tobiapoppi_Firefly-inspired-heartbeat-sync-in-Small-World-Networks/small_world_fireflies_synchronization.nlogo
globals [
  infinity         ; used to represent the distance between two turtles with no path between them

  average-path-length-of-lattice       ; average path length of the initial lattice
  average-path-length                  ; average path length in the current network

  rewiring-probability

  clustering-coefficient-of-lattice    ; the clustering coefficient of the initial lattice
  clustering-coefficient               ; the clustering coefficient of the current network (avg. across nodes)

  number-rewired                       ; number of edges that have been rewired
  rewire-one?                          ; these two variables record which button was last pushed
  rewire-all?

  lower_bound                          ; upper bound of the cycle length
  upper_bound                          ; lower bound of the cycle length

  silence_time                         ; time of silence elapsed (of nobody flashing)
                                       ; default necessary silence in the paper for sync is 200ms
  sync                                ; boolean value to tell if the network is synchronized
  sync-tick                           ; number of tick when the network has synchronized
  neighbors_to_flash
]

turtles-own [
  phase                          ; phase of a firefly ()
  reset_level
  cycle_length
  nat_cycle_length
  flash_length                   ; time duration of a flash

  distance-from-other-turtles ; list of distances of this node from other turtles
  my-clustering-coefficient   ; the current clustering coefficient of this node
]

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

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

to setup
  clear-all
  reset-ticks

  ; set the global variables
  set infinity 99999      ; this is an arbitrary choice for a large number
  set number-rewired 0    ; initial count of rewired edges

  set lower_bound 85
  set upper_bound 115

  set neighbors_to_flash 1

  ; make the nodes and arrange them in a circle in order by who number
  set-default-shape turtles "butterfly"
  create-turtles num-nodes [
    set color blue - 1
    set flash_length 2
    set reset_level flash_length

    set nat_cycle_length lower_bound + (random (upper_bound - lower_bound))
    set cycle_length nat_cycle_length
    set phase random (round cycle_length)
    set size 1.5
  ]
  layout-circle (sort turtles) max-pxcor - 1

  ; Create the initial lattice
  wire-lattice


  ; Fix the color scheme
  ask turtles [ set color blue - 1]
  ask links [ set color grey ]

  ; Calculate the initial average path length and clustering coefficient
  set average-path-length find-average-path-length
  set clustering-coefficient find-clustering-coefficient

  set average-path-length-of-lattice average-path-length
  set clustering-coefficient-of-lattice clustering-coefficient

  wire-small-world
end

;;;;;;;;;;;;;;;;;;;;;
;; Main Procedures ;;
;;;;;;;;;;;;;;;;;;;;;


to go
  check_sync
  ask turtles[
    inc_phase
    update_color
  ]
  tick
end


to inc_phase
  set phase (phase + 1)
  if phase = cycle_length [
    set phase 0
  ]
  update_length
end

to update_length
  if count link-neighbors with [color = yellow or color = green - 0.5] >= neighbors_to_flash[
    let g_plus 0   ; factor for lengthening the cycle
    let g_minus 0  ; factor for shortening the cycle
    let a (sin (360 * (phase / cycle_length))) / (2 * pi)
    if a > 0 [set g_plus a]
    if a < 0 [set g_minus a]

    let om_l (1 / upper_bound)
    let om_u (1 / lower_bound)
    let om_nat (1 / nat_cycle_length)
    let om_i (1 / cycle_length)
    set om_i (om_i + (0.01 * (om_nat - om_i)) + (g_plus * (om_l - om_i)) - (g_minus * (om_u - om_i)))
    set cycle_length round (1 / om_i)
    if cycle_length < lower_bound [set cycle_length lower_bound]
    if cycle_length > upper_bound [set cycle_length upper_bound]
  ]
end

to update_color
  ifelse (phase < flash_length)[
    show-turtle
    ifelse sync = 1 [
      set color green - 0.5
    ]
    [set color yellow]
  ]
  [set color blue - 1]

end

to check_sync
  if ticks > 150 [
    ifelse count turtles with [color = yellow or color = green - 0.5] = 0 [
      set silence_time silence_time + 1
    ]
    [
      ifelse silence_time > sync_silence_time [
        if sync = 0 [
          set sync 1
          set sync-tick ticks
        ]
      ]
      [
        set silence_time 0
      ]
    ]
  ]
end


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 [ rewire-me ]
    ; Calculate the new statistics and update the plots
    set average-path-length find-average-path-length
    set clustering-coefficient find-clustering-coefficient
    update-plots
  ]
  [ user-message "all edges have already been rewired once" ]
end


to rewire-me ; turtle procedure
  ; node-A remains the same
  let node-A end1
  ; as long as A is not connected to everybody
  if [ count link-neighbors ] of end1 < (count turtles - 1) [
    ; find a node distinct from A and not already a neighbor of "A"
    let node-B one-of turtles with [ (self != node-A) and (not link-neighbor? node-A) ]
    ; wire the new edge
    ask node-A [ create-link-with node-B [ set color grey set rewired? true ] ]

    set number-rewired number-rewired + 1
    die ; remove the old edge
  ]
end

to rewire-all
  ; confirm we have the right amount of turtles, otherwise reinitialize
  if count turtles != num-nodes [ setup ]

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

  ; we keep generating networks until we get a connected one since apl doesn't mean anything
  ; in a non-connected network
  let connected? false
  while [ not connected? ] [
    ; kill the old lattice and create new one
    ask links [ die ]
    wire-lattice
    set number-rewired 0

    ; ask each link to maybe rewire, according to the rewiring-probability slider
    ask links [
      if (random-float 1) < rewiring-probability [ rewire-me ]
    ]

    ; if the apl is infinity, it means our new network is not connected. Reset the lattice.
    ifelse find-average-path-length = infinity [ set connected? false ] [ set connected? true ]
  ]

  ; calculate the statistics and visualize the data
  set clustering-coefficient find-clustering-coefficient
  set average-path-length find-average-path-length
  update-plots
end

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

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


to-report find-clustering-coefficient

  let cc infinity

  ifelse all? turtles [ count link-neighbors <= 1 ] [
    ; it is undefined
    ; what should this be?
    set cc 0
  ][
    let total 0
    ask turtles with [ count link-neighbors <= 1 ] [ set my-clustering-coefficient "undefined" ]
    ask turtles with [ count link-neighbors > 1 ] [
      let hood link-neighbors
      set my-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 + my-clustering-coefficient
    ]
    ; take the average
    set cc total / count turtles with [count link-neighbors > 1]
  ]

  report cc
end

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

; Procedure to calculate the average-path-length (apl) in the network. If the network is not
; connected, we return `infinity` since apl doesn't really mean anything in a non-connected network.
to-report find-average-path-length

  let apl 0

  ; calculate all the path-lengths for each node
  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.
  ; If there were any "infinity" length paths between nodes, then the network is disconnected.
  ifelse num-connected-pairs != (count turtles * (count turtles - 1)) [
    ; This means the network is not connected, so we report infinity
    set apl infinity
  ][
    set apl (sum [sum distance-from-other-turtles] of turtles) / (num-connected-pairs)
  ]

  report apl
end

; 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-lattice
  ; 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)
              "default"
    ; Make the neighbor's neighbor links curved
    make-edge turtle n
              turtle ((n + 2) mod count turtles)
              "curve"
    set n n + 1
  ]

  ; Because of the way NetLogo draws curved links between turtles of ascending
  ; `who` number, two of the links near the top of the network will appear
  ; flipped by default. To avoid this, we used an inverse curved link shape
  ; ("curve-a") which makes all of the curves face the same direction.
  ask link 0 (count turtles - 2) [ set shape "curve-a" ]
  ask link 1 (count turtles - 1) [ set shape "curve-a" ]
end

to wire-small-world
  set rewiring-probability 0.1
  set rewire-one? false
  set rewire-all? true

  let connected? false
  while [ not connected? ] [
    ask links [ die ]
    wire-lattice
    set number-rewired 0

    ; ask each link to maybe rewire, according to the rewiring-probability slider
    ask links [
      if (random-float 1) < rewiring-probability [ rewire-me ]
    ]

    ; if the apl is infinity, it means our new network is not connected. Reset the lattice.
    ifelse find-average-path-length = infinity [ set connected? false ] [ set connected? true ]
    ;ifelse (find-clustering-coefficient > 0.4 and find-average-path-length < 3.2) [set connected? true] [set connected? false]
    ifelse (find-clustering-coefficient > 0.2 and find-clustering-coefficient < 0.3) [set connected? true] [set connected? false]
  ]

  ; calculate the statistics and visualize the data
  set clustering-coefficient find-clustering-coefficient
  set average-path-length find-average-path-length
  update-plots

end

; Connects two nodes
to make-edge [ node-A node-B the-shape ]
  ask node-A [
    create-link-with node-B  [
      set shape the-shape
      set rewired? false
    ]
  ]
end

to appunti
if ticks > 0 [ask turtle 0[
     create-temporary-plot-pen (word who)
     set-plot-pen-color green - 1
     plotxy ticks cycle_length
]
ask turtle 1[
     create-temporary-plot-pen (word who)
     set-plot-pen-color green - 0.8
     plotxy ticks cycle_length
]
ask turtle 2[
     create-temporary-plot-pen (word who)
     set-plot-pen-color green - 0.6
     plotxy ticks cycle_length
]
ask turtle 3[
     create-temporary-plot-pen (word who)
     set-plot-pen-color green - 0.4
     plotxy ticks cycle_length
]
ask turtle 4[
     create-temporary-plot-pen (word who)
     set-plot-pen-color green - 0.2
     plotxy ticks cycle_length
]
ask turtle 5[
     create-temporary-plot-pen (word who)
     set-plot-pen-color green
     plotxy ticks cycle_length
]
ask turtle 6[
     create-temporary-plot-pen (word who)
     set-plot-pen-color green + 0.2
     plotxy ticks cycle_length
]
ask turtle 7[
     create-temporary-plot-pen (word who)
     set-plot-pen-color green + 0.4
     plotxy ticks cycle_length
]
ask turtle 8[
     create-temporary-plot-pen (word who)
     set-plot-pen-color green + 0.6
     plotxy ticks cycle_length
]
ask turtle 9[
     create-temporary-plot-pen (word who)
     set-plot-pen-color green + 0.8
     plotxy ticks cycle_length
]
ask turtle 10[
     create-temporary-plot-pen (word who)
     set-plot-pen-color green + 1
     plotxy ticks cycle_length
]
ask turtle 11[
     create-temporary-plot-pen (word who)
     set-plot-pen-color green + 1.2
     plotxy ticks cycle_length
]
ask turtle 12[
     create-temporary-plot-pen (word who)
     set-plot-pen-color green + 1.4
     plotxy ticks cycle_length
]
ask turtle 13[
     create-temporary-plot-pen (word who)
     set-plot-pen-color green + 1.6
     plotxy ticks cycle_length
]
ask turtle 14[
     create-temporary-plot-pen (word who)
     set-plot-pen-color green + 1.8
     plotxy ticks cycle_length
]
  ]
end


; Copyright 2015 Uri Wilensky.
; See Info tab for full copyright and license.
@#$#@#$#@
GRAPHICS-WINDOW
220
10
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464
-1
-1
12.7143
1
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1
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0
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ticks
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SLIDER
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num-nodes
num-nodes
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36.0
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NIL
HORIZONTAL

MONITOR
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clustering-coefficient (cc)
clustering-coefficient
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11

MONITOR
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460
average-path-length (apl)
average-path-length
3
1
11

BUTTON
10
60
85
95
setup
setup
NIL
1
T
OBSERVER
NIL
NIL
NIL
NIL
1

BUTTON
10
165
73
198
NIL
go
NIL
1
T
OBSERVER
NIL
NIL
NIL
NIL
1

BUTTON
10
205
97
238
go forever
go
T
1
T
OBSERVER
NIL
NIL
NIL
NIL
1

MONITOR
690
10
747
55
NIL
sync
17
1
11

SLIDER
10
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182
298
sync_silence_time
sync_silence_time
40
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80.0
1
1
NIL
HORIZONTAL

PLOT
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1255
465
Flashing-nodes
t
#
0.0
1000.0
0.0
80.0
true
true
"set-plot-y-range 0 num-nodes" ""
PENS
"# flashing-nodes" 1.0 0 -16382462 true "" "plot count turtles with [color = yellow or color = green - 0.5]"
"sync?" 1.0 0 -2674135 true "" "plot sync * (num-nodes / 2)"

PLOT
690
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cycle_lengths
NIL
NIL
0.0
10.0
80.0
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true
true
"" "if ticks > 0 [ask turtle 0[\n     create-temporary-plot-pen (word who)\n     set-plot-pen-color green - 1\n     plotxy ticks cycle_length\n]\nask turtle 1[\n     create-temporary-plot-pen (word who)\n     set-plot-pen-color green - 0.8\n     plotxy ticks cycle_length\n]\nask turtle 2[\n     create-temporary-plot-pen (word who)\n     set-plot-pen-color green - 0.6\n     plotxy ticks cycle_length\n]\nask turtle 3[\n     create-temporary-plot-pen (word who)\n     set-plot-pen-color green - 0.4\n     plotxy ticks cycle_length\n]\nask turtle 4[\n     create-temporary-plot-pen (word who)\n     set-plot-pen-color green - 0.2\n     plotxy ticks cycle_length\n]\nask turtle 5[\n     create-temporary-plot-pen (word who)\n     set-plot-pen-color green\n     plotxy ticks cycle_length\n]\nask turtle 6[\n     create-temporary-plot-pen (word who)\n     set-plot-pen-color green + 0.2\n     plotxy ticks cycle_length\n]\nask turtle 7[\n     create-temporary-plot-pen (word who)\n     set-plot-pen-color green + 0.4\n     plotxy ticks cycle_length\n]\nask turtle 8[\n     create-temporary-plot-pen (word who)\n     set-plot-pen-color green + 0.6\n     plotxy ticks cycle_length\n]\nask turtle 9[\n     create-temporary-plot-pen (word who)\n     set-plot-pen-color green + 0.8\n     plotxy ticks cycle_length\n]\nask turtle 10[\n     create-temporary-plot-pen (word who)\n     set-plot-pen-color green + 1\n     plotxy ticks cycle_length\n]\nask turtle 11[\n     create-temporary-plot-pen (word who)\n     set-plot-pen-color green + 1.2\n     plotxy ticks cycle_length\n]\nask turtle 12[\n     create-temporary-plot-pen (word who)\n     set-plot-pen-color green + 1.4\n     plotxy ticks cycle_length\n]\nask turtle 13[\n     create-temporary-plot-pen (word who)\n     set-plot-pen-color green + 1.6\n     plotxy ticks cycle_length\n]\nask turtle 14[\n     create-temporary-plot-pen (word who)\n     set-plot-pen-color green + 1.8\n     plotxy ticks cycle_length\n]\n  ]"
PENS
"mean" 1.0 0 -2674135 true "" "if sync = 0 [if ticks > 0 [ plot mean [cycle_length] of turtles ]]"
"sync" 1.0 0 -13345367 true "" "ifelse sync = 1 [if ticks > 0 [ plot 90 ]] [if ticks > 0 [ plot 80 ]]"

MONITOR
750
10
812
55
NIL
sync-tick
17
1
11

@#$#@#$#@
## 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 from 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., "[Six Degrees of Kevin Bacon](https://en.wikipedia.org/wiki/Six_Degrees_of_Kevin_Bacon)"), 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 the [Watts-Strogatz model](https://en.wikipedia.org/wiki/Watts-Strogatz_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. Using this a base, we then modify the network by rewiring nodesâ€“changing one end of a connected pair of nodes and keeping the other end the same. Over time, we analyze the effect this rewiring has the on various connections between nodes and on the properties of the network.

Particularly, we're interested in identifying "small worlds." To identify small worlds, the "average path length" (abbreviated "apl") and "clustering coefficient" (abbreviated "cc") of the network are calculated and plotted after a rewiring is performed. Networks with _short_ average path lengths and _high_ clustering coefficients are considered small world networks. See the **Statistics** section of HOW TO USE IT on how these are calculated.

## HOW TO USE IT

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 in the "Network Properties Rewire-One" graph. The REWIRE-ONE button _ignores_ the REWIRING-PROBABILITY slider. It will always rewire one exactly one edge in the network that has not yet been rewired _unless_ all edges in the network have already been rewired.

Pressing the REWIRE-ALL button starts with a new lattice (just like pressing SETUP) and then rewires all of the edges edges according to the current REWIRING-PROBABILITY. In other words, it `asks` each `edge` to roll a die that will determine whether or not it is rewired. The resulting network properties are then plotted on the "Network Properties Rewire-All" graph. Changing the REWIRING-PROBABILITY slider changes the fraction of edges rewired during each run. Running REWIRE-ALL at multiple probabilities produces a range of possible networks with varying average path lengths and clustering coefficients.

When you press HIGHLIGHT and then point to a node in the view it color-codes the nodes and edges. The node itself turns white. Its neighbors and the edges connecting the node to those neighbors turn orange. Edges connecting the neighbors of the node to each other turn yellow. The amount of yellow between neighbors gives you a sort of 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.

### Statistics

**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 node in the network to another.

In order to find the shortest paths between all pairs of nodes we use the [standard dynamic programming algorithm by Floyd Warshall] (https://en.wikipedia.org/wiki/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.

**Clustering Coefficient**: The clustering coefficient of a _node_ is the ratio of existing edges connecting a node's neighbors to each other to the maximum possible number of such edges. It is, in essence, a measure of the "all-my-friends-know-each-other" property. The clustering coefficient for the entire network is the average of the clustering coefficients of all the nodes.

### Plots

1. The "Network Properties Rewire-One" visualizes the average-path-length and clustering-coefficient of the network as the user increases the number of single-rewires in the network.

2. The "Network Properties Rewire-All" visualizes the average-path-length and clustering coefficient of the network as the user manipulates the REWIRING-PROBABILITY slider.

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.

The plots for both the clustering coefficient and average path length are normalized by dividing by the values of the initial lattice. The monitors CLUSTERING-COEFFICIENT and AVERAGE-PATH-LENGTH give the actual values.

## THINGS TO NOTICE

Note that for certain ranges of the fraction of nodes rewired, 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

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

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?

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?

## EXTENDING THE MODEL

Try to see if you can produce the same results if you start with a different type of 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.

## NETLOGO FEATURES

Links are used extensively in this model to model the edges of the network. The model also uses custom link shapes for neighbor's neighbor links.

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

## RELATED MODELS

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

Check out the NW Extension General Examples model to see how similar models might implemented using the built-in NW extension.

## 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: https://oracleofbacon.org

Thanks to Connor Bain for updating this model in 2020.

## HOW TO CITE

If you mention this model or the NetLogo software in a publication, we ask that you include the citations below.

For the model itself:

* Wilensky, U. (2015).  NetLogo Small Worlds model.  http://ccl.northwestern.edu/netlogo/models/SmallWorlds.  Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.

Please cite the NetLogo software as:

* Wilensky, U. (1999). NetLogo. http://ccl.northwestern.edu/netlogo/. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.

## COPYRIGHT AND LICENSE

Copyright 2015 Uri Wilensky.

![CC BY-NC-SA 3.0](http://ccl.northwestern.edu/images/creativecommons/byncsa.png)

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.

Commercial licenses are also available. To inquire about commercial licenses, please contact Uri Wilensky at uri@northwestern.edu.

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