Memory effects on link formation in temporal networks: A fractional calculus approach

TL;DR

This paper investigates how memory influences link formation in temporal networks using a fractional calculus approach, revealing that memory causes decay in node activity and affects network dynamics.

Contribution

It introduces a fractional calculus model to analyze memory effects on temporal networks, highlighting how memory impacts node activity and network evolution.

Findings

Memory causes decay in node activity over time.

Dense networks exhibit clearer characteristic times.

Memory reduces the likelihood of nodes receiving new links as they age.

Abstract

Memory plays a vital role in the temporal evolution of interactions of complex systems. To address the impact of memory on the temporal pattern of networks, we propose a simple preferential connection model, in which nodes have a preferential tendency to establish links with most active nodes. Node activity is measured by the number of links a node observes in a given time interval. Memory is investigated using a time-fractional order derivative equation, which has proven to be a powerful method to understand phenomena with long-term memory. The memoryless case reveals a characteristic time where node activity behaves differently below and above it. We also observe that dense temporal networks (high number of events) show a clearer characteristic time than sparse ones. Interestingly, we also find that memory leads to decay of the node activity; thus, the chances of a node to receive new connections reduce with the node's age. Finally, we discuss the statistical properties of the networks for various memory-length.

Equations

$$\frac{d{k}_i(t)}{dt}=\frac{m}{N}+\frac{m{k}_i(t)}{{\sum }_{j=1}^N{k}_j(t)}$$

$$k_i(t)=\frac{2mt}{N}+c\sqrt{t}$$

$$k_i(t)\sim \{\begin{array}{ll}t,& \text{if}t\to \infty \\ \sqrt{t},& \text{if}t\to 0\end{array}$$

$$\frac{d{k}_i(t)}{dt}={\int }_{t_0}^t\kappa (t-t^{{}^{\prime }})d{t}^{{}^{\prime }}[\frac{m}{N}+\frac{m{k}_i(t^{{}^{\prime }})}{{\sum }_{j=1}^N{k}_j(t^{{}^{\prime }})}]$$

$$\frac{d{k}_i(t)}{dt}={}_{t_0}^c{D}_t^{-(\alpha -1)}[\frac{m}{N}+\frac{m{k}_i(t^{{}^{\prime }})}{{\sum }_{j=1}^N{k}_j(t^{{}^{\prime }})}]$$

$${}_{t_0}^c{D}_t^{\alpha }(k_i(t))=[\frac{m}{N}+\frac{m{k}_i(t)}{{\sum }_{j=1}^N{k}_j(t)}]$$

$${}_{t_0}^c{D}_t^{\alpha }y(t)=\frac{1}{\Gamma (1-\alpha )}{\int }_{t_0}^t(t-s{)}^{-\alpha }y(s)ds$$

$${}_{t_0}^c{D}_t^{\alpha }(k_i(t))=f(t,k_i(t))$$

$$k_i(t_0)=k_{i0}$$

$${\bar{k}}_i(t_n)=k_{i0}+h^{\alpha }\sum _{j=0}^{n-1}{b}_{n-j-1}f(t_j,k_j(t))$$

Full text

Introduction

Analyzing complex systems as compositions of entities and their interactions using network theory is a main trend in mathematical physics  [1, 2, 3]. In the real world, systems often have the self-dynamic structure; therefore, they can be better described as networks in which links among a fixed set of nodes change over time  [4, 5]. For example, activities like communication through social media  [6], market trading  [7], and online search often take place in time  [8]. To understand the temporal dimension of these phenomena, the concept of temporal networks has been employed [9, 10, 11, 12]. In this regard, it has been found that a non-Markovian dynamics is necessary to capture the non-trivial temporal patterns of real-world networks  [13, 14, 15], and can play an important role in processes occurring on temporal networks.

An important aspect in temporal networks concerns the complex behavior of agents, i.e. the intuition behind each agent’s decision to initiate connections toward certain agents as well as the strength of established links. Individuals remember their social circle of friends and are likely to perform repeated interactions within their established circle. In other words, the formation of links can be though to be a Markov process  [16, 17, 18, 19, 20, 21]. While the impact of memory has been studied in detail in static networks  [22, 23, 24], little has been devoted to study the non-Markovian aspects in the context of temporal networks. For this reason, we conduct this study by focusing on long-range memory effects, where an arbitrary length of node memory can be included in the modeling of temporal network evolution. For a detailed discussion on the origin of memory effects, several attempts have been performed, where memory is considered as exploration  [17], preferential return  [25], and social reinforcement  [19].

The field of fractional calculus represents a promising solution to model memory effects of dynamic systems [26, 27, 28, 29, 30], and has been recently used in the study of temporal networks  [31, 32, 33]. Typically, the temporal dynamics of node activity is described with a differential equation, with the derivative being of integer order. When we calculate the value of an integer-order derivative at a specific point, the result depends only on that point. This property is called locality. By replacing the ordinary time derivative with a fractional derivative, a time correlation function or memory kernel can be used, thereby making the state of the system dependent on all past states. This property is callednonlocality, which is the built-in feature of fractional derivatives that allows them to capture memory effects. For this reason, methods based on derivatives with non-integer order, as introduced by Caputo for geophysics problems  [34], constitute an important formalism for non-Markovian problems. Moreover, Caputo’s formalism provides the advantage that it is not necessary to define the fractional-order initial conditions when solving such differential equations  [34, 35, 36]. Furthermore, the time correlation function, in the definition of Caputo fractional derivative, is a power-law function, which is flexible enough to reflect the fact that the contribution of earlier states on the current state is noticeably less relevant than the contribution of more recent states.

Using fractional calculus, this paper proposes a modified preferential attachment model by incorporating memory effects into the dynamics. The rest of the paper is structured as follows. First, we offer a definition of node activity and memory in our model. Next, relying on Caputo’s approach, we convert the differential equation of our model to a fractional-order derivative to integrate memory. The main result of our work is to show that memory can play either of two opposing roles: it can slow-down or speed-up node activity depending on the strength of memory, which could be used as an alternative explanation of the findings of many other studies  [14, 17, 13, 30]. Fig.1 schematically illustrates a simple time-varying network considering aging effects in the dynamics. At each instant, we randomly choose $m$ nodes in the network and connect them with the rest of the nodes. A new edge is more likely to be established between a chosen node and the most active node(s). This illustration shows that the aging of links over time decreases the chance of high-degree nodes to receive new links.

1 Modeling temporal networks

Generating temporal networks is conceptually simple and work has been done in this area  [37, 4, 21]. Significant attention has been devoted to temporal networks generated by the interactions of individuals in agent-based models  [38, 39, 40, 17]. In the simplest setting, the researchers start with a set of nodes that connect with each other with time following a specific rule.

In what follows, we define a simple model for generating temporal networks based on the preferential attachment mechanism  [41, 42]. The hypothesis behind our model is that each node prefers to create a link with the most active nodes. Node activity is defined as the number of events in which a node participates. Active nodes in a given time slot $\delta t$ participate in the largest number of the events that occur in a specific time window. Such nodes play a key role in the formation of the network topology as well as in the dissemination of information.

Let $L_{i}(t)$ be the number of events in which node $i$ appeared in the period from $t$ to $t+\Delta t$. We consider a network with $N$ nodes and assign to each node $i$ an activity rate $k_{i}(t)=\sum_{t=0}^{t}L_{i}(t)$. The activity rate is the probability per unit time to create new contacts with other individuals (i.e. the sum of temporal events from $t=0$ to $t$). At each discrete time step $t$, we randomly choose $m$ nodes who will establish an event with a target node based on the target node’s activity rate. This model is random and Markovian in the sense that agents do not have a memory of previous time steps. The full dynamics of the network and its ensuing structure is thus completely encoded in the node activity distribution $k_{i}(t)$. Thus, in mathematical terms, the activity of node $i$ evolves as:

[TABLE]

The first term on the right side of Eq.(1) represents the percentage of nodes chosen at random to engage in new events. The second term adds a preferential connection component which we assume to be proportional to node activity. The model enables us to perform simple analytical calculations. The instantaneous network generated at each time $t$ will be composed of those nodes that were chosen randomly at that particular time, plus those who received connections from the randomly selected nodes. Since the total node activity of the network will increase over time, the total number of events will also increase as a function of $t$. By observing that each time $m$ new events are formed, and since each event has two ends, the sum of node activity increases by $2m$ at each time step. By Substituting this into (1), we obtain an analytical solution of the form:

[TABLE]

where $c$ is a constant value and depends on the initial conditions. At time $t=t_{0}$, the initial node activity is $k(t=t_{0})=k_{0}$ and so $c=\frac{k_{0}-2mt_{0}/N}{\sqrt{t_{0}}}$. Such results can be categorized into two classes of the activity as follows:

[TABLE]

To validate Eq. (2) through numerical simulation, we examined a random network with $N=1000$ nodes and simulated node activities using Eq. (1)(Fig. 2-(a)). At each time step, a total number of $m$ events are added where a source node $i$ is chosen at random to create a link with a target node $j$ with probability proportional to $k_{j}$. These steps are repeated sequentially, creating a network that grows with the passage of time. Fig.2-(a) shows the node activity per time for both the analytical calculations and the ensemble average over realizations. These observations are in agreement with the BA model [42]. The intersection of $t^{1/2}$ and $t$ is introduced as a characteristic time $t^{*}=\frac{c^{2}N^{2}}{4m^{2}}$ to detect behavior change. Fig. 2-(b) plots the node activity dynamics for various $m$ in Eq. (2). As we can see, for smaller values of $m$, the characteristic time is delayed.

2 Memory effects on node’s activity

Next we extend the modeling framework by introducing memory. Conceptually, the idea of momory comes into play when nodes prefer to connect not only to the most active nodes, but also to those nodes with whom they had interactions in the past. In order to observe the effect of memory, first we rewrite the differential Eq. (1) in terms of time-dependent integrals as follows:

[TABLE]

where $\kappa(t-t^{{}^{\prime}})$ represents a time-dependent kernel and is equal to a delta function $\delta(t-t^{{}^{\prime}})$ in a classical Markov process. In fact, any arbitrary function can be replaced by a sum of delta functions, thereby leading to some sort of time correlation. A proper choice to include long-term memory effects can be a power-law function, which exhibits a slow decay such that the state of the system at quite early times also contributes to the evolution of system. This type of kernel guarantees the existence of scaling features as it is often intrinsic in most natural phenomena. Thus, we consider the power-law correlation function for $\kappa(t-t^{{}^{\prime}})=\frac{(t-t^{{}^{\prime}})^{\alpha-2}}{\Gamma(\alpha-1)}$ where $0\leqslant\alpha<1$ and $\Gamma$ denotes the Gamma function.

The choice of the coefficient $\Gamma(\alpha-1)$ and exponent $(\alpha-2)$ allows us to rewrite Eq. (4) in the form of a Caputo-like fractional differential equation. By substituting this kernel function in Eq. (4), we obtain a fractional integral of order $(\alpha-1)$ on the interval $[t_{0},t]$, denoted by ${}^{c}_{t_{0}}D^{-(\alpha-1)}_{t}$. Thus, we arrive to the following fractional differential equation:

[TABLE]

If we set $\alpha=1$, then the fractional operator turns to unity ${}^{c}_{t_{0}}D^{0}_{t}=1$ and Eq. (5) becomes identical to our model without memory described by Eq. (1). Applying a fractional Caputo derivative of order $\alpha-1$ on both sides of the above equation, we can write it in the form of a differential equation,

[TABLE]

where ${}^{c}_{t_{0}}D^{\alpha}_{t}$ defined for an arbitrary function $y(t)$ as follows  [34]:

[TABLE]

Therefore, the fractional derivatives, when introducing a convolution integral with a power-law memory kernel, are useful to describe memory effects in dynamical systems. The decaying rate of the memory kernel (a time-correlation function) depends on $\alpha$. The upper value of $\alpha$ corresponds to a slowly decaying time-correlation function (long memory). Hence, the strength (through the ”length”) of the memory is controlled by $\alpha$. As $\alpha\to 1$, the effect of memory decreases: the system tends toward a memoryless system.

Let’s reemphasize that Eq. (6), in the following general form:

[TABLE]

Where $k_{i0}$ is a constant value that indicates the initial conditions. To deal with this equation, we use the predictor-corrector algorithm  [43, 44]. It is assumed that there exits a unique solution for $k_{i}$ on the interval $[0,t]$ for a given set of initial conditions. Considering a uniform grid $\{t_{j}=t_{0}+jh:j=0,1,2,..,N\}$ with equal space $h$, in which $N$ is an integer. Finally, the Eq.(8) can be rewritten in a discrete form as follows:

[TABLE]

Here, $\bar{k}_{i}(t)$ denotes the effective node activity and $b_{n}$’s are time-dependent coefficients which represent the aging effect and they are equal to $b_{n}=\frac{(n+1)^{\alpha}-n^{\alpha}}{\Gamma(\alpha+1)}$. This factor represents the contribution of each of the $n-1$ past events on the present event of $n$.

Numerically solving the equation system, Eq. (9), the obtained results confirm the effect of memory on the network evolution. Fig. 3 illustrates the effective node activity for various $\alpha$ values. It is obvious that for all values of $\alpha$, the effective node activity increases rapidly at the beginning and then slows down. However, it is clear that smaller values of $\alpha$, which express the strength of memory in the system, translate into a lower growth of $\bar{k}_{i}(t)$ compared with higher values of $\alpha$. In sum, we can explain the initial rapid increase of $\bar{k}_{i}(t)$ by the fact that at the beginning memory is not strong enough (there is no past) and nodes connect based on activity; With the passage of time, memory becomes more important and nodes start to prefer the nodes with whom they had connections in the past. To describe this behavior, it could be helpful to geometrically interpret the meaning of a unit of time in the fractional context. In fractional space, time slows down and expands at each unit, as we have shown in Fig. 4. At the beginning of fractional time, a unit of time is small compared to the homogeneous timescale, and with time passing, the unit length for fractional time becomes larger than in the homogeneous case. This explains the slowing down of time in Fig. 3.

In Fig. 5, we also show some statistical proprieties of the simulated networks. The BA model B predicts that after a transient period the connectivity distribution of all nodes becomes a Gaussian around its mean value. We have shown here that in presence of memory, deviation from a Gaussian occurs, see Fig. 5-(a). This deviation is of course understood as resulting from the competition between memory effects and the preferential connection mechanism. In the present case, getting older reduces the probability of nodes with old links to be selected, since memory reduces the effective degree of the nodes. This reduces the growth rate of older nodes, whence giving nodes with a smaller degree a higher chance to receive new links. Indeed, Fig. 5-(a) illustrates that as time increases, the $P(k)$ for $\alpha<1$ shifts from the initial symmetric to a broader .

Memory affects the degree distribution. For many nodes, memory increases the opportunity to develop into a hub, as opposed to the BA model where only early members have a chance of becoming a hub. Memory also impacts assortativity measure, defined as the tendency of nodes to connect to those with whom they had interactions in the past. Assortativity increases in networks with aged links. This can be seen in the normal distribution of weights in Fig. 5-(b).

3 Conclusion

Memory can have a great impact on the temporal evolution of networks. In this paper, we rely on fractional calculus to incorporate memory in a temporal network and examine its effect on link formation. We examine two models, the first is a preferential attachment mechanism without memory, and the second includes memory. The results show that the dynamics of temporal and evolving networks depends on the strength of memory effects, controlled by the order of fractional derivatives. We show also that the evolution of a temporal network with incorporated memory depends on the preferential mechanism at the early stage, while memory effects become more relevant at the later stage.

4 Acknowledgment

We thank Prof. Rosario N. Mantegna for comments that greatly improved the manuscript.