1. Navigability of Networks Dmitri Krioukov CAIDA/UCSD M. Boguñá, M. Á. Serrano, F. Papadopoulos, M. Kitsak, A. Vahdat, kc claffy May, 2010

2. Common principles of complex networks Common structure Many hubs (heterogeneous degree distributions) High probability that two neighbors of the same node are connected (many triangles, strong clustering) Small-world property (consequence of the two above + randomness) One common function Navigability

3. Navigability (or conductivity) is network efficiency with respect to: targeted information propagation without global knowledge Examples are: Internet Brain Regulatory/signaling/metabolic networks

5. Potential pitfalls with greedy navigation It may get stuck without reaching destination (low success ratio) It may travel sup-optimal paths, much longer than the shortest paths (high stretch) It may require global recomputations of node positions in the hidden space in presence of rapid network dynamics It may be vulnerable with respect to network damage

6. Results so far Hidden metric spaces do exist even in networks we do not expect them to exist Phys Rev Lett, v.100, 078701, 2008 Complex networks are navigable large numbers of hubs and triangles improve navigability do networks evolve to navigable configurations? Nature Physics, v.5, p.74-80, 2009 Regardless of metric space structure, all greedy paths are shortest in complex networks (stretch is 1) Phys Rev Lett, v.102, 058701, 2009 The success ratio and navigation robustness do depend on metric space structure

7. But if the metric space is hyperbolic then also (PRE, v.80, 035101(R), 2009) Greedy navigation almost never gets stuck (the success ratio approaches 100%) Both success ratio and stretch are very robust with respect to network dynamics and even to catastrophic levels of network damage Both heterogeneity and clustering (hubs and triangles) emerge naturally as simple consequences of hidden hyperbolic geometry

9. Agenda: mapping networks to their hidden metric spaces Mapped the Internet used maximum-likelihood techniques very messy and complicated, does not scale Need rich network data on network topological structure intrinsic measures of node similarity New mapping methods

10. If we map a network, then we can Have an infinitely scalable routing solution for the Internet Estimate distances between nodes (e.g., similarity distances between people in social networks) “soft” communities become areas in the hidden space with higher node densities Tell what drives signaling in networks, and what network perturbations drive it to failures (e.g., brain disorders, cancer, etc.)