Dynamical Growth Using Preferential Attachment and Emergent Probability

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Dynamical Growth Using Preferential Attachment
and Emergent Probability

• Rocio Ruelas
• August 1, 2007
• REU Physics

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Networks

• A network is any inter-connected group or system.
• It consists of nodes and edges.
• The edges can be directed or undirected and can
  have different weights.
• The adjacency matrix of a network contains all of
  the information of that network.
• Examples of networks include social relations,
  transportation systems, metabolic interactions
  and the Internet.

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Model of Network Growth

• Begins with a few nodes with a high degree of
  interconnectivity.
• Adds new nodes and edges using preferential
  attachment.
• Most edges are directed so that older nodes point
  to younger nodes, but there is a small
  probability that a younger node will point to an
  older node.
• After a certain amount of growth, we turn on
  nodes and make them functional.
• From the functional nodes we sort them into
  communities.
• Then we search for recurrent schemes (RS), which
  are sets of nodes that form a loop.
• Once a set is found, it is compressed into one
  node and the growth process is repeated.

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Growing the Network
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Functionality
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Functionality
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Community Splitting

• Modularity is a measure of the quality of a
  splitting.
• Communities are found through a divisive
  hierarchical greedy algorithm.
• Starting from a graph of all the functional
  nodes, each node is assigned to one of two
  communities. If the assignment increases the
  modularity then the graph is split in two.
• The same process occurs for the two communities
  and will continue for all sub-communities until
  an assignment does not correspond to an increase
  in modularity.

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Recurrent Schemes

• RS are found by taking each of the communities
  and removing trees until only the loops are left.
• Once a set is found, it is merged into one super
  node that maintains all of the connections
  between the RS and external nodes.
• We can then observe the dynamics between multiple
  RS.

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Why Use This Model?

• Original inspiration from Bernard Lonergan who
  wrote Insight A Study of Human Understanding
  in1957. Which outlined the process in which
  knowledge is acquired.
• Useful for understanding biological processes and
  chemical interactions.
• Useful for a general evolutionary model.
• It helps us understand how complexity can arise
  from simple beginnings.

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Acknowledgments

• Professor Michael Bretz
• Center for Complex Systems
• Howard Oishi
• Sarah Cherng
• REU Physics Program