Nodes join the network one by one, bringing in new edges
\(\Pr(\)existing node \(i\) gets an edge\()~\propto~\) current degree of \(i\)
In-/total degree captures the rich-gets-richer effect
Simulated network
Size according to in-degree
Marginal distribution
Why?
Easy to simulate from and fit to data
In-degrees of empirical networks seem to follow the “power law”
Essentially the (non-generalised) Pareto distribution
Some argue otherwise (e.g. Broido and Clauset, 2018)
Power law adequate? Not today’s scope
Making the model more realistic
\(\Pr(\)existing node \(i\) gets an edge\()~\propto~\) current degree of \(i\) + \(\boldsymbol{\delta}\)
A positive probability of getting connected even if the node has no in-degre to start with
Level the playing field
Reciprocity parameter \(\boldsymbol{\rho}\)
So far only either \(A\rightarrow B\) or \(B\rightarrow A\), but not both \(\quad\Rightarrow\quad\) reciprocity \(=0\)
Empirical networks usually show high reciprocity
E.g. Instagram: if \(A\) follows \(B\), \(B\) is more likely to follow \(A\) back
\(\rho\) affects how the in-degrees & out-degrees are correlated
Joint distribution
Same simulated network with 0 reciprocity
Wang & Resnick (2021)
How does incorporating reciprocity in the preferential attachment model affect the in- & out-degrees?
What is their limiting (marginal) distribution?
What is their limiting joint distribution?
Are they asymptotically independent / dependent?
Summary
A relatively restrictive preferential attachment model even with reciprocity
Possibly for the theoretical derivations
Turning the process of preferential attachment into a multitype branching process with immigration
Using results developed for branching process to arrive at the limiting distributions of in- & out-degrees
Thus answering the questions they set out
Model set up
Start with node 1 with a self-loop
Node \(n+1\) joins when there are \(n\) nodes in the network
With probability \(\alpha\), there is an edge from node \(n+1\) to an existing node
The node is chosen with probability \(~\propto~\) their current in-degree \(+~\delta\)
With probability \(\rho\), reciprocity happens i.e. an edge from the chosen node to \(n+1\) is created
With probability \(1-\rho\), reciprocity doesn’t happen
With probability \(\gamma(=1-\alpha?)\), there is an edge from an existing node to node \(n+1\)
The node is chosen with probability \(~\propto~\) their current out-degree \(+~\delta\)
With probability \(\rho\), reciprocity happens i.e. an edge from \(n+1\) to chosen node is created
With probability \(1-\rho\), reciprocity doesn’t happen
Converting to branching processes
An in-degree \(~\Leftrightarrow~\) a type 1 particle
An out-degree \(~\Leftrightarrow~\) a type 2 particle
A node joins \(~\Leftrightarrow~\) immigration event
Node \(n+1\) joins at time \(T_{n}\)
Creation of new edge \(~\Leftrightarrow~\) death of existing particle & birth of new particle(s)
While a type 1 particle can give birth to a type 2 particle, births of new type 1 particles must come with it \(~\Leftrightarrow~\)
While an in-degree can lead to a new out-degree, there must be a new in-degree first
Converting to branching processes (cont’d)
The existing node that connects with node \(n+1\)\(~\Leftrightarrow~\)\(J_{n+1}\)
Reciprocity occurs when node \(n+1\) joins \(~\Leftrightarrow~\)\(R_{n+1}\)
The evolution of the in- & out-degree of node \(k\)\(~\Leftrightarrow~\)\(\boldsymbol{\xi}_{1,\delta}(\cdot)\)
The allocation of new edge(s) from node \(n+1\) depends only on current in- & out-degrees \(~\Leftrightarrow~\)
The branching process is Markovian
Connecting the two formally
The collection of the branching processes \(\left\{\boldsymbol{\xi}^{*}_\delta(T_n):n\geq0\right\}\)
Before node \(n+1\) arrives, \(\boldsymbol{\xi}_{n+1},\delta(\cdot)\) is set to \((0,0)\)
The collection of in- & out-degree sequences \(\boldsymbol{D}(n):=\left((D_1^{in}(1), D_1^{out}(1)), \ldots, (D_1^{in}(n), D_1^{out}(n)), (0,0), \ldots\right)\)
Theorem 2: The two have the same distributions
Any result for \(\left\{\boldsymbol{\xi}^{*}_\delta(T_n):n\geq0\right\}\) can be used for the in- & out-degree sequences
Result 1 / Theorem 4
When there are \(n\) nodes in the network, the in-degree of node \(\omega\), rescaled by \(n\) raised to a certain power, converges to a finite random variable \(L_\omega\), as \(n\rightarrow\infty\).
Similar for the out-degree
\(n\) raised to a certain power – power law?
This power is a function of the parameters \((\alpha,\gamma,\rho,\delta)\)
We will come back to this
Special case: when \(\alpha=1\)
Nodes selected only according to in-degrees
Reciprocity only from existing node to new node
The rescaled out-degree converges to \(\rho~L_\omega\)
Result 2 / Theorem 5
Joint degree counts
Joint distribution of in- & out-degree in a different way
\(N_{m,l}(n)\): the number of nodes with \(m\) in-degree & \(l\) out-degree when there are \(n+1\) nodes in the network
\(N_{m,l}(n)/n\): Joint empirical pmf of \((m, l)\)
The first line of (3.10) not explicitly simplified, but leads to next result
Result 3 / Theorem 6
The in- & out-degrees are jointly regularly varying
Index \((1+\rho+\delta)/\lambda_1\)
Echoing theorem 4?
Special case: when \(\alpha=1\), \(\lambda_1=1\)
MRV with index \(1+\rho+\delta\)
Asymptotic dependence
Question: Even when \(\rho=0\)?
Full asymptotic dependence
The large \((I,O)\) pairs concentrate on the straight line \(y=ax\)
Echoed by middle plot of Figure 4.1
Calculate \(O-aI\) and investigate if there’s hidden regular variation
Simulation & Application
Without modification i.e. fixed reciprocity: middle plot of Figure 4.1
Behaviour as expected
Attempted to randomise \(\rho\) to make the model more realistic i.e. closer to Facebook data
Left & right plot of Figure 4.1
Different user pairs have different degrees of reciprocity
Some thoughts
Model quite restrictive & not very realistic
Asymptotic dependence even if there’s no reciprocity
Only one edge is added at each immigration
Presumably derivations are far more challenging when bringing in multiple edges & sampling without replacement
