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Stability in flux: community structure in dynamic networks.

Bryden J, Funk S, Geard N, Bullock S, Jansen VA - J R Soc Interface (2010)

Bottom Line: We show that if nodes rewire their edges based on fixed node states, the network modularity reaches a stable equilibrium which we quantify analytically.Furthermore, if node state is not fixed, but can be adopted from neighbouring nodes, the distribution of group sizes reaches a dynamic equilibrium, which remains stable even as the composition and identity of the groups change.These results show that dynamic networks can maintain the stable community structure that has been observed in many social and biological systems.

View Article: PubMed Central - PubMed

Affiliation: School of Biological Sciences, Royal Holloway, University of London, Egham TW20 0EX, UK. john.bryden@rhul.ac.uk

ABSTRACT
The structure of many biological, social and technological systems can usefully be described in terms of complex networks. Although often portrayed as fixed in time, such networks are inherently dynamic, as the edges that join nodes are cut and rewired, and nodes themselves update their states. Understanding the structure of these networks requires us to understand the dynamic processes that create, maintain and modify them. Here, we build upon existing models of coevolving networks to characterize how dynamic behaviour at the level of individual nodes generates stable aggregate behaviours. We focus particularly on the dynamics of groups of nodes formed endogenously by nodes that share similar properties (represented as node state) and demonstrate that, under certain conditions, network modularity based on state compares well with network modularity based on topology. We show that if nodes rewire their edges based on fixed node states, the network modularity reaches a stable equilibrium which we quantify analytically. Furthermore, if node state is not fixed, but can be adopted from neighbouring nodes, the distribution of group sizes reaches a dynamic equilibrium, which remains stable even as the composition and identity of the groups change. These results show that dynamic networks can maintain the stable community structure that has been observed in many social and biological systems.

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Related in: MedlinePlus

Size distribution of state groups. Shown is the mean size of the ith largest group across 20 snapshots from a simulation run (circles; a = 100; b = 0.001; c = 0.3), error bars indicating one standard deviation. Also shown is the distribution as predicted by equation (3.7) (crosses), obtained by sampling from y = 28 random numbers summing up to n = 1000, using the algorithm of [40], until convergence was obtained. Despite the continually changing composition of state groups in a population (figure 4), distribution of group sizes is relatively stable over time.
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RSIF20100524F5: Size distribution of state groups. Shown is the mean size of the ith largest group across 20 snapshots from a simulation run (circles; a = 100; b = 0.001; c = 0.3), error bars indicating one standard deviation. Also shown is the distribution as predicted by equation (3.7) (crosses), obtained by sampling from y = 28 random numbers summing up to n = 1000, using the algorithm of [40], until convergence was obtained. Despite the continually changing composition of state groups in a population (figure 4), distribution of group sizes is relatively stable over time.

Mentions: If we assume such exchanges of nodes between groups of states to occur completely randomly, the probability distribution Pi of groups that have i nodes is given by the Boltzmann distribution (see appendix A.2)3.7Simulations confirm that the state distribution does indeed stabilize (figure 5). However, while the shape of the distribution remains relatively constant, the identity of groups at a particular rank does not. The ongoing dynamics at the node level causes states to grow and shrink in abundance (figure 6).


Stability in flux: community structure in dynamic networks.

Bryden J, Funk S, Geard N, Bullock S, Jansen VA - J R Soc Interface (2010)

Size distribution of state groups. Shown is the mean size of the ith largest group across 20 snapshots from a simulation run (circles; a = 100; b = 0.001; c = 0.3), error bars indicating one standard deviation. Also shown is the distribution as predicted by equation (3.7) (crosses), obtained by sampling from y = 28 random numbers summing up to n = 1000, using the algorithm of [40], until convergence was obtained. Despite the continually changing composition of state groups in a population (figure 4), distribution of group sizes is relatively stable over time.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3104331&req=5

RSIF20100524F5: Size distribution of state groups. Shown is the mean size of the ith largest group across 20 snapshots from a simulation run (circles; a = 100; b = 0.001; c = 0.3), error bars indicating one standard deviation. Also shown is the distribution as predicted by equation (3.7) (crosses), obtained by sampling from y = 28 random numbers summing up to n = 1000, using the algorithm of [40], until convergence was obtained. Despite the continually changing composition of state groups in a population (figure 4), distribution of group sizes is relatively stable over time.
Mentions: If we assume such exchanges of nodes between groups of states to occur completely randomly, the probability distribution Pi of groups that have i nodes is given by the Boltzmann distribution (see appendix A.2)3.7Simulations confirm that the state distribution does indeed stabilize (figure 5). However, while the shape of the distribution remains relatively constant, the identity of groups at a particular rank does not. The ongoing dynamics at the node level causes states to grow and shrink in abundance (figure 6).

Bottom Line: We show that if nodes rewire their edges based on fixed node states, the network modularity reaches a stable equilibrium which we quantify analytically.Furthermore, if node state is not fixed, but can be adopted from neighbouring nodes, the distribution of group sizes reaches a dynamic equilibrium, which remains stable even as the composition and identity of the groups change.These results show that dynamic networks can maintain the stable community structure that has been observed in many social and biological systems.

View Article: PubMed Central - PubMed

Affiliation: School of Biological Sciences, Royal Holloway, University of London, Egham TW20 0EX, UK. john.bryden@rhul.ac.uk

ABSTRACT
The structure of many biological, social and technological systems can usefully be described in terms of complex networks. Although often portrayed as fixed in time, such networks are inherently dynamic, as the edges that join nodes are cut and rewired, and nodes themselves update their states. Understanding the structure of these networks requires us to understand the dynamic processes that create, maintain and modify them. Here, we build upon existing models of coevolving networks to characterize how dynamic behaviour at the level of individual nodes generates stable aggregate behaviours. We focus particularly on the dynamics of groups of nodes formed endogenously by nodes that share similar properties (represented as node state) and demonstrate that, under certain conditions, network modularity based on state compares well with network modularity based on topology. We show that if nodes rewire their edges based on fixed node states, the network modularity reaches a stable equilibrium which we quantify analytically. Furthermore, if node state is not fixed, but can be adopted from neighbouring nodes, the distribution of group sizes reaches a dynamic equilibrium, which remains stable even as the composition and identity of the groups change. These results show that dynamic networks can maintain the stable community structure that has been observed in many social and biological systems.

Show MeSH
Related in: MedlinePlus