# Random-walk betweenness

There is a paper on the connection by Mark Newman http://arxiv.org/abs/cond-mat/0309045 A measure of betweenness centrality based on random walks. I quote (Doug 13:19, 8 January 2008 (PST)):

"The outline of this paper is as follows. In Sec. 2 we define in detail our random-walk betweenness and show how it is calculated. We introduce the measure first using an analogy to the flow of electrical current in a circuit, and then show that this is equivalent also to the flow of a random walk. In Sec. 3 we give a number of examples of applications of our measure, first to networks artificially designed to pose a challenge for the calculation of betweenness, and then to various real-world social networks, including a collaboration network of scientists, a network of sexual contacts, and Pagdett^Rs network of intermarriages between prominent families in 15th century Florence. In Sec. 4 we give our conclusions."

"Consider, then, an electrical circuit created by placing a unit resistance on every edge of the network of interest, as shown in Fig. 2. One unit of current is injected into the network at a source vertex s and one unit extracted at a target vertex t, so that current in the network as a whole is conserved. We now define the current-flow betweenness of a vertex i to be the amount of current that flows through i in this setup, averaged over all s and t."

On Mon, 7 Jan 2008, Malcolm Dean wrote:

> Date: Mon, 7 Jan 2008 20:55:44 -0800 > From: Malcolm Dean <malcolmdean@gmail.com> > To: Doug White <drwhite@uci.edu> > Subject: Re: brain complexity ...

> Yes. That is what I've been exploring since working paper #1.

> I believe the connections are clear when a strictly Info/thermo paradigm is adopted. For example, in a general way we can relate brain architecture via cabling theory (revealed by Diffusion MRI) and show that natural information flows produce the same problems and networks as electrical systems. I think it's a safe bet that this could be extended to such things as economic networks and highway systems.

> What I mean by a strict paradigm is viewing Information as a physical transformation, ridding our thinking of any essentialism and objectification, and sandboxing misleading definitions of Information as anything other than physical. We view thermodynamic flows in the system(s) being observed. This is why I am comfortable in saying that higher-level systems have a form of cognition. Any sufficiently complex physical system will display such emergent behavior. I believe this is why, even at a lower level, Japanese researchers have shown some intelligence in slime mould populations. Cognition is an emergent result of statistical mechanics.

> As Hofstadter puts it: "Thinkodynamics is statistical mentalics."

> Malcolm
>
>
> On Jan 7, 2008 8:34 AM, Doug White <drwhite@uci.edu> wrote:
>>
>> would be interesting to see if there are any connections here between
>> brain connectivity and distributed cognition.
>>
>> http://www.scholarpedia.org/article/Complexity Olaf Sporns (2007),
>> Scholarpedia, 2(10):1623.
>>
>> http://www.scholarpedia.org/article/Brain_connectivity Olaf Sporns (2007),
>> Scholarpedia, 2(10):4695.