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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

An illustration of the evolution of state groups. This figure plots the size of eight different state groups over 200 000 time steps (p = 1; q = r = w = 0.01). The eight state groups shown (of a total of 57 that existed at some point during the simulation run) were each the largest in the population at some point in time.
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RSIF20100524F4: An illustration of the evolution of state groups. This figure plots the size of eight different state groups over 200 000 time steps (p = 1; q = r = w = 0.01). The eight state groups shown (of a total of 57 that existed at some point during the simulation run) were each the largest in the population at some point in time.

Mentions: We find that, under appropriate parameters, the model shows community structure with several concurrent groups, many of which have relatively long lifetimes (figure 3). The sizes of the groups, as well as their composition, are dynamic as nodes join and leave them in the close interplay of state changes and edge rewiring (figure 4). Again, we see that, under a wide range of parameters, some global properties, such as clustering coefficient or network modularity, stabilize as the network keeps evolving. Mathematical analysis (see appendix A.1) also predicts stability of network modularity and gives a good approximation of the corresponding topological network modularity (as with figure 2) when the state spread parameters maintain a moderate number of groups (between n/50 and n/3).


Stability in flux: community structure in dynamic networks.

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

An illustration of the evolution of state groups. This figure plots the size of eight different state groups over 200 000 time steps (p = 1; q = r = w = 0.01). The eight state groups shown (of a total of 57 that existed at some point during the simulation run) were each the largest in the population at some point in time.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

RSIF20100524F4: An illustration of the evolution of state groups. This figure plots the size of eight different state groups over 200 000 time steps (p = 1; q = r = w = 0.01). The eight state groups shown (of a total of 57 that existed at some point during the simulation run) were each the largest in the population at some point in time.
Mentions: We find that, under appropriate parameters, the model shows community structure with several concurrent groups, many of which have relatively long lifetimes (figure 3). The sizes of the groups, as well as their composition, are dynamic as nodes join and leave them in the close interplay of state changes and edge rewiring (figure 4). Again, we see that, under a wide range of parameters, some global properties, such as clustering coefficient or network modularity, stabilize as the network keeps evolving. Mathematical analysis (see appendix A.1) also predicts stability of network modularity and gives a good approximation of the corresponding topological network modularity (as with figure 2) when the state spread parameters maintain a moderate number of groups (between n/50 and n/3).

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