Network dynamics simulations

The topology and its various statistical and analytical measures provide a great deal of knowledge regarding a system. However, the analyses provide mostly what could be considered macroscopic data. The microscopic abilities of individual Nodes are of great interest in SNA. Previous authors have been able to model both cascading of properties within networks (Carreras, Lynch et al. 2002, Motter and Lai 2002, Leskovec, Singh et al. 2006, Leskovec, McGlohon et al. 2007, Buldyrev, Parshani et al. 2010). Such a model could represent information flow. There have been many different attempts at modelling microscopic interactions, with varying degrees of success. The first thing that must be acknowledged is that in network dynamics there is a propagation of some property, which may or may not influence the topology of the system itself.

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The most common approach to propagating a property is the Susceptible-Infected-Recovered (SIR) or Susceptible-Infected-Susceptible (SIS) from epidemiology, with varying degrees of success (May and Lloyd 2001, Barthélemy, Barrat et al. 2004, Volz 2008).

Pastor-Satorras and Vespignani (2001) focuses upon the SIS epidemic spreading in internet systems that can be carried through a variety of mediums (emails, FTP, etc…). These networks have been identified in previous works to be Scale-Free in nature (Barabási and Albert 1999).

Previous works on random graphs have found that there is an epidemic rate critical threshold, 𝜆𝑐.

Above this value the infection is persistent and below it the epidemic dies out (Kephart 1994, Marro and Dickman 2005).

Based on empirical data however, it has been found that the P(t) follows a power-law distribution. Furthermore, whilst most viruses die out within the first two months, many viruses survive for much longer periods. Seeing as anti-viruses respond within days or weeks of the first reported incident, this is indeed a significantly long survival period.

In simulating this, the Barabási-Albert algorithm is implemented for N=103 _{to 8.5x10}5 _{Nodes and}

half the Nodes are initially infected. What is most interesting about the results is that in Scale-Free networks, there is an absence of the critical rate threshold (viruses are prevalent), whereas in bounded networks they have a clear threshold. In fact, it was found that the steady-state density is independent of the network size. Conversely, the time to reach said steady-state is not.

It is assumed when the exponent of scale-free graphs reaches 4 (exponential tails), then the re- emergence of a threshold will exist. This seems to be confirmed in another study by the same authors.

It is an effective model that is significantly affected by the topology of the network, though it relies on states as opposed to propagation and development of a property. Analysis of the effect of perturbations by analysing the spectral density (De Aguiar and Bar-Yam 2005) demonstrate that not only does the density of states contain information regarding the topology, but also of the dynamics to external perturbations.

Other approaches vary significantly, and researchers have stated the inability to find commonality between the various papers (Barzel and Barabasi 2013). However, by adopting a framework that mimics network dynamics, a universality in network dynamics can be found. Barzel and Barabasi (2013) develop such a model based on two terms, a term that develops the property on the basis of the property itself. The second term is how the property in one Node is affected by the properties in the neighbouring Nodes.

54 𝑑𝑥𝑖 𝑑𝑡 = 𝑊(𝑥𝑖(𝑡)) + ∑ 𝐴𝑖𝑗𝑄(𝑥𝑖(𝑡), 𝑥𝑗(𝑡)) 𝑁−1 𝑗=1 (4.11)

Where W is a function concerned with altering itself, Q is a function that concerns the effect of neighbouring Nodes, and A is the adjacency matrix. Using this form, four separate studies using different dynamical methods are adapted to this form.

The models that are investigated are a Biomechanical model - B (Mass-Action Kinetics model) (Voit and Radivoyevitch 2000), a Birth-Death model - BD (Population Dynamics model) (Hayes and Babu 2004), a Regulatory dynamics of gene regulation – R (Michaelis-Menten model) (Karlebach and Shamir 2008), and an epidemiology model - E (SIS) (Pastor-Satorras and Vespignani 2001, Hufnagel, Brockmann et al. 2004, Dodds and Watts 2005).

Burstiness is a concept in Network Dynamics that the property propagation does not occur with a smooth distribution, but rather occurs in sporadic and intense bursts. This has been studied by Barabasi (2005) who analyses the emergence of heavy-tails in human dynamics. The system-wide dynamics of many systems (e.g. social, technological and economic) are driven by human dynamics. This puts understanding human behaviour at the centre of many real-world challenges. Several models for human behaviour has already been predicted by Poisson processes (Haight 1967). However, there is increasing evidence that in the context of work patterns, communication and entertainment that this is not the case. In fact, it seems to be characterised by burst of activity separated with long periods of inactivity (Barabasi 2005, Karsai, Kaski et al. 2012).

Barabasi (2005) attributes this burst activity to be a consequence of decision-based queuing. This provides a strong insight into how it is that SNA has been used to study human processes.

4.3. Chapter summary

Having reviewed the SNA literature, no studies could be found that provided direct models to identifying individuals who enable and sustain IDR. However, many models could be found that are analogous, or with relatively few tweaks could be made to identify such individuals.

Five main models were identified in the literature: 1. Degree centrality

2. Betweenness centrality 3. Eigenvector centrality 4. Structural holes (clustering)

55 5. The strength of weak ties

Furthermore, many papers focused on citation-based output metrics as a way of determining whether research was suitable.

These provide the foundations of the next steps of the research. According to the adopted research methodology, the next steps must construct an instrument for data collection, select a sample, write a research proposal, and collect the data.

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Chapter 5: The University of Bath Co-Authorship Networks