Contagion/ Social Influence- Jiawei Fu- Version 2

1 collaborator

Jiawei Fu (Author)

WHAT IS IT?

In some networks, a few "hubs" have lots of connections, while everybody else only has a few. New network members prefer to make a connection to the more popular existing members.

Such networks can be found in a surprisingly large range of real world situations, ranging from the connections between websites to the collaborations between actors.

In some other networks, actors prefer to connect to similar actors to onself.

Such networks can be found in a surprisingly large range of real world situations as well, ranging from the connections between friends, collaborations, to hyperlink networks.

This model generates these networks by a process of "preferential attachment" and "homophily" at the same time, testing whether preferential attachment and homophily mechanism work more dominant.

HOW IT WORKS

The model starts with two nodes connected by an edge.

At each step, a new node is added. A new node picks an existing node to connect to.

More specifically, a node's chance of being selected is directly proportional to the number of connections it already has, or its "degree." This is the mechanism which is called "preferential attachment."

Alternatively, there is a homophily mechanism.That is, a new node picks an existing node to connect to, which has the same color with the new node.

Depending on the probability whether preferential attachment is higher or homophily is higher, each links picks different new nodes from different mechanisms. If the probability of preferential attachment is higher, the actor has a higher probability to pick an existing popular node with high degrees, and the link between us is green. If the probability of homophily is higher, the actor has a higher probability to pick an existing similar node with the same color, then the link is set to be yellow

HOW TO USE IT

Pressing the GO ONCE button adds one new node. To continuously add nodes, press GO.

The LAYOUT? switch controls whether or not the layout procedure is run. This procedure attempts to move the nodes around to make the structure of the network easier to see.

The PLOT? switch turns off the plots which speeds up the model.

The RESIZE-NODES button will make all of the nodes take on a size representative of their degree distribution. If you press it again the nodes will return to equal size.

If you want the model to run faster, you can turn off the LAYOUT? and PLOT? switches and/or freeze the view (using the on/off button in the control strip over the view). The LAYOUT? switch has the greatest effect on the speed of the model.

If you have LAYOUT? switched off, and then want the network to have a more appealing layout, press the REDO-LAYOUT button which will run the layout-step procedure until you press the button again. You can press REDO-LAYOUT at any time even if you had LAYOUT? switched on and it will try to make the network easier to see.

THINGS TO NOTICE

The networks that result from running this model are often called "scale-free" or "power law" networks. These are networks in which the distribution of the number of connections of each node is not a normal distribution --- instead it follows what is a called a power law distribution. Power law distributions are different from normal distributions in that they do not have a peak at the average, and they are more likely to contain extreme values (see Albert & Barabási 2002 for a further description of the frequency and significance of scale-free networks). Barabási and Albert originally described this mechanism for creating networks, but there are other mechanisms of creating scale-free networks and so the networks created by the mechanism implemented in this model are referred to as Barabási scale-free networks.

You can see the degree distribution of the network in this model by looking at the plots. The top plot is a histogram of the degree of each node. The bottom plot shows the same data, but both axes are on a logarithmic scale. When degree distribution follows a power law, it appears as a straight line on the log-log plot. One simple way to think about power laws is that if there is one node with a degree distribution of 1000, then there will be ten nodes with a degree distribution of 100, and 100 nodes with a degree distribution of 10.

THINGS TO TRY

Let the model run a little while. How many nodes are "hubs", that is, have many connections? How many have only a few? Does some low degree node ever become a hub? How often?

Turn off the LAYOUT? switch and freeze the view to speed up the model, then allow a large network to form. What is the shape of the histogram in the top plot? What do you see in log-log plot? Notice that the log-log plot is only a straight line for a limited range of values. Why is this? Does the degree to which the log-log plot resembles a straight line grow as you add more nodes to the network?

EXTENDING THE MODEL

Assign an additional attribute to each node. Make the probability of attachment depend on this new attribute as well as on degree. (A bias slider could control how much the attribute influences the decision.)

Can the layout algorithm be improved? Perhaps nodes from different hubs could repel each other more strongly than nodes from the same hub, in order to encourage the hubs to be physically separate in the layout.

NETWORK CONCEPTS

There are many ways to graphically display networks. This model uses a common "spring" method where the movement of a node at each time step is the net result of "spring" forces that pulls connected nodes together and repulsion forces that push all the nodes away from each other. This code is in the layout-step procedure. You can force this code to execute any time by pressing the REDO LAYOUT button, and pressing it again when you are happy with the layout.

NETLOGO FEATURES

Nodes are turtle agents and edges are link agents. The model uses the ONE-OF primitive to chose a random link and the BOTH-ENDS primitive to select the two nodes attached to that link.

The layout-spring primitive places the nodes, as if the edges are springs and the nodes are repelling each other.

Though it is not used in this model, there exists a network extension for NetLogo that comes bundled with NetLogo, that has many more network primitives.

RELATED MODELS

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

See also Network Example, in the Code Examples section.

CREDITS AND REFERENCES

This model is based on:
Albert-László Barabási. Linked: The New Science of Networks, Perseus Publishing, Cambridge, Massachusetts, pages 79-92.

For a more technical treatment, see:
Albert-László Barabási & Reka Albert. Emergence of Scaling in Random Networks, Science, Vol 286, Issue 5439, 15 October 1999, pages 509-512.

Barabási's webpage has additional information at: http://www.nd.edu/~alb/

The layout algorithm is based on the Fruchterman-Reingold layout algorithm. More information about this algorithm can be obtained at: http://citeseer.ist.psu.edu/fruchterman91graph.html.

For a model similar to the one described in the first suggested extension, please consult:
W. Brian Arthur, "Urban Systems and Historical Path-Dependence", Chapt. 4 in Urban systems and Infrastructure, J. Ausubel and R. Herman (eds.), National Academy of Sciences, Washington, D.C., 1988.

HOW TO CITE

If you mention this model in a publication, we ask that you include these citations for the model itself and for the NetLogo software:

Wilensky, U. (2005). NetLogo Preferential Attachment model. http://ccl.northwestern.edu/netlogo/models/PreferentialAttachment. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.
Wilensky, U. (1999). NetLogo. http://ccl.northwestern.edu/netlogo/. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.

COPYRIGHT AND LICENSE

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 http://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.

Comments and Questions

Please start the discussion about this model! (You'll first need to log in.)

Click to Run Model

;;;;;;;;;;;;;;;;;;;;;;;;
;;; Setup Procedures ;;;
;;;;;;;;;;;;;;;;;;;;;;;;
globals[N]
extensions[nw]

to setup
  clear-all
  set-default-shape turtles "circle"
  nw:set-context turtles links
  ;; make the initial network of two turtles and an edge
  make-node nobody        ;; first node, unattached
  make-node turtle 0      ;; second node, attached to first node
  reset-ticks
end 

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

to go
  ;; new edge is green, old edges are gray
  while [N < number-of-links]
  [
  ask links [ set color gray ]
  make-node find-partner         ;; find partner & use it as attachment
                                 ;; point for new node
  tick
  if layout? [ layout ]
  ]
  
  ask one-of turtles [set color green]
  if any? turtles with [color = red]
  [
    ask turtles with [color = green]
    [
      ask link-neighbors [if random 100 <= probability-linked-affected [set color green]
        ask other turtles [
        if random 100 < probability-other-affected [set color green]]
      ]
    ]
  ]
end 

;; used for creating a new node

to make-node [old-node]
  crt 1
  [
    set color red
    if old-node != nobody
      [ create-link-with old-node [ set color green ]
        ;; position the new node near its partner
        move-to old-node
        fd 8
        set N N + 1
      ]
  ]
end 

to-report find-partner
  report [one-of both-ends] of one-of links
end 

;;;;;;;;;;;;;;
;;; Layout ;;;
;;;;;;;;;;;;;;

;; resize-nodes, change back and forth from size based on degree to a size of 1

to resize-nodes
  ifelse all? turtles [size <= 1]
  [
    ;; a node is a circle with diameter determined by
    ;; the SIZE variable; using SQRT makes the circle's
    ;; area proportional to its degree
    ask turtles [ set size sqrt count link-neighbors ]
  ]
  [
    ask turtles [ set size 1 ]
  ]
end 

to layout
  ;; the number 3 here is arbitrary; more repetitions slows down the
  ;; model, but too few gives poor layouts
  repeat 3 [
    ;; the more turtles we have to fit into the same amount of space,
    ;; the smaller the inputs to layout-spring we'll need to use
    let factor sqrt count turtles
    ;; numbers here are arbitrarily chosen for pleasing appearance
    layout-spring turtles links (1 / factor) (7 / factor) (1 / factor)
    display  ;; for smooth animation
  ]
  ;; don't bump the edges of the world
  let x-offset max [xcor] of turtles + min [xcor] of turtles
  let y-offset max [ycor] of turtles + min [ycor] of turtles
  ;; big jumps look funny, so only adjust a little each time
  set x-offset limit-magnitude x-offset 0.1
  set y-offset limit-magnitude y-offset 0.1
  ask turtles [ setxy (xcor - x-offset / 2) (ycor - y-offset / 2) ]
end 

to-report limit-magnitude [number limit]
  if number > limit [ report limit ]
  if number < (- limit) [ report (- limit) ]
  report number
end

There is only one version of this model, created over 10 years ago by Jiawei Fu.

Attached files

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Contagion/ Social Influence- Jiawei Fu- Version 2.png	preview	Preview for 'Contagion/ Social Influence- Jiawei Fu- Version 2'	over 10 years ago, by Jiawei Fu	Download

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