Degree Distribution and Network Performance

Next, we explore the relationship between degree distribution and network per-formance. The information depicted in Fig. 2.4 shows that the networks analysed in this paper do not only differ significantly in terms of their performance, but also with respect to the distribution of degrees. The worst performing networks, i.e.

BA and EV networks, are networks that have a highly skewed and dispersed de-gree distributions which approximately follow a power law. Even though all net-works by definition have the same average degree of four, BA and EV netnet-works are characterised by a large number of small nodes (having only a few links) and a few nodes with an extremely high degree. In contrast, WS and ER networks have more symmetric degree distributions with only small deviations from the aver-age degree of the network. These better performing networks are characterised by a less asymmetric degree distribution. The worse performing networks indeed have a more asymmetric degree distribution than the better performing networks.

To further analyse the effect of the degree distribution on diffusion perfor-mance, we show in Fig. 2.5 (left) the relationship between the variance of the degree distribution and the mean average knowledge level ¯v in the respective networks achieved after 100 simulation steps. Additionally, Fig. 2.5 (right) also shows the cumulative number of non-traders in the network over time. In contrast to path length and cliquishness, we see that the variance of the degree distribu-tion of networks shows a coherent effect on network performance. WS networks, which perform best, are characterised by the lowest variance in the nodes’ de-grees, while at the same time, showing the highest knowledge levels. The worst performing networks, i.e. EV networks, are also those networks with the highest variance in their degree distribution.

If we now look in more detail at the number of non-traders over time (Fig.

2.5 right), we can see that, in the worse performing networks, agents stop trading earlier than in the better performing networks, i.e. the diffusion process stops

5 10 15 20 25 30 35 40 45

(A) Watts-Strogatz Networks

5 10 15 20 25 30 35 40 45

(B) Erdös-Rényi Networks

5 10 15 20 25 30 35 40 45

(C) Barabási-Albert Networks

5 10 15 20 25 30 35 40 45

(D) Evolutionary Networks

FIGURE 2.4: Average degree distribution of the agents in the re-spective networks.

FIGURE2.5: Relationship between the variance of the degree dis-tribution and the mean average knowledge level ¯v in the respec-tive networks (left) and the cumularespec-tive number of non-traders over

time (right).

earlier. Comparing EV and WS networks after 40 time steps we see that, while almost 90% of the agents in EV networks have already stopped trading, 65% of the agents in WS networks are still trading. Moreover, in EV networks, almost all agents stopped trading after 70 time periods, whereas in WS networks this occurs 30 time steps later.

Figure 2.5 provides evidence that the reason why networks with asymmetric degree distribution perform poorly is that agents stop trading early and, hence, disrupt the knowledge flow. However, the question at hand is: why?

To answer this question, Figs. 2.6 and 2.7 show the relationship between the nodes’ degrees and the time when these nodes stop trading as well as the rela-tionship between the nodes’ degrees and their knowledge level acquired over an

0 2 4 6 8

FIGURE 2.6: Relationship between the time the agents in the net-work stop trading and their degree. The vertical lines represent the 0.25 quartile (Q1) and the 0.75 quartile (Q3) of the degree

distribu-tion.

average of 500 simulation runs. Figure 2.6 shows that, in general, there is a posi-tive relationship between the size of an agent and the time it stops trading, espe-cially for networks with dispersed degree distribution (i.e. BA and EV networks).

This positive relationship always holds for the first three quartiles (the lower 75%) of the distribution. So, in the lower 75% of the distribution, nodes actually stop trading earlier the fewer links they have. However, we see that, especially in net-works with asymmetric degree distribution, this relationship fails for nodes with an extremely high number of links (the fourth quartile or the upper 25% of the distribution).

From this exploration, we can conclude that nodes with a high number of links do not stop trading first. In contrast, our results indicate that small nodes (the lower two quartiles of the distribution) stop trading first. Big nodes stop trading later, however, medium-sized nodes trade the longest. As a consequence, the poor performance of networks with a dispersed degree distribution can be explained by the sheer number of small nodes. Interestingly, as indicated by the quartiles of the BA and EV networks, it is important to note that when we speak of small nodes that stop trading earlier than big nodes, we are referring to the majority of nodes, i.e. over 50% of all nodes. If we now combine the information values provided by Figs. 2.6 and 2.2, we also see that nodes with a high degree actually stop trading after the increase in knowledge levels has reached its peak and almost no knowledge is traded within the network anymore.⁵

To further investigate why nodes stop trading, we illustrate, in Fig. 2.7, the re-lationship between a node’s degree and the mean knowledge vithe nodes reaches

5See also Fig. 2.2: The point in time the knowledge stock in the network has reached its steady state ¯v∗is t∗ = 61 for Watts–Strogatz networks, t∗ = 55 for Erd˝os–Rényi networks, t∗ = 45 for Barabási–Albert networks, and t∗ =32 for networks created with the Evolutionary network algo-rithm.

0 2 4 6 8

FIGURE 2.7: Relationship between the agents’ mean knowledge levels vi, their degree and the reason why agents eventually

stopped trading (after 100 steps).

after 100 time steps averaged over 500 simulation runs. Additionally, the size and the colour of the marks indicate the reason why nodes do not trade knowledge.

The data clearly shows a positive relationship between a node’s degree and its acquired knowledge level v_i. So, in general, the more links a node has, the more knowledge it will receive. This positive effect, however, decreases for nodes with many links, indicating a saturation phenomenon for nodes with a high number of links in the network.

In addition to the relationship between degree and the time at which nodes stop trading, Fig. 2.7 also shows us why these nodes eventually stop exchanging knowledge.⁶ While white marks indicate that nodes with the respective degree centrality stop because they could not offer knowledge to their trading partners, black marks indicate that these nodes stop trading because their partners do not offer them enough knowledge as a sufficient barter object. As the figure shows, we see that especially small nodes stop trading because they cannot offer knowledge to their partners. Nodes with a high degree stop trading because their partners cannot offer new knowledge. Medium-sized nodes (with grey marks), by contrast, stop trading because they have too much knowledge for their small partners and too little knowledge for their very large partners.

In summary, we can confirm the results of Cowan and Jonard (2007) that for barter trade diffusion processes the degree distribution of nodes is of decisive im-portance, even more important than other network characteristics such as path length and cliquishness. Based on our simulation results, we come to the conclu-sion that, in fact, nodes with a high number of links acquire much more knowl-edge than most of the other agents in the network (Fig. 2.7). They stop trading

6To determine why a node stops trading we define a variable for each node which contains the information on whether its unsuccessful trades failed because the respective node had insufficient knowledge or whether its trading partner actually had insufficient knowledge. The colour marking indicates the average results over a simulation run of 100 time steps.

because their partners have nothing left to offer, however, the difference in diffu-sion performance between the four network topologies cannot be solely explained by the ‘star’ argument. In fact, our results indicate a second effect which seems to dominate the diffusion processes in the networks. Very small agents with only few links are only able to receive very little knowledge and, hence, stop trading first. Considering the sheer number of small nodes in networks with a dispersed degree distribution we have to assume that these small nodes are responsible for disconnecting the network and interrupting the knowledge flow. This, in turn, negatively influences the effectiveness of the diffusion of knowledge within the entire network.

To tackle the question about whether the absolute degree of nodes in the net-work is the decisive factor influencing diffusion performance or whether the rel-ative position (i.e. the difference between nodes and their partners) is most im-portant, Fig. 2.8 shows the relationship between the node’s knowledge level and its relative positions in the network (δ). As we see in Fig. 2.8, for the barter trade diffusion process, the node’s relative position is of decisive importance. For all network topologies, nodes with an advantageous relative position (i.e. with a positive degree difference δ between nodes and their partners δ_i > 0) demon-strate a considerably higher knowledge level compared to nodes with fewer links (δi < 0). This effect is particularly strong for degree differences of δi = −_{5 to}+_5.

However, we also see that more extreme degree differences do not considerably increase or decrease a node’s knowledge level, which again indicates a saturation effect.

Our results demonstrate that neither path length nor cliquishness are fully suf-ficient for explaining the performance of knowledge diffusion in networks. Other factors, such as the degree distribution, have to be considered in order to fully understand the relevant processes within networks. In contrast to the findings of Cowan and Jonard (2007), our results show that the dissimilarities between nodes - especially for dispersed networks with scale-free structures - can create gaps in knowledge levels. These gaps create a situation where small nodes, which make up the majority of nodes in these networks, do not gain knowledge fast enough to keep pace with the other nodes in the networks. Hence, the many small agents rapidly fall behind and stop trading, which disrupts and disconnects the network and the knowledge flow. Comparing medium-sized nodes and big nodes, how-ever, we also see that stars play a key role in networks.