## 1 Introduction  
Let \(n\) be a positive integer, \(1\leq i,j\leq n\), and \(\varepsilon(i,j)\) independent random variables such that \(\varepsilon(i,j)=\varepsilon(j,i)\) and \(\varepsilon(i,i)=0\), furthermore  
\[P(\varepsilon(i,j)=1)=p_{i,j}=p+p_{i}+p_{j},\quad 1\leq i<j\leq n \tag{1}\]  
where the sum of the \(p_{i}\)-s is zero. The least square estimate \(\hat{p}\) of \(p\) is the average of the epsilons, and the least square estimate of \(p_{i}\) is the average of the differences \(\varepsilon(i,j)-\hat{p}\). The modification of the model for non-symmetric matrices is straightforward, and in that case the statistical inference is practically a two-way analysis of variance. Perhaps this is the simplest random graph model but it shares the inconvenient property of many other random graph models that it is hard to ensure that edge probabilities remain in the interval \((0,1)\). If we use the odds  
\[r_{i,j}=\frac{p_{i,j}}{1-p_{i,j}} \tag{2}\]  
instead of the probabilities, then it is enough to ensure the positivity of \(r_{i,j}\)-s. This is the case in the model introduced by [PERSON] [1]. Historically the odds were defined as the ratios of scaling factors for rows and columns but we prefer the multiplicative form  
\[r_{i,j}=\beta_{i}\gamma_{j} \tag{3}\]  
for non-symmetric and  
\[r_{i,j}=\beta_{i}\beta_{j} \tag{4}\]  
for symmetric case. Statistical investigation of the model started with [PERSON] [2] (see also [3; 4; 5]) and later [PERSON] with his coworkers rediscovered the model and introduced the name _beta-model_ for its parameter. The model has many attractive properties (see in [6; 7; 8; 9; 10; 11]):  
* degree sequences are sufficient statistics;
* the model covers practically all possible expected degree sequence;
* the conditional distribution of the graphs on condition of a prescribed degree sequence is uniform on the set of all graphs with the given degree sequences.  
Statistical inference emerged from Gaussian distribution and later was extended to random variables in Euclidean spaces but the statistical inference on discrete structures is rather sparse [12; 13; 14; 15; 16]. Mathematical investigation of graphs has its own history. Nowadays instead of graphs, we are speaking of networks [17] where the most investigated model is the stochastic block model introduced by [PERSON], [PERSON] and [PERSON] ([18]). Here the vertices are labeled by small numbers or colors and edge probabilities depend only on the labels [19; 20]. With an eye on preferential attachment where degree sequences follow scale-free power-law, the block model was criticized because it has moderated flexibility on degree sequences. [PERSON], [PERSON], and [PERSON] [21] introduced a model with independent vertices. [PERSON], [PERSON] and [PERSON] [22] introduced the _planted partition model_ (see also [23]). [PERSON] and [PERSON] [24] proposed another extension of the block model. A natural extension of these models is the unification of the beta and block models:  
\[r_{i,j}=b(i,c(j))b(j,c(i)) \tag{5}\]  
where \(b(.,.)\) is a positive matrix with \(n\) rows and \(k\) columns, and \(c(i)\) is the label of the \(i\)-th vertex _i.e._, it is an integer between \(1\) and \(k\). We call the model _k-beta model_. The estimation of the labels in block models is possible by the spectral method [25]. It is generally believed that eigenvectors and eigenvalues of the matrix \(\varepsilon(i,j)\) tells everything of the structure of the graph [22; 26; 27; 28; 29; 30], while there are many attempts to provide more flexible models [31; 32].