Dynamic model of time-dependent complex networks

Sam A. Hill and Dan Braha

Phys. Rev. E 82, 046105 – Published 12 October 2010

Abstract

The characterization of the “most connected” nodes in static or slowly evolving complex networks has helped in understanding and predicting the behavior of social, biological, and technological networked systems, including their robustness against failures, vulnerability to deliberate attacks, and diffusion properties. However, recent empirical research of large dynamic networks (characterized by irregular connections that evolve rapidly) has demonstrated that there is little continuity in degree centrality of nodes over time, even when their degree distributions follow a power law. This unexpected dynamic centrality suggests that the connections in these systems are not driven by preferential attachment or other known mechanisms. We present an approach to explain real-world dynamic networks and qualitatively reproduce these dynamic centrality phenomena. This approach is based on a dynamic preferential attachment mechanism, which exhibits a sharp transition from a base pure random walk scheme.

Received 14 July 2010

DOI:https://doi.org/10.1103/PhysRevE.82.046105

Authors & Affiliations

Sam A. Hill

Department of Physics, University of Toledo, Toledo, Ohio 43606, USA

Dan Braha

New England Complex Systems Institute, Cambridge, Massachusetts 02138, USA and University of Massachusetts, Dartmouth, Massachusetts 02747, USA

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Issue

Vol. 82, Iss. 4 — October 2010

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Images

Figure 1
A log-log plot of the cumulative degree distributions of daily networks, averaged over 25 days, built with single chains of 1 million steps: each histogram shows the probability that a member node of the daily network has degree $k$ or larger. All curves show power-law behavior; the inset shows the difference between the power-law exponent of each daily network’s histogram, and the exponent of its underlay (approximately $- 1.9$ , though a different underlay was used for each value of $C$ ).
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Figure 2
The Top-2% overlap, as defined in the text, as a function of the number of steps $S$ in the daily network, for various values of $C$ . The overlap for $C > 0$ appears to increase at a much slower rate, suggesting that the highly connected nodes in the underlay are only modestly represented in the daily networks.
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Figure 3
The distributions of the Top-2% overlap between 10 daily networks and their common underlay (the peaks labeled “underlay”), and between the 45 pairs of those ten networks (labeled “cross”). The bar graphs were constructed by binning, with a bin size of 0.01%. The results suggest that “local” hubs vary greatly from day to day.
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Figure 4
The importance of the number of unique visited nodes. The inset shows the fraction of nodes in the underlay that are included in a daily network generated with different numbers of steps $S$ and with $C = 0$ ,1, 2, 5, and 10. Higher percentages of unique visited nodes correspond monotonically to lower values of $C$ . The main graph shows the same curves plotted versus the Top-2% overlap instead of the number of steps. The kink in the $C = 0$ curve is a finite-size effect.
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Figure 5
Comparison of the underlay network with daily networks. We show the average rank of a node in ten daily subnetworks (with 1 million steps) versus their rank in the underlay network; nodes with higher degree have lower rank, and the top-ranked node has rank 0. In the case of ties, all tied nodes are assigned the rank that is the average value of the range they cover. The inset shows, not the average, but the best daily rank that node has over the 10 days. The $C > 0$ case is remarkably egalitarian—every node has an opportunity to become a hub in some daily network.
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Figure 6
Dissimilarity of networks aggregated over times ranging from 1 to 50 days. Dissimilarity is defined as the Top-2% overlap $O$ [Eq. (2)] subtracted from 100%, and is shown for both pure random walks $(C = 0)$ and strongly weighted walks $(C = 10)$ . We show results for networks with 50 thousand, 1 million, and 2 million nodes to show how the system scales; all walks have as many steps as there are nodes $(S = N)$ .
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