Small Worlds With Diffusion

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WHAT IS IT?

This model explores the formation of networks that result in the "small world" phenomenon -- the idea that a person is only a couple of connections away any other person in the world.

A popular example of the small world phenomenon is the network formed by actors appearing in the same movie (e.g. the "six degrees of Kevin Bacon" game), but small worlds are not limited to people-only networks. Other examples range from power grids to the neural networks of worms. This model illustrates some general, theoretical conditions under which small world networks between people or things might occur.

HOW IT WORKS

This model is an adaptation of a model proposed by Duncan Watts and Steve Strogatz (1998). It begins with a network where each person (or "node") is connected to his or her two neighbors on either side. The REWIRE-ONE button picks a random connection (or "edge") and rewires it. By rewiring, we mean changing one end of a connected pair of nodes, and keeping the other end the same.

The REWIRE-ALL button creates the network and then visits all edges and tries to rewire them. The REWIRING-PROBABILITY slider determines the probability that an edge will get rewired. Running REWIRE-ALL at multiple probabilities produces a range of possible networks with varying average path lengths and clustering coefficients.

To identify small worlds, the "average path length" (abbreviated "apl") and "clustering coefficient" (abbreviated "cc") of the network are calculated and plotted after the REWIRE-ONE or REWIRE-ALL buttons are pressed. These two plots are separated because the x-axis is slightly different. The REWIRE-ONE x-axis is the fraction of edges rewired so far, whereas the REWIRE-ALL x-axis is the probability of rewiring. Networks with short average path lengths and high clustering coefficients are considered small world networks. (Note: The plots for both the clustering coefficient and average path length are normalized by dividing by the values of the initial network. The monitors give the actual values.)

Average Path Length: Average path length is calculated by finding the shortest path between all pairs of nodes, adding them up, and then dividing by the total number of pairs. This shows us, on average, the number of steps it takes to get from one member of the network to another.

Clustering Coefficient: Another property of small world networks is that from one person's perspective it seems unlikely that they could be only a few steps away from anybody else in the world. This is because their friends more or less know all the same people they do. The clustering coefficient is a measure of this "all-my-friends-know-each-other" property. This is sometimes described as the friends of my friends are my friends. More precisely, the clustering coefficient of a node is the ratio of existing links connecting a node's neighbors to each other to the maximum possible number of such links. You can see this is if you press the HIGHLIGHT button and click a node, that will display all of the neighbors in blue and the edges connecting those neighbors in yellow. The more yellow links, the higher the clustering coefficient for the node you are examining (the one in pink) will be. The clustering coefficient for the entire network is the average of the clustering coefficients of all the nodes. A high clustering coefficient for a network is another indication of a small world.

HOW TO USE IT - NETWORK GENERATION

The NUM-NODES slider controls the size of the network. Choose a size and press SETUP.

Pressing the REWIRE-ONE button picks one edge at random, rewires it, and then plots the resulting network properties. The REWIRE-ONE button always rewires at least one edge (i.e., it ignores the REWIRING-PROBABILITY).

Pressing the REWIRE-ALL button re-creates the initial network (each node connected to its two neighbors on each side for a total of four neighbors) and rewires all the edges with the current rewiring probability, then plots the resulting network properties on the rewire-all plot. Changing the REWIRING-PROBABILITY slider changes the fraction of links rewired after each run.

When you press HIGHLIGHT and then point to node in the view it color-codes the nodes and edges. The node itself turns pink. Its neighbors and the edges connecting the node to those neighbors turn blue. Edges connecting the neighbors of the node to each other turn yellow. The amount of yellow between neighbors can gives you an indication of the clustering coefficient for that node. The NODE-PROPERTIES monitor displays the average path length and clustering coefficient of the highlighted node only. The AVERAGE-PATH-LENGTH and CLUSTERING-COEFFICIENT monitors display the values for the entire network.

HOW TO USE IT - DIFFUSION

In order to use the diffusion model, you must first click the SETUP button to generate a network.

The SEED CONTAGION button randomly selects a node and activates it by turning it red.

The SPREAD ONCE button runs a single round of the diffusion model. In each round, any node with a red neighbor will also turn red. Nodes that are already red are unaffected.

The SPREAD ALL button will continue to run the diffusion model until all the nodes are red.

THINGS TO NOTICE

Note that for certain ranges of the fraction of nodes, the average path length decreases faster than the clustering coefficient. In fact, there is a range of values for which the average path length is much smaller than clustering coefficient. (Note that the values for average path length and clustering coefficient have been normalized, so that they are more directly comparable.) Networks in that range are considered small worlds.

THINGS TO TRY - NETWORK GENERATION

Try plotting the values for different rewiring probabilities and observe the trends of the values for average path length and clustering coefficient. What is the relationship between rewiring probability and fraction of nodes? In other words, what is the relationship between the rewire-one plot and the rewire-all plot?

Do the trends depend on the number of nodes in the network?

Can you get a small world by repeatedly pressing REWIRE-ONE?

Set NUM-NODES to 80 and then press SETUP. Go to BehaviorSpace and run the VARY-REWIRING-PROBABILITY experiment. Try running the experiment multiple times without clearing the plot (i.e., do not run SETUP again). What range of rewiring probabilities result in small world networks?

THINGS TO TRY - DIFFUSION

First, run the diffusion model in the default network with no rewiring. Watch the x-axis on the "Success of Contagion" plot to see how many rounds it takes for the contagion to spread through the whole network. Then, rewire just one or a small number of ties, and run the diffusion model again. How does it change the process?

One of the most widely known shapes in theories of diffusion is the "S" curve. This curve describes a diffusion process with an initially slow spread, and then a rapid takeoff, followed by a final slower spreading process. Try the diffusion model in a range of different network conditions, and watch the "Success of Contagion" plot. Can you generate an "S" curve?

EXTENDING THE MODEL

Try to see if you can produce the same results if you start with a different initial network. Create new BehaviorSpace experiments to compare results.

In a precursor to this model, Watts and Strogatz created an "alpha" model where the rewiring was not based on a global rewiring probability. Instead, the probability that a node got connected to another node depended on how many mutual connections the two nodes had. The extent to which mutual connections mattered was determined by the parameter "alpha." Create the "alpha" model and see if it also can result in small world formation.

NETWORK CONCEPTS

In this model we need to find the shortest paths between all pairs of nodes. This is accomplished through the use of a standard dynamic programming algorithm called the Floyd Warshall algorithm. You may have noticed that the model runs slowly for large number of nodes. That is because the time it takes for the Floyd Warshall algorithm (or other "all-pairs-shortest-path" algorithm) to run grows polynomially with the number of nodes. For more information on the Floyd Warshall algorithm please consult: https://en.wikipedia.org/wiki/Floyd-Warshall_algorithm

NETLOGO FEATURES

Links are used extensively in this model.

Lists are used heavily in the procedures that calculates shortest paths.

Though it is not used in this model, there exists a network extension for NetLogo.

RELATED MODELS

See other models in the Networks section of the Models Library, such as Giant Component and Preferential Attachment.

CREDITS AND REFERENCES

This model is adapted from: Duncan J. Watts, Six Degrees: The Science of a Connected Age (W.W. Norton & Company, New York, 2003), pages 83-100.

The work described here was originally published in: DJ Watts and SH Strogatz. Collective dynamics of 'small-world' networks, Nature, 393:440-442 (1998)

The small worlds idea was first made popular by Stanley Milgram's famous experiment (1967) which found that two random US citizens where on average connected by six acquaintances (giving rise to the popular "six degrees of separation" expression): Stanley Milgram. The Small World Problem, Psychology Today, 2: 60-67 (1967).

This experiment was popularized into a game called "six degrees of Kevin Bacon" which you can find more information about here: http://oracleofbacon.org/

This model is an extension of a model created by Uri Wilensky. We have extended the original network generator to add a dynamic diffusion model. The original NetLogo model can be found here: http://ccl.northwestern.edu/netlogo/models/SmallWorlds

COPYRIGHT AND LICENSE

Copyright 2017 Joshua Becker

This model has been adaopted from material copyright 2005 Uri Wilensky.

CC BY-NC-SA 3.0

This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. To view a copy of this license, visit https://creativecommons.org/licenses/by-nc-sa/3.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.

Comments and Questions

How to cite a Small world with diffusion (Question)

can someone please tell me how to cite small world with diffusion by joshua becker

Posted 4 days ago

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turtles-own
[
  node-clustering-coefficient
  distance-from-other-turtles   ;; list of distances of this node from other turtles
]

links-own
[
  rewired?                    ;; keeps track of whether the link has been rewired or not
]

globals
[
  clustering-coefficient               ;; the clustering coefficient of the network; this is the
                                       ;; average of clustering coefficients of all turtles
  average-path-length                  ;; average path length of the network
  clustering-coefficient-of-lattice    ;; the clustering coefficient of the initial lattice
  average-path-length-of-lattice       ;; average path length of the initial lattice
  infinity                             ;; a very large number.
                                         ;; used to denote distance between two turtles which
                                         ;; don't have a connected or unconnected path between them
  highlight-string                     ;; message that appears on the node properties monitor
  number-rewired                       ;; number of edges that have been rewired. used for plots.
  rewire-one?                          ;; these two variables record which button was last pushed
  rewire-all?
]

;;;;;;;;;;;;;;;;;;;;;;;;
;;; Setup Procedures ;;;
;;;;;;;;;;;;;;;;;;;;;;;;

to startup
  set highlight-string ""
end 

to setup
  ca
  set infinity 99999  ;; just an arbitrary choice for a large number
  set-default-shape turtles "circle"
  make-turtles

  ;; set up a variable to determine if we still have a connected network
  ;; (in most cases we will since it starts out fully connected)
  let success? false
  while [not success?] [
    ;; we need to find initial values for lattice
    wire-them
    ;;calculate average path length and clustering coefficient for the lattice
    set success? do-calculations
  ]

  ;; setting the values for the initial lattice
  set clustering-coefficient-of-lattice clustering-coefficient
  set average-path-length-of-lattice average-path-length
  set number-rewired 0
  set highlight-string ""
end 

to make-turtles
  crt num-nodes [ set color gray + 2 ]
  ;; arrange them in a circle in order by who number
  layout-circle (sort turtles) max-pxcor - 1
end 

;;;;;;;;;;;;;;;;;;;;;;;
;;; Main Procedure ;;;
;;;;;;;;;;;;;;;;;;;;;;;

to rewire-one

  ;; make sure num-turtles is setup correctly else run setup first
  if count turtles != num-nodes [
    setup
  ]

  ;; record which button was pushed
  set rewire-one? true
  set rewire-all? false

  let potential-edges links with [ not rewired? ]
  ifelse any? potential-edges [
    ask one-of potential-edges [
      ;; "a" remains the same
      let node1 end1
      ;; if "a" is not connected to everybody
      if [ count link-neighbors ] of end1 < (count turtles - 1)
      [
        ;; find a node distinct from node1 and not already a neighbor of node1
        let node2 one-of turtles with [ (self != node1) and (not link-neighbor? node1) ]
        ;; wire the new edge
        ask node1 [ create-link-with node2 [ set color cyan  set rewired? true ] ]

        set number-rewired number-rewired + 1  ;; counter for number of rewirings

        ;; remove the old edge
        die
      ]
    ]
    ;; plot the results
    let connected? do-calculations
    update-plots
  ]
  [ user-message "all edges have already been rewired once" ]
end 

to rewire-all

  ;; make sure num-turtles is setup correctly; if not run setup first
  if count turtles != num-nodes [
    setup
  ]

  ;; record which button was pushed
  set rewire-one? false
  set rewire-all? true

  ;; set up a variable to see if the network is connected
  let success? false

  ;; if we end up with a disconnected network, we keep trying, because the APL distance
  ;; isn't meaningful for a disconnected network.
  while [not success?] [
    ;; kill the old lattice, reset neighbors, and create new lattice
    ask links [ die ]
    wire-them
    set number-rewired 0

    ask links [

      ;; whether to rewire it or not?
      if (random-float 1) < rewiring-probability
      [
        ;; "a" remains the same
        let node1 end1
        ;; if "a" is not connected to everybody
        if [ count link-neighbors ] of end1 < (count turtles - 1)
        [
          ;; find a node distinct from node1 and not already a neighbor of node1
          let node2 one-of turtles with [ (self != node1) and (not link-neighbor? node1) ]
          ;; wire the new edge
          ask node1 [ create-link-with node2 [ set color cyan  set rewired? true ] ]

          set number-rewired number-rewired + 1  ;; counter for number of rewirings
          set rewired? true
        ]
      ]
      ;; remove the old edge
      if (rewired?)
      [
        die
      ]
    ]

    ;; check to see if the new network is connected and calculate path length and clustering
    ;; coefficient at the same time
    set success? do-calculations
  ]

  ;; do the plotting
  update-plots
end 

;; do-calculations reports true if the network is connected,
;;   and reports false if the network is disconnected.
;; (In the disconnected case, the average path length does not make sense,
;;   or perhaps may be considered infinite)

to-report do-calculations

  ;; set up a variable so we can report if the network is disconnected
  let connected? true

  ;; find the path lengths in the network
  find-path-lengths

  let num-connected-pairs sum [length remove infinity (remove 0 distance-from-other-turtles)] of turtles

  ;; In a connected network on N nodes, we should have N(N-1) measurements of distances between pairs,
  ;; and none of those distances should be infinity.
  ;; If there were any "infinity" length paths between nodes, then the network is disconnected.
  ;; In that case, calculating the average-path-length doesn't really make sense.
  ifelse ( num-connected-pairs != (count turtles * (count turtles - 1) ))
  [
      set average-path-length infinity
      ;; report that the network is not connected
      set connected? false
  ]
  [
    set average-path-length (sum [sum distance-from-other-turtles] of turtles) / (num-connected-pairs)
  ]
  ;; find the clustering coefficient and add to the aggregate for all iterations
  find-clustering-coefficient

  ;; report whether the network is connected or not
  report connected?
end 

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;; Clustering computations ;;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

to-report in-neighborhood? [ hood ]
  report ( member? end1 hood and member? end2 hood )
end 

to find-clustering-coefficient
  ifelse all? turtles [count link-neighbors <= 1]
  [
    ;; it is undefined
    ;; what should this be?
    set clustering-coefficient 0
  ]
  [
    let total 0
    ask turtles with [ count link-neighbors <= 1]
      [ set node-clustering-coefficient "undefined" ]
    ask turtles with [ count link-neighbors > 1]
    [
      let hood link-neighbors
      set node-clustering-coefficient (2 * count links with [ in-neighborhood? hood ] /
                                         ((count hood) * (count hood - 1)) )
      ;; find the sum for the value at turtles
      set total total + node-clustering-coefficient
    ]
    ;; take the average
    set clustering-coefficient total / count turtles with [count link-neighbors > 1]
  ]
end 

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;; Path length computations ;;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

;; Implements the Floyd Warshall algorithm for All Pairs Shortest Paths
;; It is a dynamic programming algorithm which builds bigger solutions
;; from the solutions of smaller subproblems using memoization that
;; is storing the results.
;; It keeps finding incrementally if there is shorter path through
;; the kth node.
;; Since it iterates over all turtles through k,
;; so at the end we get the shortest possible path for each i and j.

to find-path-lengths
  ;; reset the distance list
  ask turtles
  [
    set distance-from-other-turtles []
  ]

  let i 0
  let j 0
  let k 0
  let node1 one-of turtles
  let node2 one-of turtles
  let node-count count turtles
  ;; initialize the distance lists
  while [i < node-count]
  [
    set j 0
    while [j < node-count]
    [
      set node1 turtle i
      set node2 turtle j
      ;; zero from a node to itself
      ifelse i = j
      [
        ask node1 [
          set distance-from-other-turtles lput 0 distance-from-other-turtles
        ]
      ]
      [
        ;; 1 from a node to it's neighbor
        ifelse [ link-neighbor? node1 ] of node2
        [
          ask node1 [
            set distance-from-other-turtles lput 1 distance-from-other-turtles
          ]
        ]
        ;; infinite to everyone else
        [
          ask node1 [
            set distance-from-other-turtles lput infinity distance-from-other-turtles
          ]
        ]
      ]
      set j j + 1
    ]
    set i i + 1
  ]
  set i 0
  set j 0
  let dummy 0
  while [k < node-count]
  [
    set i 0
    while [i < node-count]
    [
      set j 0
      while [j < node-count]
      [
        ;; alternate path length through kth node
        set dummy ( (item k [distance-from-other-turtles] of turtle i) +
                    (item j [distance-from-other-turtles] of turtle k))
        ;; is the alternate path shorter?
        if dummy < (item j [distance-from-other-turtles] of turtle i)
        [
          ask turtle i [
            set distance-from-other-turtles replace-item j distance-from-other-turtles dummy
          ]
        ]
        set j j + 1
      ]
      set i i + 1
    ]
    set k k + 1
  ]
end 

;;;;;;;;;;;;;;;;;;;;;;;
;;; Edge Operations ;;;
;;;;;;;;;;;;;;;;;;;;;;;

;; creates a new lattice

to wire-them
  ;; iterate over the turtles
  let n 0
  while [n < count turtles]
  [
    ;; make edges with the next two neighbors
    ;; this makes a lattice with average degree of 4
    make-edge turtle n
              turtle ((n + 1) mod count turtles)
    make-edge turtle n
              turtle ((n + 2) mod count turtles)
    set n n + 1
  ]
end 

;; connects the two turtles

to make-edge [node1 node2]
  ask node1 [ create-link-with node2  [
    set rewired? false
  ] ]
end 

;;;;;;;;;;;;;;;;
;;; Graphics ;;;
;;;;;;;;;;;;;;;;

to highlight
  ;; remove any previous highlights
  ask turtles [ set color gray + 2 ]
  ask links [ set color gray + 2 ]
  if mouse-inside? [ do-highlight ]
  display
end 

to do-highlight
  ;; getting the node closest to the mouse
  let min-d min [distancexy mouse-xcor mouse-ycor] of turtles
  let node one-of turtles with [count link-neighbors > 0 and distancexy mouse-xcor mouse-ycor = min-d]
  if node != nobody
  [
    ;; highlight the chosen node
    ask node
    [
      set color pink - 1
      let pairs (length remove infinity distance-from-other-turtles)
      let local-val (sum remove infinity distance-from-other-turtles) / pairs
      ;; show node's clustering coefficient
      set highlight-string (word "clustering coefficient = " precision node-clustering-coefficient 3
                                 " and avg path length = " precision local-val 3
                                 " (for " pairs " turtles )")
    ]
    let neighbor-nodes [ link-neighbors ] of node
    let direct-links [ my-links ] of node
    ;; highlight neighbors
    ask neighbor-nodes
    [
      set color blue - 1

      ;; highlight edges connecting the chosen node to its neighbors
      ask my-links [
        ifelse (end1 = node or end2 = node)
        [
          set color blue - 1 ;
        ]
        [
          if (member? end1 neighbor-nodes and member? end2 neighbor-nodes)
            [ set color yellow ]
        ]
      ]
    ]
  ]
end 

to seed-contagion
  ask turtles [ set color 7 ]
  set-current-plot "Success of Contagion"
  clear-plot
  ask one-of turtles [ set color red ]
end 

to go
  ask turtles with [ any? link-neighbors with [ color = red] ] [set color red]
  set-current-plot "Success of Contagion"
  plot percent-saturated
  display
end 

to-report percent-saturated
  report ((count turtles with [color = red ] ) / (count turtles))
end 


; Copyright 2005 Uri Wilensky.
; This model has been modified from its original form
; by Joshua Becker (2017)
; The modifications have added a
; See Info tab for full copyright and license.

There is only one version of this model, created over 1 year ago by Network Dynamics Group.

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