A simple differential geometry for complex networks

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. Conver...

Differential Geometry of Complex Vector Bundles

Pinning complex Networks by A Simple Controller1

In this short paper, we point out that a single local stability controller can pin the complex network to a specified solution (or its equilibrium) of the coupled complex network. A rigorous mathematical proof is given, too.

Complex Networks from Simple Rules

Complex networks are all around us, and they can be generated by simple mechanisms. Understanding what kinds of networks can be produced by following simple rules is therefore of great importance. This issue is investigated by studying the dynamics of extremely simple systems where a “writer” moves around a network, modifying it in a way that depends upon the writer’s surroundings. Each vertex ...

Simple Dynamics on Complex Networks

An analytical approach to network dynamics is used to show that when agents copy their state randomly the network arrives to a stationary status in which the distribution of states is independent of the degree. The effects of network topology on the process are characterized introducing a quantity called influence and studying its behavior for scale-free and random networks. We show that for th...