Preferential Placement for Community Structure Formation

Abstract

Various models have been recently proposed to reflect and predict different properties of complex networks. However, the community structure, which is one of the most important properties, is not well studied and modeled. In this paper, we suggest a principle called “preferential placement”, which allows to model a realistic community structure. We provide an extensive empirical analysis of the obtained structure as well as some theoretical heuristics.

Appendices

Appendix

Proof of Theorem 1

First, recall the process of cluster formation:

• At the beginning of the process we have one vertex which forms one cluster.

• At n-th step with probability p(n) a new cluster consisting of $v_n$ is created.

• With probability $1-p(n)$ new vertex joins already existing cluster C with probability proportional to |C|.

So, we can write the following equations:

$$\begin{aligned} \mathrm {E}(F_{t+1}(1)|S_t) = F_{t}(1) \left( 1 - \frac{1-p(t)}{t}\right) + p(t)\,, \end{aligned}$$

(1)

$$\begin{aligned} \mathrm {E}(F_{t+1}(s)|S_t) = F_{t}(s) \left( 1 - \frac{s(1-p(t))}{t}\right) + F_t(s-1)\frac{(s-1)(1-p(t))}{t}\,, \,\,\, \, s>1\,. \end{aligned}$$

(2)

Now we can take expectations of the both sides of the above equations and analyze the behavior of $\mathrm {E}F_{t}(s)$ inductively.

Consider the case $\alpha =0$, i.e., $p(n) = c$. Let us prove that in this case

$$\begin{aligned} \mathrm {E}F_n(s) = \frac{c (s-1)!\,\mathrm {\Gamma }\left( 2+\frac{1}{1-c}\right) }{(2-c)\mathrm {\Gamma }\left( s+1+\frac{1}{1-c}\right) }\left( n + \theta _{n,s}\right) \,. \end{aligned}$$

(3)

where $\theta _{n,s} \le C \, s^\frac{1}{1-c}$ for some constant $C>0$.

We prove this result by induction on s and for each s the proof is by induction on n. Note that for $n=1$ Eq. (3) holds for all s. Consider now the case $s = 1$. We want to prove that

$$ \mathrm {E}F_n(1) = \frac{c}{2-c}\left( n + \theta _{n,1}\right) \,. $$

For the inductive step we use Eq. (1) and get

Since

$$ C \left( 1 - \frac{1-c}{t} \right) \le C, $$

this finishes the proof for $\alpha =0$ and $s=1$.

For $s>1$ we use Eq. (2) and get

To finish the proof we need to show that

$$ (s-1)^{\frac{1}{1-c}} \frac{s(1-c) + 1}{t} \le s^{\frac{1}{1-c}}\frac{s(1-c)}{t}\,. $$

It is easy to show that the above inequality holds.

Now we consider the case $p(n) = cn^{-\alpha }$ for $0< \alpha \le 1$. Let us prove that in this case

$$ \mathrm {E}F_n(s) = \frac{c (s-1)! \, \mathrm {\Gamma }(3-\alpha )}{(2-\alpha )\mathrm {\Gamma }(s+2-\alpha )} \left( n^{1-\alpha } + \theta _{n,s}\right) \,, $$

where $\theta _{n,s} \le C n^{\max \{0,1-2\alpha \}}s^{1-\alpha +\epsilon }$ for some constant $C>0$ and for any $\epsilon >0$.

The proof is similar to the case $\alpha = 0$. Again, for $n=1$ the theorem holds. Consider $s = 1$. We want to prove that

$$ \mathrm {E}F_n(1) = \frac{c}{2-\alpha } \left( n^{1-\alpha } + \theta _{n,1}\right) . $$

Inductive step in this case becomes

In order to finish the proof for the case $s=1$ it is sufficient to show that

$$ O\left( t^{-\alpha -1}\right) + c\, t^{-2\alpha } \le C t^{\max \{0,1-2\alpha \}} \frac{1-ct^{-\alpha }}{t}\,, $$

which holds for sufficiently large C.

For $s>1$ we have:

In order to finish the proof, it remains to show that

$$ O\left( t^{-\alpha } \right) + \frac{t^{1-2\alpha } c (1 - \alpha )}{1-ct^{-\alpha }} \le C t^{\max \{0,1-2\alpha \}} s^{1-\alpha +\epsilon }\epsilon \,, $$

which holds for sufficiently large C.