CCorrelated copying in a hidden network model
Max Falkenberg ∗ Centre for Complexity Science, Imperial College London, SW7 2AZ, U.K. (Dated: February 19, 2021)We introduce the concept of a hidden network model – a generative two-layer network in whichan observed network evolves according to the structure of an underlying hidden layer. We apply theconcept to a simple, analytically tractable model of correlated node copying. In contrast to modelswhere nodes are copied uniformly at random, we consider the case in which the set of copied nodes isbiased by the underlying hidden network. In the context of a social network, this copied set may bethought of as an individual’s inner social circle, whereas the remaining nodes are part of the widersocial circle. Correlated copying results in a stretched-exponential degree distribution, suppressingthe power-law tail observed in uniform copying, generates networks with significant clustering, and,in contrast to uniform copying, exhibits the unusual property that the number of cliques of size n grows independently of n . We suggest that hidden network models offer an alternative family ofnull models for network comparison, and may offer a useful conceptual framework for understandingnetwork heterogeneity. Node copying is an important network growth mecha-nism [1–7]. In social networks, copying is synonymouswith triadic closure, playing an important role in theemergence of high clustering [8, 9]. In biology, node copy-ing encapsulates mechanisms of duplication and deletionwhich are key in the formation of protein-interaction net-works [10–12].Despite the diversity of applications, most node copy-ing models assume uniform, or homogenous copying, i.e.,that the probability of copying any given neighbor ofa node is equal. The exact formulation varies widely,but examples include “links are attached to neighbors of[node] j with probability p ” [5], and “one node [is du-plicated]... edges emanating from the newly generated[node] are removed with probability δ ” [11]. Other pa-pers with similar uniform copying rules include [2, 3, 6–8, 10, 12–20]. A small number of models do considerheterogeneous copying, although they limited in theirtractability [4, 9, 21, 22].In this paper, we introduce a tractable model of nodecopying which breaks the uniform copying assumption,inspired by the idea that, at least for social networks,copying should be biased towards a node’s inner socialcircle. In pursuit of this goal, we introduce the “hid-den network model” as a novel modeling framework. Amultilayer network [23] where layers have identical nodestructure but different edge structure, the framework letsus build models with local heterogeneity in the rules ofnetwork growth, but where that heterogeneity is a prop-erty of a hidden network structure and not arbitrarilyencoded. The framework can also be useful for decon-structing more complex single-layer models, e.g., second-neighbor preferential attachment [24]. The concept hasloose similarities to other multilayer paradigms includingthe use of replica nodes to model hetereogeneity [25] andinterdependent networks [26]. See also [19] on multilayercopying and triadic closure. ∗ [email protected] 𝛼 𝛽𝛾 𝛾 𝛾 𝑪𝑪𝑴 𝛼 𝛽𝛾 𝛾 𝛾 𝑼𝑪𝑴
FIG. 1. A new node α is added to the network and forms anedge to a random target node, β . UCM: All neighbors of node β have an equal probability p of being copied (orange dashed).CCM: Copied edges are added deterministically; neighbors ofnode β in the hidden network are copied (solid green), theremainder are not copied (red dotted). The remainder of this paper closely follows the struc-ture of [3]. We first introduce a typical model of uniformcopying, before outlining the adaptations necessary to de-fine the correlated copying model. We derive a numberof analytical results before finally comparing the resultsfor each model.
Uniform Copying Model.
The uniform copyingmodel (UCM) was introduced in [2, 3], see Fig. 1. Attime t α , a single node, α , is added to the network,and one target node, β , is chosen uniformly at random.We label each neighbor of β with the index γ j where j ∈ { , · · · , k β } , and k β is the degree of node β . For eachneighbor γ j , the copied edge ( α, γ j ) is added to the net-work independently with probability p . The network isinitialized at t = 1 with a single node. If p = 0, no edgesare copied forming a random recursive tree. If p = 1, theUCM generates a complete graph. Hidden Network Models.
We define a hiddennetwork model as the pair of single layer graphs G =( G O , G H ), comprised of an observed network G O =( V, E O ) and a hidden network G H = ( V, E H ), where V is the set of nodes for both networks and E O and E H arethe set of edges for each network. The set V represent the a r X i v : . [ phy s i c s . s o c - ph ] F e b 𝛼 𝛽𝛾 𝛾 𝛾 𝑮 𝑯 𝛼 𝛽𝛾 𝛾 𝛾 𝑮 𝑶 FIG. 2. The correlated copying model. A new node, α , formsan edge (solid blue) to a randomly chosen target node β .Copied edges ( α, γ j ) (solid green) are formed in the observednetwork, G O , if the edge ( α, γ j ) is present in the hidden net-work, G H . Other neighbors of node β are not copied (dottedred). same entities in both G O and G H , with differences lyingexclusively in the edge structure between nodes. The keyfeature of a hidden network model is that the evolutionof G O is dependent on G H (or vice versa). Correlated Copying Model.
In the correlated copy-ing model (CCM), see Fig. 2, the observed and hiddennetworks are initialized with a single node at t = 1. At t = t α , node α is added to both networks and a singletarget node, β , is chosen uniformly at random. We labelthe k βO neighbors of β in G O with the index γ j . Then,in the observed network only, the copied edge ( α, γ j ) isformed with p = 1 if the edge ( β, γ j ) ∈ E H , p = 0 other-wise. No copied edges are added to the hidden network G H . The direct edge ( α, β ) is added to both G O and G H . G H evolves as a random recursive tree. Unlike theUCM, all copying in G O is deterministic, with the onlyprobabilistic element emerging in the choice of the targetnode β . For comparative purposes, we define the effectivecopying probability in the CCM as p eff = k βH /k βO , i.e., thefraction of the observed neighbors of node β which arecopied by node α .Framed in a social context, we might think of G O asan observed social network where individuals have manyfriends, but the quality of those friendships is unknown,with most ties being weak. In contrast, underlying ev-ery social network is a hidden structure representing theinner social circle of individuals, where a node is onlyconnected to their closest friends [27]. Copying in theCCM is biased to this inner circle. Preliminaries.
The total number of edges in G H scales as E H ( t ) ∼ t . Hence, the average degree is givenby (cid:104) k H (cid:105) = 2. Using the degree distribution of G H , seebelow, (cid:104) k H (cid:105) = 6. In the observed network, each timestep a single edge is added by direct attachment, andone copied edge is added for each neighbor of the targetnode in G H , k βH . The average change in the number ofedges is therefore (cid:104) ∆ E O ( t ) (cid:105) = 1 + (cid:104) k βH (cid:105) = 1 + (cid:104) k H (cid:105) = 3,such that (cid:104) E O ( t ) (cid:105) ∼ t and (cid:104) k O (cid:105) = 6.An alternative route is to note that the observed degree of node α can be written as( k O ) α = ( k H ) α (cid:88) β =1 ( k H ) α,β (1)where the index α, β labels the ( k H ) α unique neighboursof α in G H . Averaging both sides of Eq. (1) over allnodes we find, (cid:104) k O (cid:105) = 1 t t (cid:88) α =1 ( k H ) α (cid:88) β =1 ( k H ) α,β = 1 t t (cid:88) (cid:96) =1 n (cid:96) ( k H ) (cid:96) , (2)where n (cid:96) is the number of times that the degree of node (cid:96) appears in the expanded sum. For any tree graph, node (cid:96) will appear exactly once in Eq. (2) for each of its ( k H ) (cid:96) neighbors. Hence, n (cid:96) = ( k H ) (cid:96) and (cid:104) k O (cid:105) = (cid:104) k H (cid:105) . In thesupplement, Eq. (1) can be used to derive (cid:104) k O (cid:105) ≈ p eff = (cid:104) k H (cid:105) / (cid:104) k O (cid:105) = 1 /
3. However, for theCCM p eff = (cid:104) k βH /k βO (cid:105) (cid:54) = (cid:104) k H (cid:105) / (cid:104) k O (cid:105) . We have not found aroute to calculating this exactly, but simulations suggest p eff ≈ . Degree Distribution.
The hidden network evolvesas a random recursive tree which has a limiting degreedistribution given by p H ( k H ) = 2 − k H , for k H > . (3)In the supplement, we show that the degree distributionfor the observed network can be written as the recurrence p O ( k O ) = π O ( k O − · p O ( k O −
1) + 2 − k O π O ( k O ) , for k ≥ , (4)where the final term is the probability that at time t thenewly added node has initial degree k O and π O ( k O ) = 1 + (cid:104) k H | k O (cid:105) . (5) (cid:104) k H | k O (cid:105) is the average degree of nodes in the hiddennetwork with a fixed observed degree of k O . Here, the 1corresponds to edges that are gained from direct attach-ment, whereas (cid:104) k H | k O (cid:105) corresponds to edges gainedfrom copying. Although we have not found an exact ex-pression for (cid:104) k H | k O (cid:105) , we can make progress by consid-ering the evolution of individual nodes.Consider node α added to the network at t α . Theinitial conditions for node α are( k H ( t α )) α = 1 , (6a) (cid:104) k O ( t α ) (cid:105) α = 1 + (cid:104) k H ( t α − (cid:105) β , (6b)where the final term is the average hidden degree of thetarget node β . In G H , node α gains edges from directattachment only. Hence, at t > t α , (cid:104) k H ( t ) (cid:105) α = 1 + t − (cid:88) j = t α j = 1 + H t − − H t α − , (7)where H n is the n th harmonic number. In G O , eithernode α is targeted via direct attachment, or a copiededge is formed from the new node to node α via any ofthe ( k H ( t )) α neighbors of node α . Hence, (cid:104) k O ( t ) (cid:105) α = (cid:104) k O ( t α ) (cid:105) α + t − (cid:88) j = t α (cid:104) k H ( j ) (cid:105) α j = (cid:104) k O ( t α ) (cid:105) α + t − (cid:88) j = t α H j − H t α − − /jj , (8)where we have subbed in Eq. (7) and H j − = H j − /j .Evaluating this sum, see supplement, we find (cid:104) k O ( t ) (cid:105) α = (cid:104) k O ( t α ) (cid:105) α + 12 (cid:104) (4 + H t − − H t α − ) × ( H t − − H t α − ) − H (2) t − + H (2) t α − (cid:105) , (9)where H ( m ) n is the n th generalised Harmonic number oforder m . For t → ∞ , H (2) t → π /
6. Hence, for large t wecan drop the final two terms and substitute in Eq. (7) togive (cid:104) k O ( t ) (cid:105) α ≈ (cid:104) k O ( t α ) (cid:105) α + 12 ( (cid:104) k H ( t ) (cid:105) α + 3) ( (cid:104) k H ( t ) (cid:105) α − . (10)Noting, that Eq. (10) is a monotonically increasing func-tion of k H for k H >
1, we assume that we can drop theindex α and the time dependence giving the average ob-served degree of nodes with specific hidden degree as (cid:104) k O | k H (cid:105) ≈ (cid:104) ˜ k O | k O (cid:105) + 12 ( k H + 3) ( k H − , (11)where (cid:104) ˜ k O | k O (cid:105) denotes the average initial observed de-gree of nodes with current observed degree k O . Finally,we make the approximation that (cid:104) k H | k O (cid:105) ≈ (cid:104) k O | k H (cid:105) − where the exponent denotes the inverse functionof Eq. (11). This gives π O ( k O ) = 1 + (cid:104) k H | k O (cid:105) ≈ (cid:113) k O + 2 − (cid:104) ˜ k O | k O (cid:105) ) . (12)Let us start by solving the degree distribution at k O = 2.Although the average initial condition (cid:104) ˜ k O (cid:105) = 1+ (cid:104) k H (cid:105) =3, in this case (cid:104) ˜ k O | (cid:105) = 2. Therefore p O (2) = − π O (2) · p O (2) + 2 − = − p O (2) · (cid:112) − , (13)giving p O (2) = 1 /
6. Since (cid:104) ˜ k O | k O (cid:105) has an almost neg-ligible effect on π O ( k O ) for k O >
2, for simplicity we set (cid:104) ˜ k O | k O (cid:105) = 2. We can now rewrite Eq. (4) as p O ( k O ) = p O ( k O − (cid:112) k O −
1) + 2 − k O √ k O , for k O > . (14)Although computing this recurrence shows good agree-ment with simulations, see Fig. 3, we have not found k O p O ( k O ) CCMUCM
FIG. 3. The degree distribution for the CCM (blue crosses)and UCM ( p = p eff ; orange points) at t = 10 , averagedover 100 networks. Dashed line: CCM recurrence relation inEq. (14). Dot-dashed: stretched exponential approximation.Dotted: power-law scaling. a closed form solution to Eq. (14). As an approxima-tion, we return to Eq. (9) and note that H t − − H t α − ≈ ln( t/t α ). Substituting this into Eq. (9) and droppingsmall terms (cid:104) k O ( t > t α ) (cid:105) α ≈ t/t α ) + ln ( t/t α )2 , (15)which inverted givesln( t/t α ) ≈ − (cid:112) k O + 2) ≈ (cid:112) k O , for k (cid:29) . (16)We have dropped the expectation value and define t α as the time a node was created such that its degree attime t is approximately k O . Exponentiating each sideand taking the reciprocal, t α t ≈ e −√ k O . (17)Finally, by substituting this approximation into the cu-mulative degree distribution we find˜ p O ( k O ) = k O (cid:88) k (cid:48) O =2 p O ( k (cid:48) O ) ≈ − t α t ≈ − e −√ k O , (18)which corresponds to a Weibull (stretched exponential)distribution, suppressing the power-law scaling observedin the UCM, see Fig. 3. The approximation for the cu-mulative degree distribution stems from the observationthat, on average, nodes with k (cid:48) O > k O were added to thenetwork at t (cid:48) < t α , whereas nodes with k (cid:48) O < k O wereadded to the network at t (cid:48) > t k O . Both Eq. (16) andEq. (18) are close to the scaling expected from sub-linearpreferential attachment with an exponent 1 /
2, see [28].
Clique Distribution.
In a simple undirected graph,a clique of size n is a subgraph of n nodes which form acomplete subgraph. A clique of size n = 2 is convention-ally referred to as an edge, whereas n = 3 is a triangle.Here we calculate the exact scaling for the number ofcliques of size n , Q n ( t ), in G O .Let us first consider the case of triangles. At t = t α ,there are two mechanisms by which a new triangle forms:1. Direct triangles.
The new node, α , forms a directedge to the target node, β , and forms copied edgesto each of the k βH neighbors of node β , labeled withthe index γ j . The combination of the direct edge( α, β ), the copied edge ( α, γ j ), and the existing edge( β, γ j ) creates one triangle, ( α, β, γ j ), for each ofthe k βH neighbors.2. Induced triangles.
If node α forms copied edgesto both node γ j , and to node γ j (cid:48) , j (cid:54) = j (cid:48) , the trian-gle ( α, γ j , γ j (cid:48) ) is formed if ( γ j , γ j (cid:48) ) ∈ E O .The change in the number of triangles can be writtenas ∆ Q ( t α ) = ∆ Q D ( t α ) + ∆ Q I ( t α ) , (19)where the first and second terms on the right correspondto direct and induced triangles respectively. One new di-rect triangle is formed for each of the k βH neighbors ofnode β , ∆ Q D = k βH . For induced triangles, the copiededge ( α, γ j ) is only formed if ( β, γ j ) ∈ E H . Likewise,all pairs of nodes which are next-nearest neighbors in G H must be nearest neighbors in G O . Hence, the edge( γ j , γ j (cid:48) ) must already exist in the observed network ifboth γ j and γ j (cid:48) are copied. Therefore, one induced tri-angle is formed for each pair of copied edges ( α, γ j ) and( α, γ j (cid:48) ) such that∆ Q I = (cid:18) k βH (cid:19) = ( k βH ) − k βH . (20)Extending the argument to general n we can write∆ Q n ( t α ) = ∆ Q Dn ( t α ) + ∆ Q In ( t α ) , (21)where direct cliques are those which include the edge( α, β ), and induced cliques are formed exclusively fromcopied edges. For a clique of size n , the number of directcliques is given by the number of ways in which n − k βH nodes,∆ Q Dn ( t α ) = (cid:18) k βH n − (cid:19) , (22)whereas the number of induced cliques is given by thenumber of ways in which n − Q In ( t α ) = (cid:18) k βH n − (cid:19) . (23) As t → ∞ , the average change in clique number is (cid:104) ∆ Q n ( t ) (cid:105) = ∞ (cid:88) k H =1 p H ( k H ) (cid:20)(cid:18) k H n − (cid:19) + (cid:18) k H n − (cid:19)(cid:21) , (24)where p H ( k H ) is the probability that the randomly cho-sen target node k βH = k H . To avoid ill-defined binomials,we rewrite Eq. (24) as (cid:104) ∆ Q n ( t ) (cid:105) = p H ( n −
2) + ∞ (cid:88) k H = n − p H ( k H ) · (cid:18) k H + 1 n − (cid:19) , (25)where we have combined the two binomial terms into asingle binomial. After subbing in p H ( k H ) and solving thesum we find (cid:104) ∆ Q n ( t ) (cid:105) = 2 − n + ∞ (cid:88) k H = n − − k H · (cid:18) k H + 1 n − (cid:19) = 4 . (26)Consequently, for large t we find the curious result thatthe number of n cliques scales as Q n ( t ) ∼ t, for n > , (27)independent of the clique size. Clustering.
Transitivity is a global clustering mea-sure defined as τ G O = 3 × G O ) G O ) , (28)where a twig is any three nodes connected by two edges.The number of twigs is equivalent to the number of stargraphs of size 2, S , where a star graph of size n is asubgraph with 1 central node and n connected neighbors.The number of subgraphs of size 2 is related to the degreedistribution by S ( t ) = t (cid:88) k O ≥ (cid:18) k O (cid:19) · p O ( k O ) = t · (cid:104) k O (cid:105) − (cid:104) k O (cid:105) , (29)where we have used the property that p O ( k <
2) = 0.Recalling that (cid:104) k O (cid:105) = 6 and (cid:104) k O (cid:105) ≈
62, the number oftwigs scales as S ∼ t , such that τ G O = 3 Q S ∼ · t t = 37 . (30)The local clustering coefficient, cc ( α ), is defined fornode α as the number of edges between the ( k O ) α neigh-bors of α , normalized by the the number of edges in acomplete subgraph of size ( k O ) α . For the CCM, cc ( α ) = (cid:0) ( k H ) α (cid:1) + (cid:80) ( k H ) α β =1 (cid:0) ( k H ) α,β (cid:1)(cid:0) ( k O ) α (cid:1) , (31)where the first term corresponds to the complete sub-graph of the ( k H ) α neighbors of α in G H , and thesum contributes the edges from one complete subgraphformed by node α, β and its ( k H ) α,β − α . The global clustering coefficient, CC ( G O ), is de-fined as the average of Eq. (31) over all nodes in the net-work. Calculating this in simulations, CC ( G O ) ≈ . t . Comparing Copying Models.
Comparing theCCM to the UCM with the equivalent effective copyingprobability ( p = 0 . • Edge growth.
CCM: E ( t ) ∼ t . UCM: E ( t ) ∼ . t . • Degree distribution.
CCM: Weibull distribution, p O ( k O ) ∼ (2 k O ) − / e √ k O . UCM: Power-law tail, p ( k ) ∼ k − . . • Degree variance.
CCM: (cid:104) k O (cid:105) − (cid:104) k O (cid:105) = 26. UCM: (cid:104) k O (cid:105) − (cid:104) k O (cid:105) ≈ • Number of n cliques. CCM: Independent of n .UCM: Dependent on n . • Transitivity.
CCM: τ G O = 3 / ≈ . τ G ≈ . • Clustering coefficient.
CCM: CC ( G O ) ≈ . CC ( G ) ≈ . p , in the sparse regime0 < p < /
2, the UCM always exhibits a power-law tail in its degree distribution, and the number of n cliquesis n dependent. Similarly, the largest limiting value forthe transitivity in the UCM is τ G ≈ .
274 at p ≈ . τ G O = 3 /
7. Finally, for the clustering coef-ficient, the UCM only reaches a comparable value to theCCM when p > .
9, at which stage the UCM is alreadyfar into the dense regime.
Conclusion.
We have introduced a simple model ofcorrelated node copying, implemented using a hiddennetwork model. The CCM is the case of extreme lo-cal copying bias, but illustrates the differences that arisewhen breaking the uniform copying assumption. Thereis a natural middle ground between the uniform case andthe CCM which would warrant future investigation.The CCM is a simple hidden network model, but thereare many variants with more unusual complex behavior.The hidden network framework allows us to deconstructthe heterogeneities in the CCM in a non-arbitrary way,and poses questions regarding how hidden informationin the evolution of real networks may effect their futurestructure and dynamics. We aim to discuss these con-cepts in detail in upcoming work.
Data accessibility.
A python implementation of theCCM and UCM is available at [29].
Acknowledgements.
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