extensions [ nw ]  ;; use the network extension

globals
[
  average-path-length                  ;; average path length of the network

  tot-wire-cost
  
  old-lambda

  is-connected
  num-success
  
  current-cost
  prev-cost
]

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

to calc-cost
  ; per-turtle cost of total wire length 
  set tot-wire-cost 0.1 * (cost-of-wire * sum [link-length] of links) / count turtles
  
  set is-connected do-calculations
  
  ; if graph is still connected calculate energy of configuration
  ifelse (is-connected) [
    set current-cost cost-of-wire * tot-wire-cost + (1 - cost-of-wire) * average-path-length
  ] [  ;; otherwise set enegery to very high value to prohibit network
       ;; from becoming disconnected
    set current-cost 1000000]
end      

to setup
  clear-all
  
  create-turtles num-nodes [set size 3]
  
  ;; Layout turtles:
  layout-circle (sort turtles) max-pxcor - 8
  
  create-small-world
  
  ask links
  [set color gray + 1.5]
  
  calc-cost

  display
  reset-ticks
end

; this function gradually lowers the temperature to
; allow some jiggle, so that network is less likely to 
; get stuck in local optimumum
to find-optimum
  calc-cost
  set temperature 0.2
  while [temperature > 0.005] [
    let i 0
    while [(i < 300) and (num-success < 30)] [
      rewire-network
      set i i + 1
    ]
    set temperature temperature * 0.9
    set num-success 0
  ]
end

to rewire-network
  
  set prev-cost current-cost
  let new-link nobody
  let n1 nobody
  let n2 nobody
  
  ; calculate the current cost
  calc-cost
  set prev-cost current-cost
  
  while [new-link = nobody] [
    ;pick a random link
    ask one-of links
    [
     ; pick one of the ends at random
     ifelse (random-float 1.0 < 0.5) [
        set n1 end1
        set n2 end2
      ][
      set n2 end1
      set n1 end2
      ]
    
      ; ask one end to rewire to a random other node it is not
      ; already connected to
      if any? turtles with [(not link-neighbor? n1) and (self != n1)][
        ask n1
        [
          create-link-with one-of other turtles with [not link-neighbor? myself ]
          [
            set color red + 1.5
            set new-link self
          ]
        ]
        
        die 
      ]
    ]
  ]
  
  if show-attempts [display]
   
   ; calculate cost after rewiring
   calc-cost

   let paccept 0
   
   ; if new configuration is less costly, accept
   ifelse (current-cost <= prev-cost) [
     set paccept 1
     ] 
   [
     ; still accept some rewiring even if it's less energy efficient
     ; more likely to happen for high temperature
     set paccept 0.5 * (e ^ ( - (current-cost - prev-cost) / temperature ))
   ]
 
   ; with probability p-accept keep new link discard old
   ; otherwise discard new link, bring back old
   ifelse (((random-float 1.0) <=  paccept) and (is-connected)) [
     ask new-link [set color green + 1.5]
     set prev-cost current-cost
     set num-success num-success + 1
   ]
   [
     ;; discard new link, reinstate old one
     ask n1 [create-link-with n2]
     ask new-link [die]
     set current-cost prev-cost
   ]
  
  ifelse show-attempts  [display]
  [if (ticks mod 100 = 0) [display]
  ]
end


;; spring layout all nodes and links
to do-layout
    repeat 5 [layout-spring turtles links 0.2 4 0.9]
  display
end

;; create small world topology
to create-small-world
  ;; iterate over the nodes
  
  ; create regular lattice with links to 2 closests neighbors
  let n 0
  while [n < count turtles]
  [
    ;; connect to closest neighbor
    ask turtle n [
              if (n mod 2 = 0) [ 
                facexy 0 0
                fd 2
              ]
              create-link-with
              turtle ((n + 1) mod count turtles)
              create-link-with turtle ((n + 2) mod count turtles)
    ]
    set n n + 1
  ]
  
  ; add some random links on top
  repeat (count turtles / 10) [
    ask one-of turtles [
      create-link-with one-of other turtles with [not link-neighbor? myself]
    ]
  ]
end

to-report do-calculations
  
  ; get the average shortest path
  nw:set-snapshot turtles links
  set average-path-length nw:mean-path-length
  
  ; check if network is connected
  let giant-component first sort-by [length ?1 > length ?2] nw:weak-component-clusters
  report (length giant-component = count turtles)
end

@#$#@#$#@
GRAPHICS-WINDOW
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SLIDER
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BUTTON
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0

@#$#@#$#@
## WHAT IS IT?

This is a model of how a network topology is generated from very practical constraints.
There are two factors: the average shortest path (it's good to have a few hops to go
between nodes), and the cost of wiring (lengthy wire can get expensive).

## HOW TO USE IT

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

SETUP will set up a small world topology: a regular lattice with a few random links overlaid.

The TEMPERATURE parameter determined how likely energetically unfavorable configurations are to be accepted. The higher the temperature, the move likely more costly configurations are to occur, and the network will be more random. At lower temperatures the network will have a low energy and more optimal configuration.

WIRE-COST determines how expensive wire is.

REWIRE continuously tries to rewire the network, one link at a time, accepting a new link depending on the current WIRE-COST and TEMPERATURE. 

FIND-OPTIMUM starts with a high temperature and gradually lowers it to find a good configuration.

## THINGS TO TRY

Try adjusting the WIRE-COST parameter. When do you observe a mostly regular lattice? Is there a point in the parameter space where you observe hubs forming?

## CREDITS AND REFERENCES

Model from N. Mathias, V. Gopal, 'Small worlds: How and why', Physical Review E, 2001. http://arxiv.org/pdf/cond-mat/0002076

NetLogo model created by Lada Adamic

## COPYRIGHT AND LICENSE

Copyright 2012 Lada Adamic.

![CC BY-NC-SA 3.0](http://i.creativecommons.org/l/by-nc-sa/3.0/88x31.png)

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

@#$#@#$#@
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@#$#@#$#@
NetLogo 5.0.2
@#$#@#$#@
setup
repeat 5 [rewire-one]
@#$#@#$#@
@#$#@#$#@
<experiments>
  <experiment name="vary-rewiring-probability" repetitions="5" runMetricsEveryStep="false">
    <go>rewire-all</go>
    <timeLimit steps="1"/>
    <exitCondition>rewiring-probability &gt; 1</exitCondition>
    <metric>average-path-length</metric>
    <metric>clustering-coefficient</metric>
    <steppedValueSet variable="rewiring-probability" first="0" step="0.025" last="1"/>
  </experiment>
</experiments>
@#$#@#$#@
@#$#@#$#@
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