Asymptotic behavior of the node degrees in the ensemble average of adjacency matrix

Various important and useful quantities or measures that characterize the topological network structure are usually investigated for a network, then they are averaged over the samples. In this paper, we propose an explicit representation by the beforehand averaged adjacency matrix over samples of growing networks as a new general framework for investigating the characteristic quantities. It is applied to some network models, and shows a good approximation of degree distribution asymptotically. In particular, our approach will be applicable through the numerical calculations instead of intractable theoretical analysises, when the time-course of degree is a monotone increasing function like power-law or logarithm.

ensemble average of adjacency matrix defined as follows.
Here, without loss of generality, we set connected two nodes as the minimum initial configuration: a 11 = a 22 = 0, a 12 = a 21 = 1, and the degree k 1 (0) = k 2 (0) = 1. Note that at each time step the matrix is expanding with the inverse L shape of elements a 1n , a 2n , . . . , a n−1n , 0, a nn−1 , . . . , a n2 , a n1 at the right-bottom corner. The diagonal element a ii is always 0 due to no self-loop at each node i. Other elements are 0 ≤ a ij ≤ 1, i = j, as the average number of links from i to j over the samples of networks. The value of each element is defined in order according to the passage of time. We assume that there are no adding links between existing nodes i, j < n at any time t = n − 2. Only links between selected old nodes and new node n are added in a growing network.
In the sample-based description, the ensemble average of adjacency matrix is where A (k) (t) denotes an adjacency matrix of the k-th sample whose elements are a (k) ij = 1 or 0 corresponded to the connection or disconnection from i to j, but a (k) ij = 0 or undefined for i, j > t + 2 because of the size n = t + 2 at time t. N s denotes the number of samples. We remark that an adjacency matrix is fundamental and important because it includes the necessary and sufficient information about a network structure and is useful for a mathematical treatment.
In this explicit representation of general framework, for each i-th node, i = 1, 2, . . . , n − 1, the (out-)degree is updated from time t − 1 to t, The (out-)degree of n-th node added at time t = n − 2 is defined by the sum of links from node n to nodes i, We should remark that the iterative calculations of Eqs. (1)(2) are equivalent to the averaged values over the samples after calculating the degrees for each sample of the networks at time t. With this equivalence in mind, we investigate the asymptotic behavior of k i (t) for a large t. We note that k i (t) is a monotone non-decreasing function of time t because of k i (t) − k i (t − 1) = a in ≥ 0 from Eq. (1) in growing networks.

III. ASYMPTOTIC BEHAVIOR OF THE NODE DEGREES
As examples, we apply the ensemble average of adjacency matrix to some network models. However, our approach is applicable to other growing networks especially in a wide class, e.g. with approximately power-law or exponential degree distribution. In the following, we assume that each link is undirected: a ij = a ji .

A. Babarási and Albert model
Since the continuous-time approximation of Eq.(1) is generally our approach is applicable to the Babarási and Albert (BA) model [3] as follow. preferential attachment: uniform attachment: where m 0 denotes the initial number of nodes, and m denotes the number of adding links at each time step. For the two cases of preferential and uniform attachments, k i (t) = m t/t i and k i (t) = m(log(m 0 + t− 1)− log(m 0 + t i − 1) + 1) have been derived with the corresponding degree distributions p(k) ∼ k −3 and p(k) ∼ e −k , respectively [3]. The analysis in BA model is based on the invariant ordering property of degrees in which older nodes get more links averagely at any time in the growth of network. Under the invariant ordering property, our approach can be regarded as an extension of mathematical treatment in the BA model through the representation by the ensemble average of adjacency matrix over samples of growing networks.

B. Duplication-divergence model
We preliminary introduce a duplication-divergence (D-D) model [10,12] without mutations of random links between existing nodes, whose generation mechanism is known as fundamental in biological protein-protein interaction networks. In the D-D model, at each time step, a new node is added and links to neighbor nodes of a uniformly randomly chosen node (see Figure 1(a)). Some duplication links are deleted with probability δ > 0. Here, no mutations are to simplify the discussion and to connect to the next subsection. Although the degree distribution in the D-D model can be approximately analyzed by the approach of mean-field-rate equation [10,12], we show the applicability of our approach to the D-D model in order to extend it to more general networks. Moreover, in the next section, we reveal that older nodes tend to have higher degrees [4] in the D-D model, whose ordering of degree k i (t) for node index i was not found from the above approach. Since the n-th new node links to the neighbor node i of a chosen node j = i from existing nodes 1 ≤ i, j ≤ n − 1 in a network of D-D model, we have where we use the uniform selection probability 1/(n − 1) of each node j and the no-deletion rate 1 − δ for linking to the neighbor nodes.
From Eqs. (1)(2) (3), we obtain By applying Eq. (4) recursively, we derive In addition, the continuous-time approximation of Eq. (4) is From the separation of variables method, we obtain the solution When we denote the initial degree k i at the inserted time t i for a node i, the above solution is rewritten as From the existence of parallel curves shown in Fig. 2, the ordering of degrees is not changed. In other word, older nodes get more links averagely. Thus, we obtain where N 0 + t is the number of nodes at time t, and N 0 denotes the initial number of nodes. In the tail of degree distribution, the exponent of power-law is 1 + 1 1−δ asymptotically. Note that the slightly different exponent 1 + 1 1−δ to the conventional approximation [7,10] is not strange, since the mutations are necessary for their D-D models.

C. Copying model
A modification [5] of the D-D model [10,12] by adding a mutual link between a new node and a randomly chosen node has been proposed. The mutual link contributes to avoid the singularity called as non-self-averaging even for δ < 1/2 without mutations [5]. The growing network is constructed as shown in Fig. 1(b). It is referred to as copying model. At each time step, a new node is added. The new node links to a uniformly randomly chosen node, and to its neighbor nodes with probability 1 − δ.
In the copying model, we have since the n-th new node links to a uniformly randomly chosen node i and to the neighbor node i when other node j is chosen from existing nodes 1 ≤ i, j ≤ n − 1 in the network. These effects are in the first term 1/(n − 1) and the second term (1 − δ) By applying Eq. (9) recursively, we derive where we use n − 1 = t + 1 and the Staring formula Γ(x + 1) ≈ x x e −x . In particular, by the mathematical induction, we confirm that the case of δ = 0 generates a sequence of the complete graphs with k i (t) = t + 1 = n − 1 links at every node of i = 1, 2, . . . , n. First, k 1 (1) = k 2 (1) = 2 and k 3 (1) = 2 are obvious. Next, we assume k i (t − 1) = (t − 1) + 1 = n − 2, from Eqs.(9)(10) we derive On the other hand, the continuous-time approximation of Eq. (9) is Since this form is a 1st order linear differential equation dk i /dt + f (t)k i = g(t), by applying the solution where A and C ′ are constants of integration, and we use f (t) def = −(1 − δ)/(t + 1) and g(t) def = 1/(t + 1). Note that the solution t 1−δ is only different by O(1/t) to Eq. (11), and can be ignored for a large t. As similar to the D-D model in subsection 3.2, from Eq. (7) under the invariant ordering (6) in the parallel curves shown as Fig. 4, the degree distribution asymptotically follows a power-law with the exponent 1 + 1 1−δ . Figure 5(a)-(c) shows the time-course of k i (t) ∼ t 1−δ in the cases of δ = 0.3, 0.5, and 0.7, respectively, averaged over 100 samples. The black, orange, and magenta lines are the numerical results of Eqs. (9)(10) for the node i = 1, 10, and 100. The cyan line guides the estimated slope of 1 − δ in log-log plot. It fits to the lines of k i (t) for a large t. Moreover, as shown in Fig. 5(d), Eq. (7) gives a good approximation at the size n = 10 4 . The red, green, and blue lines show the degree distributions for δ = 0.3, 0.5, and 0.7, respectively. The magenta, cyan, and gray dashed lines guides the corresponding slopes of 1 + 1 1−δ for these δ in the fitting to the tails.

D. Copying model with positive degree-degree correlations
In this subsection, we emphasize that our approach is effective through the numerical estimation, even when an analytic derivation is intractable.
We consider a copying model with positive degree-degree correlations based on a cooperative generation mechanism by linking homophily, in which densely connected cores among high degree nodes emerge [5]. In more detail, the difference to the previously mentioned copying model is that the n-th new node links to the neighbor nodes j of a randomly chosen node i with a probability (1 − δ)f ((1 − δ)k i (t − 1), k j (t − 1)) from existing nodes in the network. Such a function is necessary to enhance the degree-degree correlations, and a > 0 is a parameter [14]. Since the degree of new node is unknown in advance due to the stochastic process, it is temporary set as (1 − δ)k i (t − 1). Thus, instead of Eq. (8), we substitute for Eqs. (1)(2). Although the theoretical analysis of Eqs. (1)(2) for Eq. (12) is difficult, the iterative calculations are possible numerically. When we assume βk i (t) ∼ log(t), we derive an exponential distribution as follows. From t ∼ e βki(t) , we obtain under the invariant ordering (6) in the parallel curves shown as Fig. 6. Figure 7(a)-(c) shows that k i (t) denoted by black, orange, and magenta lines for i = 1, 10, and 100 is approximated by log(t) in the copying model with degree-degree correlations. The cyan lines guide the estimated slopes β = 2.222, 1.818, and 1.428 for δ = 0.3, 0.5, and 0.7, respectively, in the numerical fittings for the iterative calculations of Eqs.
(1)(2) (12). Thus, as shown in Fig. 7(d), the tails in p(k) denoted by red, green, and blue lines for δ = 0.3, 0.5, and 0.7 are approximated by e −βk shown as magenta, cyan, and gray dashed lines at the size n = 10 3 . Note that p(k) is only slightly deviated but the exponential part is remained by adding shortcut links in order to self-organize a robust onion-like structure in the incrementally growing network [5].

E. More general case
We consider the asymptotic behavior of the node degrees for a general case in growing networks. When the time-course k i (t) of degree follows a monotone increasing function g(t) of time t, there exists the inverse function It is possible that the time-course k i (t) is an average of observed real data for node i, e.g. born at time t − 2. From we have Then, on the assumption of the invariant ordering (6) in parallel curves of {k i (t)}, we derive where h ′ (k) denote the derivative of h(k) by the variable k. We should remark that various degree distributions of non-power-law may appear depending on the shape of monotone increasing function k i (t) according to what type of generation in growing networks.

IV. ANALYTIC DEVIATION FOR THE INVARIANT ORDERING OF NODE DEGREES
We derive the invariant ordering (6) at any time t within parallel curves of monotone increasing functions for the D-D and copying models discussed in subsections 3.2 and 3.3. In the following, we use double mathematical induction for node index i and time t.

V. CONCLUSION
We have proposed the explicit representation by the ensemble average of adjacency matrix over samples of growing networks. The important point is that the adjacency matrix is averaged in advance before calculating a characteristic quantity about the topological structure for each sample. The ensemble average has been applied to some network models: BA [3], D-D [10,12], and copying [5] models for investigating the degree distributions in the asymptotic behavior by using the theoretical and numerical analysises for difference equations and the corresponding continuoustime approximation of differential equations with variables t and k i (t). We have derived k i (t) ∼ t 1−δ and p(k) ∼ k −(1+ 1 1−δ ) for the D-D and copying models under the invariant ordering of degrees which is supported in randomly grown networks [4]. Moreover, for the copying model with positive degree-degree correlations, we have shown that the numerical calculations of the difference equation give a good approximation of an estimated exponential distribution, even when an analytic derivation is intractable. The copying model with positive degree-degree correlations is related to the self-organization [5] of robust onion-like networks [6,11,13].
Our approach may be also applicable to data analysis for social networks, when the observed time-course of degree is a monotone increasing function like power-law or logarithm in the average over samples by ignoring short-time fluctuations, and a node index i or j represents an ordering of its birth time. This expectation is supported as follows. It is helpful for grasping a trend to study the average behavior of many users (nodes) added at a same (sampling interval) time into a network community. Random growth in a social network probably corresponds to encountered chances among people. Moreover, as time goes by, the number of his/her friends for a member of social networks is usually increasing. It is natural that the connections to friends are maintained. However, we must consider the effect of rewirings between old members in the definition of adjacency matrix. Also from a practical viewpoint, O(n 2 ) space to store an adjacency matrix may cause a problem for dig data.
From the conventional analysis for special network models to a general framework, the representation by the ensemble average will open a door for investigating the characteristic quantities, e.g. node degrees in growing networks. In particular, the time-course of a quantity depending on the birth time of node is considered as a key point. The discussion about other quantities such as clustering coefficient or the number of paths of a given length requires further studies of how to analyze the average behavior related to the transitivity.