Autocatalytic reaction on low-dimensional substrates

Agliari, Elena; Burioni, Raffaella; Cassi, Davide; Neri, Franco Maria

doi:10.1007/s00214-007-0323-5

Cited by 20 publications

(16 citation statements)

References 30 publications

(42 reference statements)

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“…The following investigation is just aimed at the study of its topological features, which, as well known, are intimately connected with the dynamical properties of phenomena occurring on the network itself (e.g. diffusion [26,2,5], transport [9,7], critical properties [31,8], coherent propagation [6], relaxation [44], just to cite a few). We first focus on the topology neglecting the role of weights and we say that two nodes i and j are connected whenever J ij is strictly positive; disorder on couplings will be addressed in Sec.…”

Section: The Emergent Networkmentioning

confidence: 99%

Equilibrium statistical mechanics on correlated random graphs

Barra¹,

Agliari²

2011

J. Stat. Mech.

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Biological and social networks have recently attracted enormous attention between physicists. Among several, two main aspects may be stressed: A non trivial topology of the graph describing the mutual interactions between agents exists and/or, typically, such interactions are essentially (weighted) imitative. Despite such aspects are widely accepted and empirically confirmed, the schemes currently exploited in order to generate the expected topology are based on a-priori assumptions and in most cases still implement constant intensities for links. Here we propose a simple shift [−1, +1] → [0, +1] in the definition of patterns in an Hopfield model to convert frustration into dilution: By varying the bias of the pattern distribution, the network topology -which is generated by the reciprocal affinities among agents (the Hebbian kernel)-crosses various well known regimes (fully connected, linearly diverging connectivity, extreme dilution scenario, no network), coupled with small world properties, which, in this context, are emergent and no longer imposed a-priori. The model is investigated at first focusing on these topological properties of the emergent network, then its thermodynamics is analytically solved (at a replica symmetric level) by extending the double stochastic stability technique, and presented together with its fluctuation theory for a picture of criticality: both a statistical mechanics and a topological phase diagrams are obtained. Overall the picture depicted from statistical mechanics is quite intuitive: at least at equilibrium, dilution (of whatever kind) simply decreases the strength of the coupling felt by the spins, but leaves the paramagnetic/ferromagnetic flavors unchanged. The main difference with respect to previous investigations and a naive picture is that within our approach replicas do not appear: instead of (multi)-overlaps as order parameters, we introduce a class of magnetizations on all the possible sub-graphs belonging to the main one investigated: As a consequence, for these objects a closure for a self-consistent relation is achieved. 1As we will see, small values of a give rise to highly correlated, diluted networks, while, as a gets larger the network gets more and more connected and correlation among links vanishes.Even though the theory is defined at each finite V and L, as standard in statistical mechanics, we are interested in the large V behavior (such that, under central limit theorem permissions, deviations from averaged values become negligible and the theory predictive). To this task we find meaningful to let even L diverge linearly with the system size (to bridge conceptually to high storage neural networks), such that lim V →∞ L/V = α defines α as another control parameter. Finally, since we are interested in the regime of large V and large L we will often confuse V with V − 1 and L with L − 1. 4

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Section: The Emergent Networkmentioning

confidence: 99%

Equilibrium statistical mechanics on correlated random graphs

Barra¹,

Agliari²

2011

J. Stat. Mech.

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“…In fact, the trapping problem of multiple targets has already received great attention [11][12][13][14][15] .…”

Section: Introductionmentioning

confidence: 99%

Multitarget search on complex networks: A logarithmic growth of global mean random cover time

Weng

Zhang

Small

et al. 2017

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We investigate multitarget search on complex networks and derive an exact expression for the mean random cover time that quantifies the expected time a walker needs to visit multiple targets. Based on this, we recover and extend some interesting results of multitarget search on networks. Specifically, we observe the logarithmic increase of the global mean random cover time with the target number for a broad range of random search processes, including generic random walks, biased random walks, and maximal entropy random walks. We show that the logarithmic growth pattern is a universal feature of multi-target search on networks by using the annealed network approach and the Sherman-Morrison formula. Moreover, we find that for biased random walks, the global mean random cover time can be minimized, and that the corresponding optimal parameter also minimizes the global mean first passage time, pointing towards its robustness. Our findings further confirm that the logarithmic growth pattern is a universal law governing multitarget search in confined media.

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“…The mean first-passage time is also intrinsically related to a number of different problems [12]. In the context of reaction-diffusion processes it represents the mean time to react for a particle diffusing in the presence of an active site located in i 0 [8,16,17,18], which is sometimes referred to as target problem. Not only: the mean firstpassage time defined above also describes the asymptotic behaviour of the average time for two diffusive particles to first encounter [19].…”

Section: Introductionmentioning

confidence: 99%

Exact mean first-passage time on the T-graph

2008

Self Cite

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We consider a simple random walk on the T-fractal and we calculate the exact mean time taug to first reach the central node i0. The mean is performed over the set of possible walks from a given origin and over the set of starting points uniformly distributed throughout the sites of the graph, except i0. By means of analytic techniques based on decimation procedures, we find the explicit expression for taug as a function of the generation g and of the volume V of the underlying fractal. Our results agree with the asymptotic ones already known for diffusion on the T-fractal and, more generally, they are consistent with the standard laws describing diffusion on low-dimensional structures.

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Autocatalytic reaction on low-dimensional substrates

Cited by 20 publications

References 30 publications

Equilibrium statistical mechanics on correlated random graphs

Equilibrium statistical mechanics on correlated random graphs

Multitarget search on complex networks: A logarithmic growth of global mean random cover time

Exact mean first-passage time on the T-graph

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