Dynamics of node influence in
network growth models^††thanks: This document does not contain technology or technical data controlled under either the U.S. International Traffic in Arms Regulations or the U.S. Export Administration Regulations.

Over the past two decades, complex networks have been used to model real-world systems across different domains ranging from social, biological, information, and technological systems [26, 25]. Investigating the behavior of influential entities or leaders in these real-world networks would help us understand how they are able to gather and maintain prominence over time. For instance, influential papers in citation networks continue to acquire new citations every year [31, 11]. Likewise, celebrities in online social networks keep increasing their follower count over time. These influential entities act as potential spreaders of information in networks. Therefore, keeping a track of their characteristics in an evolving network could have significance in applications ranging from viral marketing and target advertisement to rumor and epidemic control, and protection from spam attacks [13], [18], [20]. To understand and model the dynamics of how a leader node maintains its influence over time, one needs to study the temporal behavior of nodes in a network. In this paper, we introduce a notion, called visibility of a node which is defined as the probability of the node to form new connections in a growing network. For instance, in a preferential attachment model [4, 5], the visibility of a node is proportional to its degree, and inversely proportional to the number of edges in the network. An essential aspect of the study is to investigate the visibility profile of a node which characterizes the temporal evolution of the node’s influence as the network grows. We argue that studying the visibility profile of nodes leads to a better understanding of network evolution due to attachment dynamics, which might not be possible to obtain by simply analysing global network properties such as degree distribution or local node-centric properties such as degree, clustering coefficient, etc. Similar to node persistence over time studied in [28], our approach allows to make headway into this understanding by characterizing the visibility behavior of leaders in the network. While the framework is applicable to arbitrary nodes as well, it is more interesting to first understand the leaders’ behavior. For example, Chakraborty et al. [11] argued that the growth of the degree of a node (its visibility) in a citation network follows one of the five patterns – early rise, late rise, frequent rise, steady rise and steady drop. In a subsequent study [12], they also concluded that highly-cited papers and authors (leaders) follow steady rising pattern. However, it was not clear whether existing network growth models are able to describe such patterns particularly for leader nodes [24]. This motivates us to study the temporal evolution of the visibility of nodes, in particular leaders in networks generated by different network growth models.

We study the visibility of influential nodes¹¹1We use “leaders” and “influential nodes” interchangeably to denote high-degree nodes that can keep attracting new edges over long periods of time. in the graphs simulated by the following network growth models. Barabási-Albert (BA) model [5], aka the preferential attachment model, was able to explain power law behavior in real-world networks using the idea of network growth and the “rich-get-richer” phenomenon. However, the BA model could only capture the “old-get-rich” phenomenon or the “first-mover-advantage” whereby older nodes increase their connectivity and become dominant at the expense of younger nodes in the network. It does not take into account the competitive characteristics of a node that help them flourish in a very short period of time [1, 21]; for instance, in citation networks a few research papers are able to gain lot of citations within a short span of time [3]. Using this as a motivation, Bianconi and Barabási [7, 8] introduced a new class of network growth models in which the incoming nodes form connections based on inherent characteristics of nodes such as novelty, usefulness, etc., captured through a fitness value [27, 10, 16]. This was inspired by the “fit-get-richer” phenomenon observed in real-world networks [7]. Following this, Ergun and Rodgers [14] analysed the degree distribution of network growth models with an attachment mechanism combining the degree and fitness information of nodes in an additive and multiplicative manner.

There are several real-world networks in which the aspect of space plays an important role to understand dynamics of network evolution. For instance, in the biological domain the regions in brain that are spatially closer have a higher probability of being connected as compared to the far-off regions [9]. Similar significance of space can also be observed in online social networks capturing spatial features [2, 22], transportation networks [17], and communication networks. To model networks incorporating spatial features, the class of spatial network models has been proposed. A basic spatial model incorporating notions of preferential and spatial attachment was proposed by Yook et al. [32] to capture underlying mechanisms driving the evolution of the Internet topology. Subsequently, Kaiser et al. [19] analysed a spatial growth model where the edge connection probability decreases with node distance either in an exponential or a power-law manner to explain multiple, interconnected clusters that emerge in real-world networks. See Barthélemy [6] for a comprehensive study on spatial networks. Recently, we proposed a new spatial growth model [24] that was better able to capture the five growth patterns presented in [11], compared to the preferential attachment model and its variants (i.e., additive and multiplicative fitness models [10, 30]).

In this paper, we extend our previous studies [29, 24] and address the following problem statement: Given an influential node with high visibility at a certain point in time, how would its visibility evolve over time? We study this phenomenon across three popular and diverse network growth models – Barabási-Albert (BA) model, (additive, multiplicative, and general) fitness based model and spatial models. One of the primary theoretical findings that we continue to build on from our previous work [29] is that leaders are able to gain more visibility in the multiplicative fitness model setting as compared to the BA and additive fitness models. Along with this, in the multiplicative fitness model, the influential nodes are able to increase their visibility over time, given that their fitness value remains high in comparison to the rest of the network. On the other hand, the visibility values always decrease over time for the BA and additive fitness models (see Section 3 for details). In Section 3.1, we study a general framework of fitness models and observe that a non-linear attachment rule based on degree and fitness would lead to highly dominant nodes and make it exceedingly difficult for new nodes to gain influence in the network. Experimental analysis provided in Section 4 supports our theoretical analysis that suggest multiplicative fitness models best explain the prolonged influence of leader nodes in certain real-world networks, while at the same time allowing new high fitness nodes to gain influence over time. However, we observe that multiplicative fitness models do not allow multiple influential nodes to exist at the same time. In Section 5, we give theoretical insights on how spatial growth models can allow a multiplicity of leaders to coexist, while Section 6 provides experimental justifications for the same.

Reproducibility: To encourage reproducible research, the codes are publicly available at https://github.com/mittalshravika/Network-Growth-Models.

Following our previous work [29], here we set up the notations and problem definition. Consider the following sequence of graphs $\{{\mathbb{G}}_{t},\ t=0,1,\ldots\}$, where ${\mathbb{G}}_{t}=(V_{t},{\mathbb{E}}_{t})$, with $V_{t}$ and ${\mathbb{E}}_{t}$ being the set of nodes and edges in ${\mathbb{G}}_{t}$ respectively. In a network growth model, we have $V_{t}\subset V_{t+1}$ and ${\mathbb{E}}_{t}\subseteq{\mathbb{E}}_{t+1}$ for every $t=0,1,\ldots$. In other words, new nodes arrive at every time step $t$, and form connections with existing nodes, thus adding to the edge set of the previous graph ${\mathbb{G}}_{t-1}$. For purposes of simplicity, here we consider the basic model where a single node enters at any time step $t$, and forms a connection with one node in the existing graph ${\mathbb{G}}_{t-1}$. Therefore, we can label the incoming node by the time index of its entry to the network, which leads to $V_{t}=\{0,1,\ldots,t\}$ for $t=0,1,\ldots$. Note that all our results can be easily extended to more general scenarios where multiple nodes can enter the network and incoming nodes can form multiple connections. At time $t$, let the degree of the node $i$ in $V_{t}$ be denoted by $D_{t}(i)$. Also, let the rv $S_{t+1}$ denote the node with which an incoming node $t+1$ connects.

Barabási-Albert (BA) model: In the preferential attachment mechanism [5], new nodes connect preferentially to existing nodes with higher degree. Let ${\bf p}^{BA}(t+1)=(p_{i}^{BA}(t+1),\ i\in V_{t})$ be the pmf with which the new node indexed as $t+1$ connects with the existing graph ${\mathbb{G}}_{t}$, i.e., $p_{i}^{BA}(t+1)$ is the probability with which node $t+1$ connects with an existing node $i$. This is given by:

Note that we term node $i$’s visibility in the graph ${\mathbb{G}}_{t}$ by $p_{i}^{BA}(t+1)$.

Fitness based attachment rules: In fitness based models [7, 14, 23], every node is assumed to have a fitness value independently drawn from a distribution, and new nodes connect preferentially on the basis of the fitness and degree values of the existing nodes. We describe multiple ways in which such an attachment could occur.

Assume a sequence of i.i.d. fitness rvs $(\xi,\xi_{i},\ i=0,1,\ldots)$ with $\xi_{i}$ denoting the fitness value of node $i$. A generic fitness model can be described by the following attachment rule

for an attachment function $g:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ which determines the relative importance of the fitness and degree values. In the additive fitness model, new nodes connect preferentially to existing nodes having a higher sum of degree and fitness value. For $t=0,1,\ldots$, let the pmf delineating formation of new connections at time $t+1$ be given by ${\bf p}^{AF}(t+1)$, where

Similarly, the attachment rule for multiplicative fitness (MF) model is given by

Therefore, the visibilities of node $i$ in graph ${\mathbb{G}}_{t}$ are given by $p_{i}^{AF}(t+1)$ and $p_{i}^{MF}(t+1)$ for the additive and multiplicative fitness models, respectively. Note that the influential nodes as described in Section 1 relate to nodes having high visibility as defined for the particular network growth model in question.

Spatial attachment rules: In spatial models [32, 19, 15], every node is assumed to have a location vector drawn from a distribution over a location space $A$. Assume a sequence of i.i.d. location rvs $(\chi,\ \chi_{i},\ i=0,1,\ldots)$ and i.i.d. fitness rvs $(\xi,\xi_{i},\ i=0,1,\ldots)$ with $\chi_{i}$ and $\xi_{i}$ denoting the location and fitness values of node $i$ respectively. A generic spatial attachment model can be given by the following attachment rule

The attachment probability now depends on the location vector of the new node $\chi_{t}$. This is different from the models described previously. Therefore, we could have multiple definitions of visibility. A global variant of visibility is given below

while a local version is given as

where $h:A\times A\times\mathbb{R}\times\mathbb{N}\to\mathbb{R}$. We find it useful to consider the following separable form of attachment function

where $\alpha:A\times A\to\mathbb{R}$ and $\beta:\mathbb{R}\times\mathbb{N}\to\mathbb{R}$. While the notion of global visibility models the overall attractivity of a node in the entire attribute space, the local visibility considers only its attractivity from the local neighborhood in the attribute space of a node. This model allows for nodes whose global attractivity is low, with their local attractivity being high.

In this section, we study and compare the evolution of visibility of a node over time for the BA model and two fitness models, namely the additive and multiplicative fitness models. The following lemma describes the change in visibility with time for the three growth models.

First, we introduce some notation: Define $\Xi_{t}=\sum_{i\in V_{t}}\xi_{i}$ and $\psi_{t}=\sum_{i\in V_{t}}\xi_{i}D_{t}(i)$, for $t=0,1,\ldots$.

Proof.  Fix $t=0,1,\ldots$, and $i$ in $V_{t}$.

Preferential Attachment model: The difference in the visibility of node $i$ in the BA model between time $t+1$ and $t$ is given as

The above follows by noting that the sum of degree rvs $D_{t}(i)$ for all the nodes in the vertex set $V_{t}$ equals $2t$. Furthermore, by noting that when looking at the expected difference in visibility conditioned on the graph at time $t-1$, $S_{t}$ is the only random variable in (10), we obtain

and (7) follows.

Additive Fitness model: Similarly in the additive fitness model, the difference in the visibility of node $i$ can be written as

Taking expectation on both sides conditioned on ${\mathbb{G}}_{t-1}$ and $\xi_{t}$ leads to (8).

Multiplicative Fitness model: The difference in the visibility of node $i$ can be written for the multiplicative model as follows

Furthermore, we lower bound the expected change in visibility as follows

and the result follows.  

We observe from (10) that a node’s visibility increases if it forms a new edge connection in the BA model. However, it is also evident from Lemma 3.1 that the visibility of the node decreases in expectation, in a manner that is directly proportional to its degree $D_{t-1}(i)$. This can be understood from the fact that higher degree nodes in the network have higher visibility values as a result of which, their decrease in visibility would be more as compared to nodes that have lower visibility values. In the additive fitness model, we can infer from (12) that a node’s visibility increases when it forms a new edge, provided $\Xi_{t-1}+2(t-1)>(\xi_{t}+2)[\xi_{i}+D_{t-1}(i)]$. This condition is expected to hold for large values of $t$, unless the fitness value $\xi_{t}$, or $\xi_{i}$, or both, are very large. Similar to the BA model, node visibility decreases in expectation, with the magnitude of decrease being directly proportional to the sum of degree and fitness values, and the fitness value of the new incoming node $\xi_{t}$.

In contrast with the above, we can see from (9) that in expectation, the nodes are able to increase their visibility over time, given that their fitness value remains large with respect to the network. In addition to this, the expected change in visibility directly depends on the product of fitness and the present degree of the node, boosting the visibility of a leader much more as compared to the BA and additive fitness models. Note that we derive results for change in visibility values over a single time step. The results can be easily generalized to any fixed number of time steps.

In order to illustrate Lemmas 3.1 and 3.2 and depict the change in visibility of influential nodes, we carry out two sets of simulation experiments. We compare how the visibility of leaders or influential nodes change over time in the Barabasi-Albert (BA), additive fitness (AF), multiplicative fitness (MF) and general fitness (GF) models. Throughout, the fitness variable $\xi$ is taken to be Pareto distributed with parameter $\alpha_{p}$.

For each given growth model and parameter value of the Pareto distribution, we generate a graph ${\mathbb{G}}_{T_{0}}^{X}$, where $X\in\{BA,AF,MF,GF\}$ and $T_{0}=10000$. We define $p_{(k)}(T_{0};{\mathbb{G}}_{t}^{X})$ to be the visibility of the node in graph ${\mathbb{G}}_{t}^{X}$ which had the $k$-highest visibility in graph ${\mathbb{G}}_{T_{0}}^{X}$. For each growth model, starting from ${\mathbb{G}}_{T_{0}}^{X}$, we generate $R$ realizations ${\mathbb{G}}_{T}^{X,(1)},{\mathbb{G}}_{T}^{X,(2)},\ldots{\mathbb{G}}_{T}^{X,(R)}$ with $T=100000$, which are mutually independent conditioned on ${\mathbb{G}}_{T_{0}}^{X}$. Since we are interested in the evolution of the visibility of influential nodes, for the purpose of this experiment we track the visibility of the top 50 nodes starting from $t=10000$ to $t=100000$. However, conditioned on the graph at time $T_{0}$, the visibility values are random variables; therefore, we average the visibility values across all the realizations at any given time. We define $\bar{p}_{(k)}(T_{0};t;X)=\frac{1}{R}\sum_{r=1}^{R}p_{(k)}(T_{0};{\mathbb{G}}_{t}^{X,(r)})$ as the averaged visibility of the node at time $t$ ($t>T_{0}$) which had the k-highest visibility at time $T_{0}$ for growth model $X$.

In other words, we track $p_{(k)}\left(T_{0};{\mathbb{G}}_{t}^{X,(r)}\right)$ for $k=1,2,\ldots,50$, $r=1,2,\ldots,R$, $t=10000,10001,\ldots,100000$, and $X\in\{BA,AF,MF,GF\}$. For large enough independent runs $R$, we expect $\bar{p}_{(k)}(T_{0};t;X)$ to be a reasonable approximation of ${\mathbb{E}}\left[{p_{(k)}\left(T_{0};{\mathbb{G}}_{t}^{X,(1)}\right)\big{|}{\mathbb{G}}_{T_{0}}^{X}}\right]$, which is the expected value of visibility at time $t$ for the node which had the $k$-highest visibility value at time $T_{0}$ conditioned on the graph at time $T_{0}$.

Figures 1 and 2 show the change in visibility of top 50 nodes at $t=T_{0}=10000$ when the graph is allowed to grow for 90000 iterations until $t=100000$. Visibility values averaged over $R=50$ independent runs from $T_{0}=10000$ are shown. We observe from the box plots in the two figures that the highest visibility nodes in the BA and AF slowly reduce in their visibility values over time as was predicted by Lemma 3.1. We also observe that lot more nodes in BA and AF models have significantly higher visibility values. However, for the multiplicative fitness model only 2 nodes exhibit high visibility values (for $\alpha_{p}=1,2$), with one node dominating the entire network at any point of time for most of the duration. In the MF model, for $\alpha_{p}=1,2$, we observe that a node with lower visibility replaces one with higher visibility between $t=T_{0}=10000$ and $t=100000$, because the lower visibility node joined the network later but with a significantly higher fitness value. For $\alpha_{p}=3$, we do not see this behavior because it is less likely that a node with a significantly high fitness value will enter the network.

While we observe that in both the MF and GF models, nodes with high visibility are able to maintain their influence (or, visibility) over time, we next investigate how easy it is for new nodes with high fitness to gain influence over time. For this purpose, we conduct an experiment where we introduce a node at time $t=T_{0}+1=10001$, with fitness value $\xi_{T_{0}+1}=2\max_{t\in V_{T_{0}}}\xi_{t}$, i.e., twice the fitness value of the maximum fitness value among all nodes until time $T_{0}$. For growth model $X\in\{AF,MF,GF\}$, we generate $R=50$ realizations beyond time $T_{0}$ by setting the fitness value of node $T_{0}+1$ as mentioned above. We define $p_{i}\left({\mathbb{G}}_{t}^{X}\right)$ to be the visibility of the node $i$ in graph ${\mathbb{G}}_{t}^{X}$. We average the visibility values across $R$ realizations, ${\mathbb{G}}_{T}^{X,(1)},{\mathbb{G}}_{T}^{X,(2)},\ldots,{\mathbb{G}}_{T}^{X,(R)}$, and compute the averaged visibility of node $T_{0}+1$ defined as $\bar{p}_{T_{0}+1}(t)=\frac{1}{R}\sum_{r=1}^{R}\left({\mathbb{G}}_{t}^{X,(r)}\right)$, $t>T_{0}$. Subsequently, we track the visibility of this newly introduced node in the three fitness models and present the results in Figure 3. We observe that in the AF and GF models, the visibility of the newly introduced node decreases with time. This concurs with Lemma 3.1 where we showed that the visibility of nodes in AF models decreases with time; and with Lemma 3.2 where we argue that in the general fitness model with a nonlinear attachment rule, it becomes progressively difficult for new nodes to get visible in the network. For the MF model with $\alpha_{p}=1$, we observe that $\bar{p}_{T_{0}+1}(t)$ increases slightly but decays beyond $t=30000$. This is because nodes with even higher fitness values enter the network after node $T_{0}+1$. However, for $\alpha_{p}=2,3$ the visibility of node $T_{0}+1$ keeps increasing until $t=100000$. Note that for $\alpha_{p}=3$, node $T_{0}+1$ becomes dominant in the network very quickly because as the node increases its degree, the degree-fitness product becomes large compared to that of other nodes in the network because having a large fitness node is a rarer event for larger value of $\alpha_{p}$.

From the experiments we reaffirm that multiplicative models allow high visibility nodes to maintain their influence in the network for a longer period of time, while at the same time allowing high fitness nodes that are introduced later in the network to become influential. However, we observe that only a few number of nodes can be influential in the network at any given moment of time. This leads us to consider spatial attachment rules in conjunction with the multiplicative model to enforce regions of influence for each individual node such that multiple influential nodes can coexist in a network at the same time.

In the previous sections, we investigated the node visibility dynamics of the fitness models. A major takeaway was that multiplicative fitness models allow influential nodes to maintain their visibility while still permitting newly introduced nodes with high fitness to gain influence over time. However, a shortcoming of the MF model was that it could not support a multiplicity of influential nodes. We investigate whether a configuration of spatial models exists that retains the positive aspects of the MF model while addressing this shortcoming.