The dynamics of an agreement protocol interacting with a disagreement process over a common random network is considered. The model can represent the spreading of true and false information over a communication network, the propagation of faults in a large-scale control system, or the development of trust and mistrust in a society. At each time instance and with a given probability, a pair of network nodes are selected to interact. At random each of the nodes then updates its state towards the state of the other node (attraction), away from the other node (repulsion), or sticks to its current state (neglect). Agreement convergence and disagreement divergence results are obtained for various strengths of the updates for both symmetric and asymmetric update rules. Impossibility theorems show that a specific level of attraction is required for almost sure asymptotic agreement and a specific level of repulsion is required for almost sure asymptotic disagreement. A series of sufficient and or necessary conditions are then established for agreement convergence or disagreement divergence. In particular, under symmetric updates, a critical convergence measure in the attraction and repulsion update strength is found, in the sense that the asymptotic property of the network state evolution transits from agreement convergence to disagreement divergence when this measure goes from negative to positive. The result can be interpreted as a tight bound on how much bad action needs to be injected in a dynamic network in order to consistently steer its overall behavior away from consensus.
When a networked information is executed on top of an underlying network, nodes are endowed with internal states that evolve as nodes interact. The dynamics of the node states depend on the particular problem under investigation. For instance, the boids model was introduced in @cite_11 to model swarm behavior and animal groups, followed by Vicsek's model in @cite_9 . Models of opinion dynamics in social networks were considered in @cite_45 @cite_41 @cite_30 and the dynamics of communication protocols in @cite_32 . Distributed averaging or consensus algorithms have relative simple dynamics for the network state evolution and serve as a basic model for the complex interaction between node state dynamics and the dynamics of the underlying communication graph.
@cite_30 @cite_41 @cite_9 @cite_32 @cite_45 @cite_11
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The emergence of order in natural systems is a constant source of inspiration for both physical and biological sciences. While the spatial order characterizing for example the crystals has been the basis of many advances in contemporary physics, most complex systems in nature do not offer such high degree of order. Many of these systems form complex networks whose nodes are the elements of the system and edges represent the interactions between them. Traditionally complex networks have been described by the random graph theory founded in 1959 by Paul Erdohs and Alfred Renyi. One of the defining features of random graphs is that they are statistically homogeneous, and their degree distribution (characterizing the spread in the number of edges starting from a node) is a Poisson distribution. In contrast, recent empirical studies, including the work of our group, indicate that the topology of real networks is much richer than that of random graphs. In particular, the degree distribution of real networks is a power-law, indicating a heterogeneous topology in which the majority of the nodes have a small degree, but there is a significant fraction of highly connected nodes that play an important role in the connectivity of the network. The scale-free topology of real networks has very important consequences on their functioning. For example, we have discovered that scale-free networks are extremely resilient to the random disruption of their nodes. On the other hand, the selective removal of the nodes with highest degree induces a rapid breakdown of the network to isolated subparts that cannot communicate with each other. The non-trivial scaling of the degree distribution of real networks is also an indication of their assembly and evolution. Indeed, our modeling studies have shown us that there are general principles governing the evolution of networks. Most networks start from a small seed and grow by the addition of new nodes which attach to the nodes already in the system. This process obeys preferential attachment: the new nodes are more likely to connect to nodes with already high degree. We have proposed a simple model based on these two principles wich was able to reproduce the power-law degree distribution of real networks. Perhaps even more importantly, this model paved the way to a new paradigm of network modeling, trying to capture the evolution of networks, not just their static topology. Predicting the diffusion of information on social networks is a key problem for applications like Opinion Leader Detection, Buzz Detection or Viral Marketing. Many recent diffusion models are direct extensions of the Cascade and Threshold models, initially proposed for epidemiology and social studies. In such models, the diffusion process is based on the dynamics of interactions between neighbor nodes in the network (the social pressure), and largely ignores important dimensions as the content of the piece of information diffused. We propose here a new family of probabilistic models that aims at predicting how a content diffuses in a network by making use of additional dimensions: the content of the piece of information diffused, user's profile and willingness to diffuse. These models are illustrated and compared with other approaches on two blog datasets. The experimental results obtained on these datasets show that taking into account the content of the piece of information diffused is important to accurately model the diffusion process. In real-world and online social networks, individuals receive and transmit information in real time. Cascading information transmissions (e.g. phone calls, text messages, social media posts) may be understood as a realization of a diffusion process operating on the network, and its branching path can be represented by a directed tree. The process only traverses and thus reveals a limited portion of the edges. The network reconstruction inference problem is to infer the unrevealed connections. Most existing approaches derive a likelihood and attempt to find the network topology maximizing the likelihood, a problem that is highly intractable. In this paper, we focus on the network reconstruction problem for a broad class of real-world diffusion processes, exemplified by a network diffusion scheme called respondent-driven sampling (RDS). We prove that under realistic and general models of network diffusion, the posterior distribution of an observed RDS realization is a Bayesian log-submodular model.We then propose VINE (Variational Inference for Network rEconstruction), a novel, accurate, and computationally efficient variational inference algorithm, for the network reconstruction problem under this model. Crucially, we do not assume any particular probabilistic model for the underlying network. VINE recovers any connected graph with high accuracy as shown by our experimental results on real-life networks. The dynamic behavior of a network in which information is changing continuously over time requires robust and efficient mechanisms for keeping nodes updated about new information. Gossip protocols are mechanisms for this task in which nodes communicate with one another according to some underlying deterministic or randomized algorithm, exchanging information in each communication step. In a variety of contexts, the use of randomization to propagate information has been found to provide better reliability and scalability than more regimented deterministic approaches.In many settings, such as a cluster of distributed computing hosts, new information is generated at individual nodes, and is most "interesting" to nodes that are nearby. Thus, we propose distance-based propagation bounds as a performance measure for gossip mechanisms: a node at distance d from the origin of a new piece of information should be able to learn about this information with a delay that grows slowly with d, and is independent of the size of the network.For nodes arranged with uniform density in Euclidean space, we present natural gossip mechanisms, called spatial gossip, that satisfy such a guarantee: new information is spread to nodes at distance d, with high probability, in O(log1 + e d) time steps. Such a bound combines the desirable qualitative features of uniform gossip, in which information is spread with a delay that is logarithmic in the full network size, and deterministic flooding, in which information is spread with a delay that is linear in the distance and independent of the network size. Our mechanisms and their analysis resolve a conjecture of [1987].We further show an application of our gossip mechanisms to a basic resource location problem, in which nodes seek to rapidly learn the location of the nearest copy of a resource in a network. This problem, which is of considerable practical importance, can be solved by a very simple protocol using Spatial Gossip, whereas we can show that no protocol built on top of uniform gossip can inform nodes of their approximately nearest resource within poly-logarithmic time. The analysis relies on an additional useful property of spatial gossip, namely that information travels from its source to sinks along short paths not visiting points of the network far from the two nodes.