Constrained Markovian Dynamics of Random Graphs

Coolen, Anthony C. C.; Martino, Andrea De; Annibale, Alessia

doi:10.1007/s10955-009-9821-2

Cited by 49 publications

(97 citation statements)

References 42 publications

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“…The algorithm is based on a random rewiring algorithm whose elementary moves are the so-called ‘edge-swaps’ [48]. An edge swap consists of the following steps:

(a) Randomly, select four distinct nodes, namely ( i , j , k , l ).

(b) If then swap the edges, so that the adjacency matrix entries become .

…”

Section: Methodsmentioning

confidence: 99%

Using Pareto optimality to explore the topology and dynamics of the human connectome

Avena-Koenigsberger

Goñi

Betzel

et al. 2014

Phil. Trans. R. Soc. B

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Graph theory has provided a key mathematical framework to analyse the architecture of human brain networks. This architecture embodies an inherently complex relationship between connection topology, the spatial arrangement of network elements, and the resulting network cost and functional performance. An exploration of these interacting factors and driving forces may reveal salient network features that are critically important for shaping and constraining the brain's topological organization and its evolvability. Several studies have pointed to an economic balance between network cost and network efficiency with networks organized in an ‘economical’ small-world favouring high communication efficiency at a low wiring cost. In this study, we define and explore a network morphospace in order to characterize different aspects of communication efficiency in human brain networks. Using a multi-objective evolutionary approach that approximates a Pareto-optimal set within the morphospace, we investigate the capacity of anatomical brain networks to evolve towards topologies that exhibit optimal information processing features while preserving network cost. This approach allows us to investigate network topologies that emerge under specific selection pressures, thus providing some insight into the selectional forces that may have shaped the network architecture of existing human brains.

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“…The algorithm is based on a random rewiring algorithm whose elementary moves are the so-called ‘edge-swaps’ [48]. An edge swap consists of the following steps:

(a) Randomly, select four distinct nodes, namely ( i , j , k , l ).

(b) If then swap the edges, so that the adjacency matrix entries become .

…”

Section: Methodsmentioning

confidence: 99%

Using Pareto optimality to explore the topology and dynamics of the human connectome

Avena-Koenigsberger

Goñi

Betzel

et al. 2014

Phil. Trans. R. Soc. B

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show abstract

“…As a consequence, the biological network can be realistically approximated by a member of such ensemble. As an additional test, we generate synthetically a member of the maximum entropy ensemble asymptotically tailored to the production of graphs with the same degree sequence and degree correlations as the PPIN of C. elegans by using the MCMC algorithm proposed in Coolen et al [11]. The degree correlations of the resulting graph are shown in the top right panel of figure 7 and are in good agreement with the degree correlations of the PPIN that are being targeted (top left panel).…”

Section: Degree Correlations For Random Samplingmentioning

confidence: 76%

What you see is not what you get: how sampling affects macroscopic features of biological networks

Annibale

Coolen

2011

Interface Focus.

Self Cite

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We use mathematical methods from the theory of tailored random graphs to study systematically the effects of sampling on topological features of large biological signalling networks. Our aim in doing so is to increase our quantitative understanding of the relation between true biological networks and the imperfect and often biased samples of these networks that are reported in public data repositories and used by biomedical scientists. We derive exact explicit formulae for degree distributions and degree correlation kernels of sampled networks, in terms of the degree distributions and degree correlation kernels of the underlying true network, for a broad family of sampling protocols that include random and connectivity-dependent node and/or link undersampling as well as random and connectivity-dependent link oversampling. Our predictions are in excellent agreement with numerical simulations.

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“…Using an algorithm presented in Ref. [32], this allows for a uniform sampling of the graphs. Unfortunately, this approach creates correlations between subsequent graphs; hence one has to put forth additional effort to estimate mixing (decorrelation) times and a much larger numerical effort is needed.…”

Section: Discussionmentioning

confidence: 99%

Bias in generation of random graphs

Klein-Hennig

Hartmann

2012

Phys. Rev. E

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We study the statistical properties of the generation of random graphs according to the configuration model, in which one assigns randomly degrees to nodes. This model is often used, for example, for the scale-free degree distribution ~d(-γ). For the efficient variant, where nonfeasible edges are rejected and the construction of a graph continues, there exists a bias, which we calculate explicitly for a small sample ensemble. We find that this bias does not disappear with growing system size. This becomes visible, for example, also for scale-free graphs when measuring quantities such as the graph diameter. Hence the efficient generation of general scale-free graphs with a very broad distribution (γ < 2) remains an open problem.

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Constrained Markovian Dynamics of Random Graphs

Cited by 49 publications

References 42 publications

Using Pareto optimality to explore the topology and dynamics of the human connectome

Using Pareto optimality to explore the topology and dynamics of the human connectome

What you see is not what you get: how sampling affects macroscopic features of biological networks

Bias in generation of random graphs

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