On the Voting Time of the Deterministic Majority Process

In the deterministic binary majority process we are given a simple graph where each node has one out of two initial opinions. In every round, every node adopts the majority opinion among its neighbors. By using a potential argument first discovered by Goles and Olivos (1980), it is known that this process always converges in $O(|E|)$ rounds to a two-periodic state in which every node either keeps its opinion or changes it in every round. It has been shown by Frischknecht, Keller, and Wattenhofer (2013) that the $O(|E|)$ bound on the convergence time of the deterministic binary majority process is indeed tight even for dense graphs. However, in many graphs such as the complete graph, from any initial opinion assignment, the process converges in just a constant number of rounds. By carefully exploiting the structure of the potential function by Goles and Olivos (1980), we derive a new upper bound on the convergence time of the deterministic binary majority process that accounts for such exceptional cases. We show that it is possible to identify certain modules of a graph $G$ in order to obtain a new graph $G^\Delta$ with the property that the worst-case convergence time of $G^\Delta$ is an upper bound on that of $G$. Moreover, even though our upper bound can be computed in linear time, we show that, given an integer $k$, it is NP-hard to decide whether there exists an initial opinion assignment for which it takes more than $k$ rounds to converge to the two-periodic state.Comment: full version of brief announcement accepted at DISC'1

Geometric vulnerability of democratic institutions against lobbying: a sociophysics approach

An alternative voting scheme is proposed to fill the democratic gap between a president elected democratically via universal suffrage (deterministic outcome, the actual majority decides), and a president elected by one person randomly selected from the population (probabilistic outcome depending on respective supports). Moving from one voting agent to a group of r randomly selected voting agents reduces the probabilistic character of the outcome. Building r such groups, each one electing its president, to constitute a group of the groups with the r local presidents electing a higher-level president, does reduce further the outcome probabilistic aspect. Repeating the process n times leads to a n-level bottom-up pyramidal structure. The hierarchy top president is still elected with a probability but the distance from a deterministic outcome reduces quickly with increasing n. At a critical value n_{c,r} the outcome turns deterministic recovering the same result a universal suffrage would yield. The scheme yields several social advantages like the distribution of local power to the competing minority making the structure more resilient, yet preserving the presidency allocation to the actual majority. An area is produced around fifty percent for which the president is elected with an almost equiprobability slightly biased in favor of the actual majority. However, our results reveal the existence of a severe geometric vulnerability to lobbying. A tiny lobbying group is able to kill the democratic balance by seizing the presidency democratically. It is sufficient to complete a correlated distribution of a few agents at the hierarchy bottom. Moreover, at the present stage, identifying an actual killing distribution is not feasible, which sheds a disturbing light on the devastating effect geometric lobbying can have on democratic hierarchical institutions.Comment: 52 pages, 22 figures, to appear in Mathematical Models and Methods in Applied Science

Deterministic voting in distributed systems using error-correcting codes

Distributed voting is an important problem in reliable computing. In an N Modular Redundant (NMR) system, the N computational modules execute identical tasks and they need to periodically vote on their current states. In this paper, we propose a deterministic majority voting algorithm for NMR systems. Our voting algorithm uses error-correcting codes to drastically reduce the average case communication complexity. In particular, we show that the efficiency of our voting algorithm can be improved by choosing the parameters of the error-correcting code to match the probability of the computational faults. For example, consider an NMR system with 31 modules, each with a state of m bits, where each module has an independent computational error probability of 10^-3. In, this NMR system, our algorithm can reduce the average case communication complexity to approximately 1.0825 m compared with the communication complexity of 31 m of the naive algorithm in which every module broadcasts its local result to all other modules. We have also implemented the voting algorithm over a network of workstations. The experimental performance results match well the theoretical predictions