Observers agree that a citizen of Ohio had much larger voting power than a citizen of Texas or California in the recent US presidential election. Why is it so? A brief introduction to the theory of voting will be provided. We analyze the voting power of a member of a voting body, or of a person which elects his representative, who will take part in the voting on her behalf. The notion of voting power is illustrated by examples of the systems of voting in the European Council. We propose a representative voting system based on the square root law of Penrose. Using statistical approach and considering fictitious countries with randomly chosen populations we study the problem of selecting an optimal quota.
In the first part of the talk, we will discuss our recent paper, "How Famous is a Scientist? Famous to Those Who Know Us". Our findings show that fame and merit in science are linearly related, and that the probability distribution for a certain level of fame falls off exponentially. This is in sharp contrast with more popularly famous groups of people, for which fame is exponentially related to merit (number of downed planes), and the probability of fame decays in power-law fashion. We will define fame in terms of the type of popularity growth model as a rich-get-richer scheme which leads to a scale-free graph. We will discuss the statistics and ergodicity properties of cycles in the topology of a large scale graph, and likewise the roles of communities and subcommunities to understanding the large scale graphs. In, "Statistics of Cycles: How Loopy is your Network?" we study the distribution of cycles of length h in large networks (of size N>>1) and find it to be an excellent ergodic estimator, even in the extreme inhomogeneous case of scale-free networks. The distribution is sharply peaked around a characteristic cycle length, h ~Ná. Finally, we will analyze Communication and Synchronization in Disconnected Networks with Dynamic Topology: Moving Neighborhood Networks.