Formation of Modularity in a Model of Evolving Networks

Modularity structures are common in various social and biological networks. However, its dynamical origin remains an open question. In this work, we set up a dynamical model describing the evolution of a social network. Based on the observations of real social networks, we introduced a link-creating/deleting strategy according to the local dynamics in the model. Thus the coevolution of dynamics and topology naturally determines the network properties. It is found that for a small coupling strength, the networked system cannot reach any synchronization and the network topology is homogeneous. Interestingly, when the coupling strength is large enough, the networked system spontaneously forms communities with different dynamical states. Meanwhile, the network topology becomes heterogeneous with modular structures. It is further shown that in a certain parameter regime, both the degree and the community size in the formed network follow a power-law distribution, and the networks are found to be assortative. These results are consistent with the characteristics of many empirical networks, and are helpful to understand the mechanism of formation of modularity in complex networks.Comment: 6 pages, 4 figur

Artin's Conjecture on Average for Composite Moduli

AbstractLet a be an integer ≠−1 and not a square. Let Pa(x) be the number of primes up to x for which a is a primitive root. Goldfeld and Stephens proved that the average value of Pa(x) is about a constant multiple of x/lnx. Carmichael extended the definition of primitive roots to that of primitive λ-roots for composite moduli n, which are integers with the maximal order modulo n. Let Na(x) be the number of natural numbers up to x for which a is a primitive λ-root. In this paper we will prove that the average value of Na(x) oscillates. That is, limx→∞1/x2∑1⩽a⩽xNa(x)>0 and limx→∞1/x2∑1⩽a⩽xNa(x)=0

Parallel Machine Scheduling with Nested Processing Set Restrictions and Job Delivery Times

The problem of scheduling jobs with delivery times on parallel machines is studied, where each job can only be processed on a specific subset of the machines called its processing set. Two distinct processing sets are either nested or disjoint; that is, they do not partially overlap. All jobs are available for processing at time 0. The goal is to minimize the time by which all jobs are delivered, which is equivalent to minimizing the maximum lateness from the optimization viewpoint. A list scheduling approach is analyzed and its approximation ratio of 2 is established. In addition, a polynomial time approximation scheme is derived