A measure of centrality based on the spectrum of the Laplacian

We introduce a family of new centralities, the k-spectral centralities. k-Spectral centrality is a measurement of importance with respect to the deformation of the graph Laplacian associated with the graph. Due to this connection, k-spectral centralities have various interpretations in terms of spectrally determined information. We explore this centrality in the context of several examples. While for sparse unweighted networks 1-spectral centrality behaves similarly to other standard centralities, for dense weighted networks they show different properties. In summary, the k-spectral centralities provide a novel and useful measurement of relevance (for single network elements as well as whole subnetworks) distinct from other known measures.Comment: 12 pages, 6 figures, 2 table

Opinion Dynamics in Heterogeneous Networks: Convergence Conjectures and Theorems

Recently, significant attention has been dedicated to the models of opinion dynamics in which opinions are described by real numbers, and agents update their opinions synchronously by averaging their neighbors' opinions. The neighbors of each agent can be defined as either (1) those agents whose opinions are in its "confidence range," or (2) those agents whose "influence range" contain the agent's opinion. The former definition is employed in Hegselmann and Krause's bounded confidence model, and the latter is novel here. As the confidence and influence ranges are distinct for each agent, the heterogeneous state-dependent interconnection topology leads to a poorly-understood complex dynamic behavior. In both models, we classify the agents via their interconnection topology and, accordingly, compute the equilibria of the system. Then, we define a positive invariant set centered at each equilibrium opinion vector. We show that if a trajectory enters one such set, then it converges to a steady state with constant interconnection topology. This result gives us a novel sufficient condition for both models to establish convergence, and is consistent with our conjecture that all trajectories of the bounded confidence and influence models eventually converge to a steady state under fixed topology.Comment: 22 pages, Submitted to SIAM Journal on Control and Optimization (SICON

The Impact of Network Flows on Community Formation in Models of Opinion Dynamics

We study dynamics of opinion formation in a network of coupled agents. As the network evolves to a steady state, opinions of agents within the same community converge faster than those of other agents. This framework allows us to study how network topology and network flow, which mediates the transfer of opinions between agents, both affect the formation of communities. In traditional models of opinion dynamics, agents are coupled via conservative flows, which result in one-to-one opinion transfer. However, social interactions are often non-conservative, resulting in one-to-many transfer of opinions. We study opinion formation in networks using one-to-one and one-to-many interactions and show that they lead to different community structure within the same network.Comment: accepted for publication in The Journal of Mathematical Sociology. arXiv admin note: text overlap with arXiv:1201.238