Memory effects on link formation in temporal networks: A fractional calculus approach

Memory plays a vital role in the temporal evolution of interactions of complex systems. To address the impact of memory on the temporal pattern of networks, we propose a simple preferential connection model, in which nodes have a preferential tendency to establish links with most active nodes. Node activity is measured by the number of links a node observes in a given time interval. Memory is investigated using a time-fractional order derivative equation, which has proven to be a powerful method to understand phenomena with long-term memory. The memoryless case reveals a characteristic time where node activity behaves differently below and above it. We also observe that dense temporal networks (high number of events) show a clearer characteristic time than sparse ones. Interestingly, we also find that memory leads to decay of the node activity; thus, the chances of a node to receive new connections reduce with the node's age. Finally, we discuss the statistical properties of the networks for various memory-length.