A citation network splits into two similarly sized subfields

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Complex and collective systems

Many Parts, One Behaviour, Ch Communities: finding the groups in a network

Problem

A citation network splits into two similarly sized subfields,

A

and

B

, each with

n_A=n_B=500

authors. A snapshot of the citation network shows

m_{AA}=760

links between two subfield-

A

authors,

m_{BB}=745

links between two subfield-

B

authors, and

m_{AB}=520

links between a subfield-

A

and a subfield-

B

author. ((a)) Using the stochastic block model maximum-likelihood result

\hat p_{rs}=m_{rs}/n_{rs}

, where

n_{rs}

is the number of possible pairs of that type (

n_r(n_r-1)/2

for

r=s

, and

n_rn_s

for

r\ne s

), estimate

\hat p_{AA}

\hat p_{BB}

and

\hat p_{AB}

, and convert them into the planted-partition parameters of the detectability threshold,

\hat c_{\rm in}=N\hat p_{\rm in}

and

\hat c_{\rm out}=N\hat p_{AB}

, with

N=n_A+n_B

and

\hat p_{\rm in}

the average of

\hat p_{AA}

and

\hat p_{BB}

. ((b)) Apply the detectability threshold

(c_{\rm in}-c_{\rm out})^2>2(c_{\rm in}+c_{\rm out})

to the estimated

\hat c_{\rm in}

and

\hat c_{\rm out}

, and decide whether the two subfields are, in principle, detectable from this network. ((c)) A researcher runs only the local-moving phase of the Louvain algorithm once on this network, starting from every author in its own group, and reports the resulting partition's modularity. Using the conclusion of part (b), and the pattern that even a structureless Erdos-Renyi graph of comparable size and mean degree yields partitions of substantial modularity from local moving alone, explain what such a reported modularity value would and would not tell the researcher about the two subfields.

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