Network science describes how entities in complex systems interact, and argues that the structure of the network influences processing. Clustering coefficient, C – one measure of network structure – refers to the extent to which neighbors of a node are also neighbors of each other. Previous simulations suggest that networks with low C dissipate information (or disease) to a large portion of the network, whereas in networks with high C information (or disease) tends to be constrained to a smaller portion of the network (Newman, 2003). In the present simulation we examined how C influenced the spread of activation to a specific node, simulating retrieval of a specific lexical item in a phonological network. The results of the network simulation showed that words with lower C had higher activation values (indicating faster or more accurate retrieval from the lexicon) than words with higher C. These results suggest that a simple mechanism for lexical retrieval can account for the observations made in Chan and Vitevitch (2009), and have implications for diffusion dynamics in other fields.

The cover time and mixing time of graphs has much relevance to algorithmic applications and has been extensively investigated. Recently, with the advent of ad hoc and sensor networks, an interesting class of random graphs, namely random geometric graphs, has gained new relevance and its properties have been the subject of much study. A random geometric graph G(n, r ) is obtained by placing n points uniformly at random on the unit square and connecting two points iff their Euclidean distance is at most r . The phase transition behavior with respect to the radius r of such graphs has been of special interest. We show that there exists a critical radius r opt such that for any r ≥ r opt G(n, r ) has optimal cover time of Θ(n log n) with high probability, and, importantly, r opt = Θ(r con ) where r con denotes the critical radius guaranteeing asymptotic connectivity. Moreover, since a disconnected graph has infinite cover time, there is a phase transition and the corresponding threshold width is O(r con ). On the other hand, the radius required for rapid mixing r rapid = ω(r con ), and, in particular, r rapid = Θ(1/poly(log n)). We are able to draw our results by giving a tight bound on the electrical resistance and conductance of G(n, r ) via certain constructed flows.

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This paper examines a schema for graph-theoretic clustering using node-based resilience measures. Node-based resilience measures optimize an objective based on a critical set of nodes whose removal causes some severity of disconnection in the network. Beyond presenting a general framework for the usage of node based resilience measures for variations of clustering problems, we experimentally validate the usefulness of such methods in accomplishing the following: (i) clustering a graph in one step without knowing the number of clusters a priori; (ii) removing noise from noisy data; and (iii) detecting overlapping communities. We demonstrate that this clustering schema can be applied successfully using a wide range of data, including both real and synthetic networks, both natively in graph form and also expressed as point sets.

The purpose of this work is four-fold: (1) We propose a new measure of network resilience in the face of targeted node attacks, vertex attack tolerance, represented mathematically as τ (G) = minS⊂V |S| |V −S−Cmax(V −S)|+1 , and prove that for d-regular graphs τ (G) = Θ(Φ(G)) where Φ(G) denotes conductance, yielding spectral bounds as corollaries. (2) We systematically compare τ (G) to known resilience notions, including integrity, tenacity, and toughness, and evidence the dominant applicability of τ for arbitrary degree graphs. (3) We explore the computability of τ , first by establishing the hardness of approximating unsmoothened vertex attack tolerance τ (G) = minS⊂V |S| |V −S−Cmax(V −S)|under various plausible computational complexity assumptions, and then by presenting empirical results on the performance of a betweenness centrality based heuristic algorithm applied not only to τ but several other hard resilience measures as well. (4) Applying our algorithm, we find that the random scale-free network model is more resilient than the Barabasi−Albert preferential attachment model, with respect to all resilience measures considered.