). JFD thanks the Stordalen Foundation and BMBF (project GLUES) for financial support. JK acknowledges the IRTG 1740 funded by DFG and FAPESP. MT Gastner is acknowledged for providing his data on the airline, interstate, and Internet network. P Menck thankfully provided his data on the Scandinavian power grid. We thank S Willner on behalf of the entire zeean team for providing the data on the world trade network. All computations have been performed using the Python package pyunicorn [41] that is available at https://github.com/pik-copan/pyunicorn.Peer reviewedPreprin

Evolving Clustered Random Networks

We propose a Markov chain simulation method to generate simple connected random graphs with a specified degree sequence and level of clustering. The networks generated by our algorithm are random in all other respects and can thus serve as generic models for studying the impacts of degree distributions and clustering on dynamical processes as well as null models for detecting other structural properties in empirical networks

The multiplex structure of interbank networks

The interbank market has a natural multiplex network representation. We employ a unique database of supervisory reports of Italian banks to the Banca d'Italia that includes all bilateral exposures broken down by maturity and by the secured and unsecured nature of the contract. We find that layers have different topological properties and persistence over time. The presence of a link in a layer is not a good predictor of the presence of the same link in other layers. Maximum entropy models reveal different unexpected substructures, such as network motifs, in different layers. Using the total interbank network or focusing on a specific layer as representative of the other layers provides a poor representation of interlinkages in the interbank market and could lead to biased estimation of systemic risk.Comment: 41 pages, 8 figures, 10 table

Hyperbolic Geometry of Complex Networks

We develop a geometric framework to study the structure and function of complex networks. We assume that hyperbolic geometry underlies these networks, and we show that with this assumption, heterogeneous degree distributions and strong clustering in complex networks emerge naturally as simple reflections of the negative curvature and metric property of the underlying hyperbolic geometry. Conversely, we show that if a network has some metric structure, and if the network degree distribution is heterogeneous, then the network has an effective hyperbolic geometry underneath. We then establish a mapping between our geometric framework and statistical mechanics of complex networks. This mapping interprets edges in a network as non-interacting fermions whose energies are hyperbolic distances between nodes, while the auxiliary fields coupled to edges are linear functions of these energies or distances. The geometric network ensemble subsumes the standard configuration model and classical random graphs as two limiting cases with degenerate geometric structures. Finally, we show that targeted transport processes without global topology knowledge, made possible by our geometric framework, are maximally efficient, according to all efficiency measures, in networks with strongest heterogeneity and clustering, and that this efficiency is remarkably robust with respect to even catastrophic disturbances and damages to the network structure

Ground states of stealthy hyperuniform potentials: I. Entropically favored configurations

Systems of particles interacting with "stealthy" pair potentials have been shown to possess infinitely degenerate disordered hyperuniform classical ground states with novel physical properties. Previous attempts to sample the infinitely degenerate ground states used energy minimization techniques, introducing algorithmic dependence that is artificial in nature. Recently, an ensemble theory of stealthy hyperuniform ground states was formulated to predict the structure and thermodynamics that was shown to be in excellent agreement with corresponding computer simulation results in the canonical ensemble (in the zero-temperature limit). In this paper, we provide details and justifications of the simulation procedure, which involves performing molecular dynamics simulations at sufficiently low temperatures and minimizing the energy of the snapshots for both the high-density disordered regime, where the theory applies, as well as lower densities. We also use numerical simulations to extend our study to the lower-density regime. We report results for the pair correlation functions, structure factors, and Voronoi cell statistics. In the high-density regime, we verify the theoretical ansatz that stealthy disordered ground states behave like "pseudo" disordered equilibrium hard-sphere systems in Fourier space. These results show that as the density decreases from the high-density limit, the disordered ground states in the canonical ensemble are characterized by an increasing degree of short-range order and eventually the system undergoes a phase transition to crystalline ground states. We also provide numerical evidence suggesting that different forms of stealthy pair potentials produce the same ground-state ensemble in the zero-temperature limit. Our techniques may be applied to sample this limit of the canonical ensemble of other potentials with highly degenerate ground states

Path Similarity Analysis: a Method for Quantifying Macromolecular Pathways

Diverse classes of proteins function through large-scale conformational changes; sophisticated enhanced sampling methods have been proposed to generate these macromolecular transition paths. As such paths are curves in a high-dimensional space, they have been difficult to compare quantitatively, a prerequisite to, for instance, assess the quality of different sampling algorithms. The Path Similarity Analysis (PSA) approach alleviates these difficulties by utilizing the full information in 3N-dimensional trajectories in configuration space. PSA employs the Hausdorff or Fr\'echet path metrics---adopted from computational geometry---enabling us to quantify path (dis)similarity, while the new concept of a Hausdorff-pair map permits the extraction of atomic-scale determinants responsible for path differences. Combined with clustering techniques, PSA facilitates the comparison of many paths, including collections of transition ensembles. We use the closed-to-open transition of the enzyme adenylate kinase (AdK)---a commonly used testbed for the assessment enhanced sampling algorithms---to examine multiple microsecond equilibrium molecular dynamics (MD) transitions of AdK in its substrate-free form alongside transition ensembles from the MD-based dynamic importance sampling (DIMS-MD) and targeted MD (TMD) methods, and a geometrical targeting algorithm (FRODA). A Hausdorff pairs analysis of these ensembles revealed, for instance, that differences in DIMS-MD and FRODA paths were mediated by a set of conserved salt bridges whose charge-charge interactions are fully modeled in DIMS-MD but not in FRODA. We also demonstrate how existing trajectory analysis methods relying on pre-defined collective variables, such as native contacts or geometric quantities, can be used synergistically with PSA, as well as the application of PSA to more complex systems such as membrane transporter proteins.Comment: 9 figures, 3 tables in the main manuscript; supplementary information includes 7 texts (S1 Text - S7 Text) and 11 figures (S1 Fig - S11 Fig) (also available from journal site