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Abstracts and Talk Materials
International Workshop on Complex Systems & Networks
September 5 - 7, 2012

Keywords of the presentation: Synchronization, bursting, network topology

We study the influence of individual neuronal dynamics and network topology on synchronization of bursting neurons. We demonstrate that the type and duration of bursting are the critical characteristics that determine the synchronization properties of the network. We show that the onset of synchrony in an excitatory-inhibitory network with any coupling topology admitting spike synchronization is ensured by one single condition.

Keywords of the presentation: complex network, synchronization, Laplacian spectrum

The synchronizability of a connected undirected network is essentially determined by the spectrum of its Laplacian matrix, which reflects most topological characteristics of the network such as degree distribution, shortest-path length, betweenness centrality, among others. Recently, we found that networks with best possible synchronizability are in some sense “homogenous” and “symmetric”, with several common features such as an identical degree sequence, a longest girth, and a shortest path-sum. We have verified this observation by degree-3 regular networks of small sizes, and conjectured this be true in general.

Keywords of the presentation: Causality, complex networks, seismicity, nerve cell cultures

Inferring cause-effect relationships from observations is one of the fundamental challenges in natural sciences and beyond. Due to the technological advances over the last decade, the amount of observations and data available to characterize complex systems and their dynamics has increased substantially, making scientists face this challenge in many different areas. Specific examples of general importance include seismicity as well as nerve cell cultures and even the brain. In this talk, I will discuss new methods from nonlinear sciences and complex network theory to infer causal interactions and characterize spatio-temporal clustering of point processes with a particular focus on the aforementioned applications. In particular, I will present a method that identifies triggering relationships between earthquakes such that seismicity can be described by triggering cascades and mapped to a complex network. This novel approach allows one to tackle many open questions related to earthquakes and seismicity.

This talk surveys recent results on phase-shift synchrony for periodic solutions of coupled systems of differential equations. The mathematical questions were motivated by previous work on quadrupedal gaits and help study a generalized model for binocular rivalry.

How many large eigenvalues can a graph have? An answer depends on the interpretation of what it means for en eigenvalue to be large. This question and some related problems in extremal algebraic graph theory will be discussed.

Keywords of the presentation: Networks, neurons, epilepsy

Many neurological diseases, such as epilepsy and Parkinson's disease, present symptoms of pathological neuronal activity. Large bursts of neuronal activity or large scale synchronous oscillations may arise from changes in the dynamics of the neurons and/or how they are connected. How changes in network topology result in these pathological behaviors, as well as how do neurons refine their connections to prevent these behaviors from emerging normally, are fundamental questions in neuroscience. There are many limitations in predicting network activity provided the neuronal dynamics and the network topology. This workshop could make a significant impact in addressing many of these problems.

Keywords of the presentation: network topology, neuronal networks, mean-field equations

We introduce a random network model in which one can prescribe the frequency of second order edge motifs. We derive effective equations for the activity of spiking neuron models coupled via such networks. A key consequence of the motif-induced edge correlations is that one cannot derive closed equations for average activity of the nodes (the average firing rate neurons) but instead must develop the equations in terms of the average activity of the edges (the synaptic drives). As a result, the network topology increases the dimension of the effective dynamics and allows for a larger repertoire of behavior. We demonstrate this behavior through simulations of spiking neuronal networks.

Keywords of the presentation: Genes, Boolean

We consider Boolean models of the dynamics of interacting genes. Stability is defined for a large Boolean network by imagining two system states that are initially close in the sense of Hamming distance and asking whether or not their evolutions lead to subsequent divergence or convergence. We find a general method for answering this question and investigate the impact on stability of various factors,such as network topology (including assortitivity, community structure, small motifs, etc.), nodal dynamics, link delay times, and correlation between topological and nodal properties. We also introduce and discuss a hypothesis that orbital instability might be a causal contributor to the occurence of cancer. Reference: A.Pomerance, E.Ott, M.Girvan, W.Losert, Proc. Nat. Acad. Sci vol. 106, p. 8209 (2009).

We explore a geometric and topological approach to understanding the structural significance of edges in a complex network. To do so, we embed the complex network (or the graph $G(V, E)$ representing it) into a Euclidean space determined by the eigen-space of the Moore-Penrose pseudo-inverse of the combinatorial laplacian (denoted by $bb L^+(G)$). The element $l^+_{ij}$ in $bb L^+(G)$ ($i^{th} ~row-j^{th} ~column)$ determines the angular distance between the position vectors of nodes $i$ and $j$ in this space and is thus a measure of angular affinity between the end points of an edge $e_{ij}$; whereas the Euclidean distance between nodes $i$ and $j$, called the {em effective resistance} distance ($Omega_{ij} = l^+_{ii} + l^+_{jj} - l^+_{ij} - l^+_{ji}$), is a measure of the separation between the end-points of the edge $e_{ij}$. Our emphasis is on the topological characteristics reflected in these quantities ($l^+_{ij}$ and $Omega_{ij}$) with respect to the set of connected bi-partitions of the network (spanning sub-networks with exactly two components). In particular, $l^+_{ij}$ determines the number of nodes that the node pair $(i, j)$ joined through the edge $e_{ij}$, gets dissociated from when the network breaks into two. Higher the value of $l^+_{ij}$, greater the loss in connectedness of the node pair $(i, j)$ in the bi-partitions, and more peripheral $e_{ij}$ is in the network. Therefore, $l^+_{ij}$ captures the topological centrality of the edge $e_{ij}$. On the other hand, $Omega_{ij}$ determines the strength of connectivities in the two sub-graphs representing a partition (in terms of the number of spanning trees in each part), when the edge $e_{ij}$ is eliminated. It, in fact, is related to the fraction of spanning trees of the network that $e_{ij}$ is present in. Based on these topological characteristics, we motivate several important applications, such as network core identification, greedy spanning tree extraction etc, that are relevant to complex network analysis. We demonstrate the properties of our metrics with the help of example as well as real world networks from diverse domains.

We have studied large, heterogeneous populations of discrete chemical oscillators (~100,000) to characterize two different types of density-dependent transitions to synchronized oscillatory behavior. For different chemical exchange rates between the oscillators and the surrounding solution, we find, with increasing oscillator number density, (1) the gradual Kuramoto synchronization of oscillatory activity or (2) the sudden quorum sensing "switching on" of synchronized oscillatory activity. We also describe the formation of phase clusters, where each cluster has the same frequency but is phase shifted with respect to other clusters, giving rise to a global signal that is more complex than that of the individual oscillators. Finally, we describe experimental studies of chimera states and their relation to other synchronization states in populations of coupled chemical oscillators.

A. F. Taylor et al., Science 323, 614 (2009). A. F. Taylor et al., Angewandte Chemie Int. Ed. 50, 10161 (2011). M. R. Tinsley et al., Nature Physics, doi:10.1038/nphys2371 (2012).

Keywords of the presentation: machine learning, feature selection, classification, traffic anomaly detection, network intrusion

Traffic anomalies in communication networks greatly degrade network performance. In this talk, I will survey statistical and machine learning techniques that are used to classify and detect network anomalies such as Internet worms that affect performance of routing protocols. Various classification features are used to design anomaly detection mechanisms. They are used to classify test datasets and identify the correct anomaly types.

We consider a class of models that generalizes the Bak Sneppen model that can be used to study evolution. Agents with random fitnesses are located at the vertices of a graph $G$. At every time-step the agent with the worst fitness emph{and its neighbors on $G$} are replaced by new agents with random fitnesses.

By using Order Statistics and Dynamical Systems, we succeeded in solving a number of questions about how the distribution of the fitnesses evolves under this process. In particular we can prove that for the models that describe in-species evolution, all initial conditions converge to a (not necessarily unique) discrete measure. In contrast for models that describe speciation, every initial probability measure will converge to a (unique) absolutely continuous measure.

The conclusion is that in-species evolution can optimize fitness but all agents tend to become identical (no diversity). Speciation on the other hand also improves fitness a little less dramatically, but diversity is retained: a range of fitnesses is preserved.

This is a preliminary report of research in progress.

Joint work with F. J. Prieto(Univ Carlos III, Madrid, Spain) and M. Orhai (Portland State University).

Keywords of the presentation: control, complex networks, graph, synchronization

We look at complex networks of coupled dynamical systems where an external forcing control signal is applied to the network in order to align the state of all the individual systems to the forcing signal. We study how the eff ectiveness of such control is related to the topology of the underlying graph. For instance, we show that for the cycle graph, the best way to achieve control is by applying control to systems that are approximately equally spaced apart.

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