Synchronization Properties of Network Motifs

We address the problem of understanding the variable abundance of 3-node and 4-node subgraphs (motifs) in complex networks from a dynamical point of view. As a criterion in the determination of the functional significance of a n-node subgraph, we propose an analytic method to measure the stability of the synchronous state (SSS) the subgraph displays. We show that, for undirected graphs, the SSS is correlated with the relative abundance, while in directed graphs the correlation exists only for some specific motifs.Comment: 7 pages, 3 figure

Gershgorin disks for multiple eigenvalues of non-negative matrices

Gershgorin's famous circle theorem states that all eigenvalues of a square matrix lie in disks (called Gershgorin disks) around the diagonal elements. Here we show that if the matrix entries are non-negative and an eigenvalue has geometric multiplicity at least two, then this eigenvalue lies in a smaller disk. The proof uses geometric rearrangement inequalities on sums of higher dimensional real vectors which is another new result of this paper

Stochastic dynamics of model proteins on a directed graph

A method for reconstructing the energy landscape of simple polypeptidic chains is described. We show that we can construct an equivalent representation of the energy landscape by a suitable directed graph. Its topological and dynamical features are shown to yield an effective estimate of the time scales associated with the folding and with the equilibration processes. This conclusion is drawn by comparing molecular dynamics simulations at constant temperature with the dynamics on the graph, defined by a temperature dependent Markov process. The main advantage of the graph representation is that its dynamics can be naturally renormalized by collecting nodes into "hubs", while redefining their connectivity. We show that both topological and dynamical properties are preserved by the renormalization procedure. Moreover, we obtain clear indications that the heteropolymers exhibit common topological properties, at variance with the homopolymer, whose peculiar graph structure stems from its spatial homogeneity. In order to obtain a clear distinction between a "fast folder" and a "slow folder" in the heteropolymers one has to look at kinetic features of the directed graph. We find that the average time needed to the fast folder for reaching its native configuration is two orders of magnitude smaller than its equilibration time, while for the bad folder these time scales are comparable. Accordingly, we can conclude that the strategy described in this paper can be successfully applied also to more realistic models, by studying their renormalized dynamics on the directed graph, rather than performing lengthy molecular dynamics simulations.Comment: 15 pages, 12 figure

The road to deterministic matrices with the restricted isometry property

The restricted isometry property (RIP) is a well-known matrix condition that provides state-of-the-art reconstruction guarantees for compressed sensing. While random matrices are known to satisfy this property with high probability, deterministic constructions have found less success. In this paper, we consider various techniques for demonstrating RIP deterministically, some popular and some novel, and we evaluate their performance. In evaluating some techniques, we apply random matrix theory and inadvertently find a simple alternative proof that certain random matrices are RIP. Later, we propose a particular class of matrices as candidates for being RIP, namely, equiangular tight frames (ETFs). Using the known correspondence between real ETFs and strongly regular graphs, we investigate certain combinatorial implications of a real ETF being RIP. Specifically, we give probabilistic intuition for a new bound on the clique number of Paley graphs of prime order, and we conjecture that the corresponding ETFs are RIP in a manner similar to random matrices.Comment: 24 page

Synchronisation in networks of delay-coupled type-I excitable systems

We use a generic model for type-I excitability (known as the SNIPER or SNIC model) to describe the local dynamics of nodes within a network in the presence of non-zero coupling delays. Utilising the method of the Master Stability Function, we investigate the stability of the zero-lag synchronised dynamics of the network nodes and its dependence on the two coupling parameters, namely the coupling strength and delay time. Unlike in the FitzHugh-Nagumo model (a model for type-II excitability), there are parameter ranges where the stability of synchronisation depends on the coupling strength and delay time. One important implication of these results is that there exist complex networks for which the adding of inhibitory links in a small-world fashion may not only lead to a loss of stable synchronisation, but may also restabilise synchronisation or introduce multiple transitions between synchronisation and desynchronisation. To underline the scope of our results, we show using the Stuart-Landau model that such multiple transitions do not only occur in excitable systems, but also in oscillatory ones.Comment: 10 pages, 9 figure