Background

• Directed networks here
• Preferential attachment model:
  • 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?

1. What is their limiting (marginal) distribution?
2. What is their limiting joint distribution?
3. 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

1. Start with node 1 with a self-loop
2. Node $n+1$ joins when there are $n$ nodes in the network
3. 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
4. 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
• No connecting between existing nodes
  • They did mention this in extensions