Temporal Dynamics of Scale-Free Networks Erez Shmueli, Yaniv Altshuler, and Alex ”Sandy” Pentland MIT Media Lab fshmueli,yanival,[email protected] Abstract. Many social, biological, and technological networks display substan- tial non-trivial topological features. One well-known and much studied feature of such networks is the scale-free power-law distribution of nodes’ degrees. Several works further suggest models for generating complex networks which comply with one or more of these topological features. For example, the known Barabasi-Albert ”preferential attachment” model tells us how to create scale-free networks. Since the main focus of these generative models is in capturing one or more of the static topological features of complex networks, they are very limited in cap- turing the temporal dynamic properties of the networks’ evolvement. Therefore, when studying real-world networks, the following question arises: what is the mechanism that governs changes in the network over time? In order to shed some light on this topic, we study two years of data that we received from eToro: the world’s largest social financial trading company. We discover three key findings. First, we demonstrate how the network topology may change significantly along time. More specifically, we illustrate how popular nodes may become extremely less popular, and emerging new nodes may become extremely popular, in a very short time. Then, we show that although the network may change significantly over time, the degrees of its nodes obey the power- law model at any given time. Finally, we observe that the magnitude of change between consecutive states of the network also presents a power-law effect. 1 Introduction Many social, biological, and technological networks display substantial non-trivial topological features. One well-known and much studied feature of such networks is the scale-free power-law distribution of nodes’ degrees [4]. That is, the degree of nodes is distributed according to the following formula: P [d] = c·d−λ. As the study of complex networks has continued to grow in importance and popularity, many other features have attracted attention as well. Such features include among the rest: short path lengths and a high clustering coefficient [12, 2], assortativity or disassortativity among vertices [10], community structure [8] and hierarchical structure [11] for undirected networks and reciprocity [7] and triad significance profile [9] for directed networks. Several works further suggested models for generating complex networks which comply with one or more of these topological features. For example, the known Barabasi-Albert model [4] tells us how to create scale-free networks. It incorpo- rates two important general concepts: growth and preferential attachment. Growth means that the number of nodes in the network increases over time and prefer- ential attachment means that the more connected a node is, the more likely it is to receive new links. More specifically, the network begins with an initial con- nected network of m0 nodes. New nodes are added to the network one at a time. Each new node is connected to m ≤ m0 existing nodes with a probability that is proportional to the number of links that the existing nodes already have. More sophisticated models for creating scale-free networks exist. For example, in [6], at each time step, apart of m new edges between the new node and the old nodes, mc new edges are created between the old nodes, where the probability that a new edge is attached to existing nodes of degrees d1 and d2 is proportional to d1 · d2. A very similar effect produces a rewiring of edges [1]. That is, instead of the creation of connections between nodes in the existing network, at each time step, mr randomly chosen vertices loose one of their connections. In mrr cases, a free end is attached to a random vertex. In the rest mrp = mr − mrr cases, a free end is attached to a preferentially chosen vertex. The main focus of these generative models is in capturing one or more of the static topological features of complex networks. However, these models are very lim- ited in capturing the temporal dynamic properties of the networks’ evolvement. Therefore, when studying real-world networks, the following question arises: what is the mechanism that governs changes in the network over time? In order to shed some light on this question, we studied two years of data (from 2011/07/01 to 2013/06/30) that we received from eToro: the worlds largest social financial trading company. We discover three key findings. First, we demonstrate how the network topology may change significantly along time. More specifically, we illustrate how popular nodes may become extremely less popular, and emerging new nodes may become extremely popular, in a very short time. Then, we show that although the network may change significantly over time, the degrees of its nodes obey the power- law model at any given time. Finally, we observe that the magnitude of change between consecutive states of the network also presents a power-law effect. 2 Datasets Our data come from eToro: the world’s largest social financial trading company (See http://www.etoro.com). eToro is an on line discounted retail broker for for- eign exchanges and commodities trading with easy-to-use buying and short sell- ing mechanisms as well as leverages up to 400 times. Similarly to other trading platforms, eToro allows users to trade between cur- rency pairs individually (see Fig ??). In addition, eToro provides a social network platform which allows users to watch the financial trading activity of other users (displayed in a number of statistical ways) and copy their trades (see Fig. 1). More specifically, users in eToro can place three types of trades: (1) Single trade: The user places a normal trade by himself, (2) Copy trade: The user copies one single trade of another user and (3) Mirror trade: The user picks a target user to copy, and eToro automatically places all trades of the target user on behalf of the user. Our data contain over 67 million trades that were placed between 2011/07/01 and 2013/06/30. More than 53 million of these trades are automatically executed mirror trades, less than 250 thousands are copy trades and roughly 13 million are single trades. The total number of unique traders is roughly 275 thousands and the total number of unique mirror operations is roughly 850 thousands (one mirror operation may result in several mirror trades). eToro . The world’s largest social eToro financial trading company. Watch the financial trading activity of other users and copy them. Serving 3 million users worldwide. Roughly two years of data. The platform allows users to trade between currency pairs (individually) or… 1 . All trades are automatically uploaded to the network where they Fig. 1. The eToro platform. Illustratingcan thebe displayed trading portfolio in a number of a of single statistical user ways. (left) and the trading activity of all users (right). 2 In the remainder of this paper, we use these trades to construct snapshot networks as we proceed to describe. Given a start time s and an end time e, the snapshot network’s nodes consist of all users that had at least one trade open at some point in time between s and e. An edge from user u to user v exists, if and only if, user u was mirroring user v at some point in time between s and e. Figure 2 illustrates how the size of the eToro network grows along time terms of both the number of nodes and the number of edges. For each day during the two years period, a snapshot network is constructed, and the number of nodes and edges for that network are counted. 50000 100000 40000 80000 30000 60000 20000 40000 Number of nodes Number of edges 10000 20000 0 0 0 100 200 300 400 500 600 700 800 0 100 200 300 400 500 600 700 800 Day Day Fig. 2. The size of the eToro network in terms of the number of nodes (left) and the number of edges (right) along time. 3 Results First, we examined the in-degrees of nodes in the eToro network, over the entire period of two years. As can be seen in Figure 3, the degree distribution presents a strong power-law pattern. Although, quite expected, this result is non-trivial. One might expect to see a bunch of users that are mirrored by the others, but what we actually witness is a heavy tail of users with only a few followers each. This result is consistent with the observation in [3] where the authors demonstrate by simulation that the degree distribution of social-learning networks converges to a power-law distribution, regardless of the underlying social network topology. 10-1 10-2 -3 γ=1.64 10 10-4 Density 10-5 10-6 10-7 101 102 103 104 Degree Fig. 3. In-degree distribution of nodes in the entire eToro network. (The in-degree of a node depicts the number of mirroring traders for the trader represented by that node) Next, we investigated how the popularity of traders in eToro, in terms of the num- ber of mirroring traders, changes along time. Fig. 4 illustrates the popularity of four traders. As can be seen in the figure, popular traders may become extremely less popular, and emerging new traders may become extremely popular, in a very short time. Note how this behavior differs significantly from the state-of-the-art ”rich get richer” behavior. 1600 700 400 1400 1500 600 1200 300 500 1000 1000 400 800 200 300 600 500 200 400 100 200 100 Number of mirroring traders Number of mirroring traders Number of mirroring traders Number of mirroring traders 0 0 0 0 0 100 200 300 400 500 600 700 800 0 100 200 300 400 500 600 700 800 0 100 200 300 400 500 600 700 800 0 100 200 300 400 500 600 700 800 Day Day Day Day Fig.