Navigability of Networks

Dmitri KrioukovCAIDA/UCSD

M. Boguñá, M. Á. Serrano,F. Papadopoulos, M. Kitsak,

A. Vahdat, kc claffy

May, 2010

Common principlesof 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

Navigability

Navigability (or conductivity) is network efficiency with respect to: targeted information propagation without global knowledge

Examples are: Internet Brain Regulatory/signaling/metabolic networks

Potential pitfallswith 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 dynamicsIt may be vulnerable with respect to network damage

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

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

Agenda: mapping networksto 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

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.)