Signed graph embedding: when everybody can sit closer to friends than enemies

Signed graphs are graphs with signed edges. They are commonly used to represent positive and negative relationships in social networks. While balance theory and clusterizable graphs deal with signed graphs to represent social interactions, recent empirical studies have proved that they fail to reflect some current practices in real social networks. In this paper we address the issue of drawing signed graphs and capturing such social interactions. We relax the previous assumptions to define a drawing as a model in which every vertex has to be placed closer to its neighbors connected via a positive edge than its neighbors connected via a negative edge in the resulting space. Based on this definition, we address the problem of deciding whether a given signed graph has a drawing in a given $\ell$-dimensional Euclidean space. We present forbidden patterns for signed graphs that admit the introduced definition of drawing in the Euclidean plane and line. We then focus on the $1$-dimensional case, where we provide a polynomial time algorithm that decides if a given complete signed graph has a drawing, and constructs it when applicable

System Stability Under Adversarial Injection of Dependent Tasks

Technological changes (NFV, Osmotic Computing, Cyber-physical Systems) are making very important devising techniques to efficiently run a flow of jobs formed by dependent tasks in a set of servers. These problem can be seen as generalizations of the dynamic job-shop scheduling problem, with very rich dependency patterns and arrival assumptions. In this work, we consider a computational model of a distributed system formed by a set of servers in which jobs, that are continuously arriving, have to be executed. Every job is formed by a set of dependent tasks (i. e., each task may have to wait for others to be completed before it can be started), each of which has to be executed in one of the servers. The arrival of jobs and their properties is assumed to be controlled by a bounded adversary, whose only restriction is that it cannot overload any server. This model is a non-trivial generalization of the Adversarial Queuing Theory model of Borodin et al., and, like that model, focuses on the stability of the system: whether the number of jobs pending to be completed is bounded at all times. We show multiple results of stability and instability for this adversarial model under different combinations of the scheduling policy used at the servers, the arrival rate, and the dependence between tasks in the jobs

Ensuring Uniformity in Random Peer Sampling Services

The peer sampling service is a core building block for gossip protocols in peer-to-peer networks. Ideally, a peer sampling service continuously provides each peer with a sample of peers picked uniformly at random in the network. While empirical studies have shown that uniformity was achieved, analysis proposed so far assume strong restrictions on the topology of the overlay network it continuously generates. In this work, we analyze a Generic Random Peer Sampling Service (GRPS) that satisfies the desirable properties for any peer sampling service –small views, uniform sample, load balancing, and independence– and relieve strong degree connections in the nodes assumed in previous works. The main result we prove is: starting from any simple (without loops and parallel edges) directed graph with out-degree equal to c for all nodes, and recursively applying GRPS, eventually results in a random simple directed graph with out-degree equal to c for all nodes. We test empirically convergence time and independence time for GRPS. We use this empirical evaluation to show that GRPS performs better than previously presented peer sampling services. We also present a variant of GRPS that ensures that the in and out-degrees of nodes in the initial network are maintained in the resulting graph. Finally, we discuss on how to deal with new nodes in both settings