Proximity index

This is metric function that we use in Netwalk method

Proximity Index is determined as $\Delta (i,j)={\frac {{\sqrt {(}}\sum _{k!=j}\delta ^{2}(i;j;k))}{N-2}}$

where $\delta (i;j;k)=d(i,k)-d(j,k)$ and </math>d(i,k)</math> is known as Mean First Passage Time.

There has been a lot of research to determine the mean-first-passage-time of variable diffusive brownian motion. For an instance following papers are of interest.

Explanation of proximity index

The node node distance is defined as the average number of steps needed for the particle to move from node i to node j in the network. The particle could stay in that node depends on the sum of the forces at the position (ignore gravity). sum (i Akl)/ sum (mn,A mn)

Local Attractor

for any $j\in E$ if it is the most proximal node, then it is considered as local attractor of that community.

Global Attractor

for any $j\in V$ where $V$ is set of all nodes; if the node attracts itself at global level (as opposed to nearest neighbours) then it is considered as global attractor.

I think, it is to account for substratum influences between global communities in a huge network. Eg. language families.

Centrality of the community

the global attractor of itself

Hard clusters

The notion of mutually exclusive communities $L_{a}$ and $L_{b}$ are either identical or distinct there is overlap.

Properties of communities

1. size of community
2. instability index $I_{c}$ - determined by degree of interaction between node i in a comm A and nodes in comm B.
3. centrality of the community

• Klein, MFPT of Brownian Motion and Related Problems. Royal society of London. 1951

• M. Gitterman, Mean first passage time for anomalous diffusion. Phys Rev E 62. 2000