(Color online) Schematic diagram representing (a) complete node duplication (Eq. 1) and (b) partial node duplication (Eq. 2) in networks. 

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Context 1

... isolated node in the network leads to all zero entries in the corresponding row and column, thus lowering the rank of the matrix by one. Conditions (a) and (b) lead to linear dependence of row (column), reducing the rank of the matrix. Note that we consider a connected network in order to rule out the trivial possibility (c) of occurrence of zero eigenvalues. Further there are N ( N − 1) / 2 possible ways in which condition (a) of complete duplication can be re- alized, while for the partial duplication (b) among ‘ x ’ number of nodes with ‘ y ’ number of nodes, there can be 2 . ( N − N x ! − y )! possibilities. Hence, for a given network, checking the existence of condition (b) becomes compu- tationally exhaustive as with increase in network size the number of possibilities becomes very large. In order to demonstrate the effect of duplication on zero degeneracy, we construct an Erdös-Renýi (ER) random network for size N and connection probability p using ER model [2] such that it has no duplicates and no zero eigenvalues (row 1 of Table I). Next we add a node to the existing network in a way that it satisfies the complete node duplication criteria, i.e. condition (a). This leads to exactly one zero eigenvalue corresponding to one duplicate node. Addition of one more node mimicking the previous node leads to two zero eigenvalues (row 2 and 3 of Table I). This demonstrates how complete node duplication leads to zero eigenvalues (Fig. 1 (a)). Further, we consider another situation where we devise our algorithm such that two new nodes are added to the ex- isting random network in a way that in coalition they mimic the neighbors of an existing node (condition (b)), i.e. they duplicate an existing node (row 4 of Table I) as demonstrated in Fig. 1 (b). Impact of duplications of conditions (a) and (b) on the zero eigenvalues are pre- sented in the subsequent rows of Table I. Thus, we observe that with entry of every new node in the network satisfying condition (a) or (b) of complete or partial duplication, there is an addition of exactly one zero eigenvalue in the spectra. The number of duplicates (complete or partial) equals the number of zero eigenvalues. The density distribution at very low average degree yields a peak at zero eigenvalue. With an increase in k , the peak of the density distribution flattens (Fig. 2 (a)). In order to demonstrate the impact of network archi- tecture on the duplication phenomenon, we present results for ensemble average of the scale-free (SF) networks as they are known to have high degeneracy at zero eigenvalue. We generate the SF network using the preferential attachment mechanism [21], where each new node gets attached to the existing nodes with the probability pro- portional to their respective degrees. This phenomenon gives rise to power law degree distribution. Here at each time step, a new node enters which is most likely to connect with the highest degree nodes owing to the preferential attachment algorithm. The next entry also has a tendency to attach with the highest degree nodes. From the power law degree distribution of SF networks it is ev- ident that there are very few high degree nodes which are known as the hubs of the network and a large number of low degree nodes. At low values of k , there is a high degeneracy at zero eigenvalue indicating high duplication. This is because at low average degree, most of the low degree nodes attain very few connections. By virtue of preferential attachment property, these low degree nodes have the highest probability to connect with the hubs of the network, which increases the likelihood of any two nodes to have the same neighbors, leading to a pair of duplicate nodes. Although even with increase in average degree, the density distribution remains triangular, there is flattening of the peak (Fig. 2 (b)). This might be because the low degree nodes also tend to acquire connections with nodes other than the hubs. All these findings indicate that low average degree favors duplication. The explanation behind this can be given in terms of the possible number of ways of duplication which ...

Context 2

... isolated node in the network leads to all zero entries in the corresponding row and column, thus lowering the rank of the matrix by one. Conditions (a) and (b) lead to linear dependence of row (column), reducing the rank of the matrix. Note that we consider a connected network in order to rule out the trivial possibility (c) of occurrence of zero eigenvalues. Further there are N ( N − 1) / 2 possible ways in which condition (a) of complete duplication can be re- alized, while for the partial duplication (b) among ‘ x ’ number of nodes with ‘ y ’ number of nodes, there can be 2 . ( N − N x ! − y )! possibilities. Hence, for a given network, checking the existence of condition (b) becomes compu- tationally exhaustive as with increase in network size the number of possibilities becomes very large. In order to demonstrate the effect of duplication on zero degeneracy, we construct an Erdös-Renýi (ER) random network for size N and connection probability p using ER model [2] such that it has no duplicates and no zero eigenvalues (row 1 of Table I). Next we add a node to the existing network in a way that it satisfies the complete node duplication criteria, i.e. condition (a). This leads to exactly one zero eigenvalue corresponding to one duplicate node. Addition of one more node mimicking the previous node leads to two zero eigenvalues (row 2 and 3 of Table I). This demonstrates how complete node duplication leads to zero eigenvalues (Fig. 1 (a)). Further, we consider another situation where we devise our algorithm such that two new nodes are added to the ex- isting random network in a way that in coalition they mimic the neighbors of an existing node (condition (b)), i.e. they duplicate an existing node (row 4 of Table I) as demonstrated in Fig. 1 (b). Impact of duplications of conditions (a) and (b) on the zero eigenvalues are pre- sented in the subsequent rows of Table I. Thus, we observe that with entry of every new node in the network satisfying condition (a) or (b) of complete or partial duplication, there is an addition of exactly one zero eigenvalue in the spectra. The number of duplicates (complete or partial) equals the number of zero eigenvalues. The density distribution at very low average degree yields a peak at zero eigenvalue. With an increase in k , the peak of the density distribution flattens (Fig. 2 (a)). In order to demonstrate the impact of network archi- tecture on the duplication phenomenon, we present results for ensemble average of the scale-free (SF) networks as they are known to have high degeneracy at zero eigenvalue. We generate the SF network using the preferential attachment mechanism [21], where each new node gets attached to the existing nodes with the probability pro- portional to their respective degrees. This phenomenon gives rise to power law degree distribution. Here at each time step, a new node enters which is most likely to connect with the highest degree nodes owing to the preferential attachment algorithm. The next entry also has a tendency to attach with the highest degree nodes. From the power law degree distribution of SF networks it is ev- ident that there are very few high degree nodes which are known as the hubs of the network and a large number of low degree nodes. At low values of k , there is a high degeneracy at zero eigenvalue indicating high duplication. This is because at low average degree, most of the low degree nodes attain very few connections. By virtue of preferential attachment property, these low degree nodes have the highest probability to connect with the hubs of the network, which increases the likelihood of any two nodes to have the same neighbors, leading to a pair of duplicate nodes. Although even with increase in average degree, the density distribution remains triangular, there is flattening of the peak (Fig. 2 (b)). This might be because the low degree nodes also tend to acquire connections with nodes other than the hubs. All these findings indicate that low average degree favors duplication. The explanation behind this can be given in terms of the possible number of ways of duplication which ...