Random multi-hooking networks

We introduce a broad class of multi-hooking networks, wherein multiple copies of a seed are hooked at each step at random locations, and the number of copies follows a predetermined building sequence of numbers. We analyze the degree profile in random multi-hooking networks by tracking two kinds of node degrees -- the local average degree of a specific node over time and the global overall average degree in the graph. The former experiences phases and the latter is invariant with respect to the type of building sequence and is somewhat similar to the average degree in the initial seed. We also discuss the expected number of nodes of the smallest degree. Additionally, we study distances in the network through the lens of the average total path length, the average depth of a node, the eccentricity of a node, and the diameter of the graph.


Introduction
Trees have long been in the focus of research on random graphs.The classic types, such as those that appear in data structures [3,10,14,15] and digital processing [6,12,20], grow incrementally, one node at a time.In more recent times, authors considered more complex types of random graphs grown by adjoining entire graphs to a growing network [1,2,5,7,11,13,16,21].We consider a growing network model in which the number of components attached at a stage follows a predetermined building sequence of numbers.
Societies and social networks grow and change over time in multiple random ways, which include growth patterns that add "components" at each step.Networks grown by adding components reflect these dynamics better than networks evolving on single node additions.One can embed a graph in a predetermined growth structure leading to multiple scenarios of growing networks.
In this paper, we develop a model where networks grow by hooking multiple copies of the seed at multiple nodes of the growing network chosen in a random fashion and study the theoretical and statistical properties of the networks so generated.

The building sequence
We assume that a network grows by attaching a number of components at each step to the existing structure, which starts with  0 ≥ 2 vertices.In the next subsection, we give a formal definition.Here, we only say a word on the number of components added at each step.After  steps of growth, the number of components attached to obtain the next network is   , a predetermined sequence of nonnegative numbers.

Regularity conditions
Let  0 ≥ 2. This represents the number of nodes in a building block (a seed).We grow the network by adding a number of copies of the seed at places called latches.At each latch, a designated vertex in the seed (called the hook) is fused with the latch.A formal definition of this process is given in the sequel.
We shall consider adding   ≥ 1 copies of the seed to construct the ( + 1)st network, under the following regularity conditions: A sequence of nonnegative integers {  } ∞ =0 satisfying (R1)-(R3) is called a building sequence.Condition (R1) is to guarantee the feasibility of choosing latches.At no point in time does the process require more (distinct) latches than the number of nodes existing in the network.Conditions (R2)-(R3) facilitate the existence of limits for properties of interest and expedite finding their values.Note that  = 0 and  = 0 are both allowed.For instance, for a constant sequence   =  ∈ N, we have  = 0, and  =  0 / > 0, whereas when   =  + 1, we have both  = 0 and  = 0.
Regularity conditions (R1)-(R3) are not too restrictive and the class covered by the investigation remains very broad.The examples that come up in practice satisfy these regularity conditions.For example, at one extreme, the building sequence   = 1 builds networks of linear growth, including trees.At the other extreme, the case of equality in Condition (R1) builds a deterministic network where the entire vertex set is chosen at each step (a take-all model); such extremal case grows the network exponentially fast.

The multi-hooking network
A network grows as follows.We start with a connected seed graph  0 with vertex set of size  0 and edge set of size .One of the vertices in the seed is designated as a hook (vertex ℎ).When a copy of the seed is adjoined to the network, it is the seed's hook that latches into that larger graph.The hooking is accomplished by fusing together the hook and a latch (vertex) chosen from the network.
At step ,  −1 copies of the seed are hooked into the graph,  −1 = ( −1 , E −1 ), with vertex set  −1 and edge set E −1 , that exists at time  − 1.To complete the th hooking step, we sample  −1 latches from the graph  −1 .The selection mechanism can take a number of forms, such as choosing distinct hooks as opposed to allowing repetitions.
We use the notation | | for the cardinality of a set .We consider a uniform model that selects  −1 distinct nodes in the network, with all | −1 |  −1 subsets being equally likely.In the language of statistics, this boils down to sampling without replacement.
Figure 1 illustrates a seed and a network grown from it in three steps under the building sequence   =  +1.So,  1 grows by choosing a latch from  0 (the starred node in  0 ),  2 grows by choosing the two starred nodes from  1 , and  3 grows by choosing the three starred nodes from  2 .The networks in Figure 1 have loops and multiple edges, as we do not restrict the study to simple graphs.

Notation
The notation Hypergeo(, , ) stands for the hypergeometric random variable associated with the random sampling of  objects out of a total of  objects, of which  objects are of a special type.So, the hypergeometric random variable counts the number of special objects in the sample.
It is customary to call the cardinality of the vertex set of a graph the order of the graph and reserve the term size of the graph to the cardinality of the set of edges in the graph.Let   be the set of vertices  of the graph   , and E  be the set of edges of that graph.Thus, the seed  0 = ( 0 , E 0 ) is a connected graph with the set  0 of vertices and the set E 0 of edges.
Let   be the order of the graph at age .Hence, the cardinality of the vertex set  0 of the seed is | 0 | =  0 .The th hooking step adds  −1 copies of the seed at  −1 distinct latches chosen uniformly at random from  −1 .Each copy contributes  0 − 1 new vertices to the network.The reason for subtracting 1 is the absorption of the hook.This gives the recurrence Unwinding this recurrence, we obtain We use the notation deg() to denote the degree of node  in a given graph, and we set ℎ * = deg(ℎ).

Useful limits
By the regularity conditions, we can argue from (2) that Reorganize (1) as to find the limit

A degree profile of the network
Various aspects of the degrees of nodes in a network are of interest in different contexts.For example, in the language of epidemiology, the degree of a node may be a useful representation of a highly infective person.From a health policy point of view, having knowledge about the degrees in conjunction with other graph parameters may help in identifying hot spots that trigger outbreaks and may be useful in controlling and mitigating the contagion.In the context of a social network, the degree of a node may represent the popularity and social skills of the person represented by the node.
Equally interesting are the global overall average degree in the entire graph (where we look at all the nodes), the local degree of a specific node during its temporal evolution, and the number of nodes of the smallest degree.We deal with the average behavior of each of these in a separate subsection.The different aspects of the degree complete a profile of the graph.

Evolution of the degree of a specific node
Suppose a node appears for the first time at step .What will become of its degree at step ?At step , several copies are added.To avoid a heavy notation identifying the time of appearance , the copy number, which node within the copy to be tracked, and , we use a simpler notation that needs only  and , for after all nodes of the same degree in the seed have the same distribution over time.Theorem 4.1.Suppose {  } ∞ =0 is a building sequence of the family of graphs {  } ∞ =1 .Let  : be the degree of a node at time  that had appeared for the first time at step .If initially its degree (in the seed) is , then we have Proof.Suppose a node  appears at time  for the first time.So, it belongs to one of the copies adjoined to the graph at that time.As the graph evolves, in any single step, the degree of  can increase, if it is one of the nodes selected as latches in that step; otherwise its degree stays put, and when it does increase, it goes up by ℎ * = deg(ℎ), the degree of the hook in the seed.This gives rise to a recurrence: where I −1 () is an indicator of the event of choosing  among the  −1 latches of that step of growth.
On average, we have Unwinding the recurrence, we obtain the exact average: By the limits in Subsection 3.2, we obtain Remark 4.2.Consider the case  > 0. The average in Theorem 4.1 indicates that the degree of a specific node experiences phases.The degree of a node in the early phase with  =  () = () grows linearly with its age in the network.When  () ∼ , for 0 <  < 1, we still get a linear growth, but the coefficient of linearity is attenuated to In this case, a finer analysis is needed to identify the leading order of the average degree of a node that appears at time .For instance, in the case of a tree grown from the complete graph  2 , we have  = 1, ℎ * = 1,   = 1, and  = 0.The exact formula in this case yields Whence, we have the phases

The overall average degree
The main result about the overall average degree in the graph is developed in this section.The result is expressed in terms of , the number of edges in the seed graph.
Theorem 4.2.Suppose {  } ∞ =0 is a building sequence of the family of graphs {  } ∞ =1 .Let   be the degree of a randomly chosen node in the graph   at age .We have Proof.Upon hooking  −1 copies of the seed to  −1 distinct nodes of  −1 = ( −1 , E −1 ), we add  −1 edges to the graph.Therefore, we have This recurrence has the solution https://doi.org/10.1017/S0269964822000523Published online by Cambridge University Press Using the classical First Theorem of Graph Theory, we obtain Scaling the equation by   , we get Taking limits, and using Eq. ( 2), we obtain Remark 4.4.The average degree in the seed is 2/ 0 .For any building sequence, the asymptotic average degree in the graph is 2/( 0 − 1), only slightly higher than the average degree in the initial seed.This should be anticipated because the additions introduce a number of copies of the seed, each of which has the degree properties of the seed with the hook eliminated.

Nodes of the smallest degree
We study only the nodes of the smallest degree.Let  * be the smallest degree in the seed.Note that the smallest admissible degree in the graph is  * .After the network grows, the smallest degree in it may be  * or higher.Let   be the number of nodes of degree  * at time .Thus,  0 is the number of nodes of degree  * in the seed.Later graphs can have more nodes of degree  * .The seed in Figure 1 has  * = 2, and  0 = 1,  1 = 2,  2 = 3, and  3 = 3.

Stochastic recurrence
In the evolution at step , we hook  −1 copies of the seed to the graph  −1 .Let  0 be the event deg(ℎ) =  * and I  0 be an indicator that assumes value 1, if deg(ℎ) =  * , otherwise, it assumes the value 0. A latch of degree  * in the sample will have a higher degree (namely, its degree goes up to  * + deg(ℎ)) in   .So, we lose such vertices in the count of   .If the hook degree is  * , every hooked copy contributes only  0 − 1 vertices of degree  * .For the case when  −1 = 1 and the latch is ℓ, the change from  −1 to   for the four cases can be seen as shown in the table below: Thus, for any value of  −1 , the count   therefore satisfies a (conditional) stochastic recurrence: https://doi.org/10.1017/S0269964822000523Published online by Cambridge University Press 4.3.2.The average proportion of nodes of degree  * Take (conditional) expectation of (3) to get Theorem 4.3.Suppose {  } ∞ =0 is a building sequence of the family of graphs {  } ∞ =1 , starting from a seed with  0 nodes of the smallest degree  * .Let   be the number of vertices of this degree in the graph after  steps of evolution according to the building sequence.We have Subsequently, the average proportion converges to a limit independent of the limits  and ; namely we have the convergence Proof.Taking a double expectation of ( 4) yields This recurrence equation is of the standard linear form with solution So, the sought solution for the average of the number of nodes of degree  * (for  ≥ 1) is The strategy for the asymptotic part of the statement is two-fold: We prove the existence of a limit (under any building sequence) for the proportion from the exact solution.We then find the value of the limit from the recurrence under the mild regularity conditions imposed on the building sequence.First, express the expected proportion as https://doi.org/10.1017/S0269964822000523Published online by Cambridge University Press at  = , the first product does not exist, and is taken to be 1, as usual.Let We manipulate this to turn it into a recurrence as follows: Rearrange the recurrence in the form leading to the inequality Noting that the sum in   is empty at  = 0, we have  0 = 0 and the bounds simplify to 0 ≤ |  − 1/ 0 | ≤ 1/  .So, both inferior and superior limits of |  − 1/ 0 | are equal to 0, which furnishes the existence of a limit for   equal to 1/ 0 , too.As for the remainder part , it clearly converges to 0, as   is increasing, and the product is bounded from above by 1. Plugging in the limits lim →∞   = 1/ 0 and lim →∞   = 0 in (8), we reach the conclusion that lim Remark 4.5.In the case when the hook is not of the smallest degree  * , we have E[  /  ] →  0 / 0 .The initial proportion of nodes of the smallest degree in the seed is preserved on average in larger subsequent graphs.
Remark 4.6.In the case when the hook is of the smallest degree  * , we have E[  /  ] → ( 0 − 1)/ 0 .The long-term proportion of nodes of the smallest degree is less than the proportion of nodes of degree  * in the seed.
Remark 4.7.In the case when the hook is the only node of the smallest degree  * in the seed, we have  0 = 1, and E[  /  ] → 0, for all  ≥ 1.Indeed, the degree  * disappears after the first latching at the initial hook and never reappears.
Remark 4.8.The limit in Theorem 4.3 is more than just an ultimate value in the take-all case.In this case, it is the actual value for each  ≥ 0, which can be seen from the recurrence.The only term that does not vanish is the last term in sum, yielding ( 0 −   0 ) −1 /  = ( 0 −   0 )/ 0 .

Distances in the network
We measure node distances in   relative to a reference point (vertex).We take the reference to be the hook of  0 .We look at two (related) kinds of distances: The total path length and the average distance in the graph.Let the nodes of the th graph be labeled with the numbers 1, 2, . . .,   , with 1 being reserved for the reference vertex and the rest of the nodes are arbitrarily assigned distinct numbers from the set {2, . . .,   }.The depth of a node in the network is its distance from the reference vertex (i.e., the length of the shortest path from the node to the reference vertex measured in the number of edges).We denote the depth of the th node in the th network by Δ , The total path length is the sum of all depths; namely it is For instance, the networks  0 ,  1 ,  2 , and  3 in Figure 1 have total path lengths  0 = 3,  1 = 6,  2 = 18, and  3 = 45, respectively.

Average total path length
As the network grows, at step , a sample of size  −1 latches is chosen from  −1 to grow  −1 into   .Suppose these latches are ℓ 1 , . . ., ℓ  −1 at depths  1 , . . .,   −1 .In view of the absorption of the hooks of the added graphs, a copy's hook fused at the th latch adds  0 − 1 nodes, which appear in   at depths equal to their distance from the hook of the copy translated by an additional distance from the latch to the reference vertex.So, collectively, the vertices of the copy hooked to ℓ  increase the total path length by  0 + ( 0 − 1)  .We have a conditional recurrence: Averaging over the graphs and the choices of the latches within, we get https://doi.org/10.1017/S0269964822000523Published online by Cambridge University Press Lemma 5.1.
Proof.Condition on the event The subsets of size  −1 latches that appear in a sample of vertices from  −1 are all equally likely, and we get Let us write out the inner sum in expanded form: Upon a reorganization collecting similar terms, we get Plugging this expression in the expectation, we proceed to Theorem 5.1.Suppose {  } ∞ =0 is a building sequence of the family of graphs {  } ∞ =1 , starting from a seed of total path length  0 .Let   be the total path length after  steps of evolution according to the building sequence.We have https://doi.org/10.1017/S0269964822000523Published online by Cambridge University Press Proof.By Lemma 5.1 and the recurrence (9), we have a recurrence for the average total path length: Again, the recurrence is of the standard form (6) with the solution (7).In the specific case at hand, this solution is The recurrence (1) on the order of the graph simplifies the solution into telescopic products

Average depth
Theorem 5.1 provides a benchmark for the calculation of the average depth.Let the depth of a randomly selected node in the th network be   .

E[𝐷
Proof.Given a specific development leading to   , the average depth in that graph is Upon taking expectation, it follows that E[  ] = E[  ]/  .The form given in the statement ensues from Theorem 5.1.
Corollary 5.2.Under the regularity conditions (R1)-(R3), we have the asymptotic equivalent When  = 0, as in the case of trees for example, one needs to sharpen the argument to find the leading asymptotic term, as we do in some specific cases below.

Distances under specific building sequences
At one extreme, there is the sequence of least possible growth (  = 1).At the other extreme, we have a take-all model (  =   ) in which all the nodes of   are taken as latches for   copies of the seed.
In the case of   = , for fixed  ∈ N, of nearly the least growth, the average depth is The limit  in regularity condition (R2) is 0, and Corollary 5.2 only tells us that E[  ] = ().However, we can sharpen the asymptotic equivalence from the specific construction of the case.
Here, we have   = ( 0 − 1) +  0 , which gives In terms of the generalized harmonic numbers1 the depth in the near-least-growth is compactly expressed as Remark 5.2.The case  = 1 and  0 = 2 grows a recursive tree.The seed is a rooted tree on two vertices, in which  0 = 1 and  0 = 1 2 .In this case, the average depth becomes which recovers a known result [19].
Remark 5.3.At the other end of the spectrum, there is the take-all model, in which   =   , leading at step  to a graph of order   =  +1 0 .Here, the limit  is ( 0 − 1)/ 0 and the limit  is 0. According to Corollary 5.2, we have E[  ] ∼  0 , as  → ∞.This asymptotic estimate can be sharpened as the case is amenable to exact calculation:

Eccentricity
The eccentricity  () of a node  in a graph G is the distance between  and a vertex farthest from  in G.The eccentricity is instrumental in constructing a notion of the diameter of a graph (extreme distances).We use the eccentricity of the hook and the various latches selected in  −1 to determine the diameter of the graph   .The eccentricity is technically defined as follows.If Q is a path in a graph G, we denote its length by |Q| (the number of edges in it).There can be several paths joining two vertices  and  in G, and the distance between  and , denoted by  (, ), is the length of the shortest such path.That is, with P (, ) denoting the collection of paths between  and , the distance between these two nodes is given by The eccentricity  () of a vertex  in a graph with vertex set V is: For instance, the eccentricity in Figure 1 of the reference vertex of  0 is 2, of the reference vertex in  1 is 2 as well, but of the reference vertex in  2 is 4 and becomes 6 in  3 .

Eccentricity of a node in 𝑮 𝒏
The  −1 nodes selected as latches from the graph  −1 = ( −1 , E −1 ) are vertices that play a key role in designing the network at stage  and onward and contribute significantly in determining the diameter of the graph at the next stage.
As a node's eccentricity changes over time, its value at step  in   may be different from its value at step  − 1 in  −1 .We need an eccentricity notation reflecting the possible change over time.For that we use   () to speak of the eccentricity of a vertex  in   .
If  ∈   is a vertex in a copy of  0 latched at a vertex ℓ  ∈  −1 , we express that by saying  ∈    0 , otherwise we say  ∈  −1 .We now introduce some notation: Proof.The graph   is obtained by attaching a copy of the seed  0 at each of the latches ℓ We denote the vertex set of the th copy of the seed, for  = 1, . . .,  −1 , by  co  0 .We now compute the distance from a node  to a vertex  in   by considering the four cases: For a given  ∈   , the maximum (over the range of  and ) in each block is The result now follows.
Remark 6.1.Suppose a vertex ℓ is chosen as a latch from  −1 .From Theorem 6.1, we pick up the top line and write

Diameter of the graph 𝑮 𝒏
The diameter of a connected graph with vertex set V is the longest distance between any two nodes in it [4].That is, the diameter is the maximum eccentricity, max  ∈V  ().For example, the diameters of the graphs  0 ,  1 ,  2 , and  3 in Figure 1 are, respectively, 2, 4, 6, and 10.We now introduce some additional notation: 1. Д  = Д  |  −1 , L −1 .This is the conditional diameter Д  of the graph   given  −1 and the  −1 latches in L −1 (see Subsection 6.1 for the definition of L −1 ). 2. Only for   > 1, we define   = max ℓ, l ∈L   (ℓ, l) = max ℓ ∈L  ℓ # .Thus,   computes the maximum distance between any two latches in   .3.   = max ℓ ∈L    (ℓ).Thus,   is the maximum eccentricity of a latch in   .The longest distance in the graph is the maximum of the three possibilities discussed.Remark 7.1 shows that, under this special hooking mechanism, the diameter Д  at step  only requires the knowledge of the seed graph and .It does not take into consideration how many latches were picked at stages 1 through  − 1 as long as there are two latches ℓ, l picked at each stage such that  (ℓ, l) is maximum.

Figure 1 .
Figure 1.A seed (top) with a hook and three networks grown from it (second row) under the building sequence   =  + 1.The white vertices in the network  1 ,  2 , and  3 represent the reference vertex.

Remark 7 . 1 .
Consider the case where, at stage  (for each  ≥ 1), we pick among the  −1 latches two, say ℓ, l in  −1 , such that  (ℓ, l) is the diameter of  −1 .Note that this selection mechanism is no longer random in the sense discussed in all the preceding sections.Let us call the diameter of the graph so constructed Д * −1 .This is only possible if  −1 > 1, for each .By arguments similar to what we used in the proof of Theorem 7.1, we get Д *  = 2 0 (ℎ) + Д * −1 .Unwinding we get Д *  = 2 0 (ℎ) + Д 0 .

Theorem 6.1. Suppose
This is the conditional eccentricity   () of the node  in the graph   , given  −1 and the  −1 latches in it.3.For any  ∈  −1 , we define  #  = max ℓ  ∈L −1  (, ℓ  ).So,  #  is the maximum distance from  to the nodes in L −1 .Also, in what follows we use the notation I C to indicate a predicate (condition) C. So, it is 1, when C holds, and is 0, otherwise.{  } ∞ =0 is a building sequence of the family of graphs {  } ∞ =1 .Let  be a node in the graph   .Conditional upon the choice of the latches ℓ 1 , . . ., ℓ  −1 in  −1 , the eccentricity   () is given by