Dynamic Complex Networks: Models, Algorithms, and Applications

Dear Colleagues,

Dynamical complex networks (DCNs) are ubiquitous as models for many real-world systems whose network properties evolve in time. The present Special Issue is devoted to the network evolution theory, making use of an analogy with the evolutionary models of stabilizing and diversifying selection.  

Defining network behaviors is challenging because the global dynamics of DCNs at the ensemble (macroscopic) level are the manifestation of the coupled constituent dynamics at the individual constituent (microscopic) level. Microscopic dynamics and macroscopic dynamics together define the emergence of collective behaviors such as synchronization and asynchronization at the network level. Significant efforts have been made to correlate the complex interactions between ensemble constituents with simultaneous collective behaviors as well as critical and abrupt failures when the system is perturbed.  

The first attempt to understand the relations between the local (i.e., pertaining to a single node) and global (of the entire network) properties of a complex system in the framework of a thermodynamic approach to graphs was initiated in complex network theory (CNT). In the thermodynamic limit of infinitely large graphs →∞ considered in CNT, any functionally relevant structural features of a graph appear to be asymptotically negligible “fluctuations”; the entire network is viewed as an ever-growing collection of structurally homogeneous random graphs; and the famous power law statistics for nodes’ degree distributions may result from a superposition of the binomial or Poisson degree distributions typical for random graphs—true scale-free graphs are rare. The evolution of complex networks is explained in CNT by the preferential attachment mechanism under the spell of Matthew’s principle of accumulated advantage making use of an analogy with evolutionary models of stabilizing selection. By assigning an intrinsic fitness value to each node—the higher the fitness, the higher the probability of attracting new edges—an operator initiates the Bose–Einstein condensation mechanism driving a phase transition of the network topology to the appearance of a few “super hubs” of the maximal centrality and of the highest fitness accumulating all branches of the network as →∞.  

Such a theory certainly was not complete, since it did not explain another important type of natural selection: diversifying, or disruptive selection, describing changes in population genetics favorable to the extreme values for a trait over the normative values formed in the course of stabilizing the selection process. The related thermodynamic approach to the study of complex networks concerns the thermodynamic limit of very long walks →∞ in graphs. The local fluctuations of a path’s growth rate around the graph topological entropy that rise due to the graph structure modifications follow Fermi–Dirac statistics originally describing identical particles that obey the Pauli exclusion principle that does not allow for condensation (of particles at a single state). The thermodynamics limit of infinite walks subjected to the Fermi–Dirac statistics neither requires a network to grow, nor predicts any “eventual stage” of its structural evolution. As a DCN is open to random structural modifications, centrality flips its role in the network dynamics. Namely, the nodes of minimal centrality loosely integrated into the fabric of the network would most likely feature its structural modifications. While central hubs accumulate most traffic in normally operating transport networks, traffic congestion may catch it at the nodes of low centrality and structural bottlenecks due to the random fluctuations of transportation capacity in networks. 

In the proposed Special Issue, we aim to organize a broad discussion on network evolution theory and its applications in studies of real-world complex systems.

Prof. Dr. Dimitri Volchenkov
Guest Editor

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• dynamic complex networks
• statistics of long walks
• graph structural modifications
• stability and synchronization in networks
• entropic force and pressure
• graph node fugacity
• graph node navigability
• graph navigation
• delocalization–localization