This paper introduces a new variant of the popular n-dimensional hypercube network Q _n , known as the n-dimensional locally twisted cube LTQ _n , which has the same number of nodes and the same number of connections per node as Q _n . Furthermore, LTQ _n is similar to Q _n in the sense that the nodes can be one-to-one labeled with 0–1 binary sequences of length n, so that the labels of any two adjacent nodes differ in at most two successive bits. One advantage of LTQ _n is that the diameter is only about half of the diameter of Q _n . We develop a simple routing algorithm for LTQ _n , which creates a shortest path from the source to the destination in O(n) time. We find that LTQ _n consists of two disjoint copies of Q _{n −1} by adding a matching between their nodes. On this basis, we show that LTQ _n has a connectivity of n.

Due to widespread applications, the multi-virus competing spreading dynamics has recently aroused considerable interests. To our knowledge, all previous competing spreading models assume infection rates that are each linear in the virus occupancy probabilities of the individuals in a population. As linear infection rates are overestimation of real infection rates, in some situations these models cannot accurately predict the spreading process of multiple competing viruses. This work takes the first step toward enhancing the accuracy of multi-virus competing spreading models. A continuous-time bilayer-network-based bi-virus competing spreading model with generic infection rates is proposed. Criteria for the extinction of both viruses and for the survival of only one virus are presented, respectively. Numerical examples show that (1) if the generic bi-virus spreading model with linear infection rates predicts that the fraction of nodes infected with some virus would approach zero, the prediction of the fraction is accurate, and (2) if the scenario-relevant generic infection rates could be estimated accurately, the resulting model would be able to accurately forecast the evolutionary process of a pair of competing viruses.

Advanced persistent threat (APT) is a new kind of cyber attack that poses a serious threat to modern society. When an APT campaign on an organization has been identified, the available repair resources must be reasonably allocated to the potentially insecure hosts to mitigate the potential loss of the organization. We refer to the feasible repair resource allocation strategies as repair strategies. This paper focuses on the APT repair problem, i.e., the problem of developing effective repair strategies for organizations. First, for an organization with timevarying communication relationship, we establish an evolution model of the organization’s expected state, in which the impact of lateral movement of APT is accommodated. On this basis, we model the APT repair problem as a differential Nash game problem (the APT repair game) in which the attacker attempts to maximize his potential benefit and the organization manages to minimize its potential loss. Second, we derive a system (the potential system) for calculating a potential Nash equilibrium of an APT repair game, and we examine the structure of the potential attack and repair strategies in a potential Nash equilibrium. Next, we solve some potential systems to get the corresponding potential Nash equilibria. Finally, by comparison with a large number of randomly generated attack and repair strategies, we conclude that the potential Nash equilibrium of each APT repair game is a Nash equilibrium of the game. Therefore, we recommend to organizations their respective potential repair strategies. Our findings help to better understand and effectively defend against APT.