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Community Detection Algorithm and Community Quality Metric Mingming Chen & Boleslaw K. Szymanski Department of Computer Science Rensselaer Polytechnic Institute

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Community Structure Many networks display community structure Groups of nodes within which connections are denser than between them Community detection algorithms Community quality metrics

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Two Related Community Detection Topics Community detection algorithm LabelRank: a stabilized label propagation community detection algorithm LabelRankT: extended algorithm for dynamic networks based on LabelRank A new community quality metric solving two problems of Modularity M. E. J. Newman, 2006; Newman and Girvan, 2004. Xie, Chen, and Symanski, 2013. Xie and Symanski, 2013.

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LabelRank Algorithm Four operators applied to the labels Label propagation operator Inflation operator Cutoff operator Conditional update operator 2 4 1 3 1 1 1 1 Question: NP=P ? Node 1: No; Node 2: No; Node 3: No; Node 4: Yes. P 1 (No)=3/4; P 1 (Yes)=1/4. Node 1: No. No Yes 97 P 1 (No)=3/100; P 1 (Yes)=97/100. Node 1: Yes.

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Label Propagation Operator where W is the n x n weighted adjacent matrix. P is the n x n label probability distribution matrix which is composed of n (1 x n) row vectors P i, one for each node Each element P i (c) holds the current estimation of probability of node i observing label, where C is the set of labels (here, suppose C={1, 2, …, n}) Ex. P i =(0.1, 0.2, …, 0.05, …) To initialize P, each node is assigned a distribution of probabilities of all incoming edges

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Label Propagation Operator Each node receives the label probability distribution from its neighbors and computes the new distribution P 3 = (0.25, 0, 0.25, 0, 0, 0, 0.25, 0.25, 0, 0) P 2 = (0.25, 0.25, 0, 0, 0.25, 0.25, 0, 0, 0, 0)P 4 = (0.25, 0, 0, 0.25, 0, 0, 0, 0, 0.25, 0.25) P 1 = (0.25, 0.25, 0.25, 0.25, 0, 0, 0, 0, 0, 0) P 1 = (0.25, 0.125, 0.125, 0.125, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625)

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Inflation Operator Each element P i (c) rises to the in th power: It increases probabilities of labels with high probability but decreases that of labels with low probabilities during label propagation. P 1 = (0.25, 0.125, 0.125, 0.125, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625, 0.0625) P 1 = (0.129, 0.0323, 0.0323, 0.0323, 0.00806, 0.00806, 0.00806, 0.00806, 0.00806, 0.00806)

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Cutoff Operator The cutoff operator on P removes labels that are below the threshold with the help from Inflation Operator that decreases probabilities of labels with low probabilities during propagation. efficiently reduces the space complexity from quadratic to linear. P 1 = (0.129, 0.0323, 0.0323, 0.0323, 0.00806, 0.00806, 0.00806, 0.00806, 0.00806, 0.00806) P 1 = (0.129) With r = 0.1, the average number of labels in each node is less than 3.

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Conditional Update Operator At each iteration, it updates a node i only when it is significantly different from its incoming neighbors in terms of labels: where is the set of maximum probability labels at node i at the last step. returns 1 if and 0 otherwise. k i is the node degree and q ∈ [0,1]. isSubset can be viewed as a measure of similarity between two nodes.

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Effect of Conditional Update Operator

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Running time of LabelRank O(Tm): m is the number of edges and T is the number of iterations. LabelRank is a linear algorithm

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Performance of LabelRank

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LabelRankT It is a LabelRank with one extra conditional update rule by which only nodes involved changes will be updated. Changes are handled by comparing neighbors of node i at two consecutive steps, and.

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Two Problems of Modularity Maximization Split large communities Favor small communities Resolution limit problem Modularity optimization may fail to discover communities smaller than a scale even in cases where communities are unambiguously defined. This scale depends on the total number of edges in the network and the degree of interconnectedness of the communities. Favor large communities Fortunato et al, 2008; Li et al, 2008; Arenas et al, 2008; Berry et al, 2009; Good et al, 2010; Ronhovde et al, 2010; Fortunato, 2010; Lancichinetti et al, 2011; Traag et al, 2011; Darst et al, 2013.

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Modularity Modularity (Q): the fraction of edges falling within communities minus the expected value in an equivalent network with edges placed at random Equivalent definition M. E. J. Newman, 2006. Newman and Girvan, 2004.

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Modularity with Split Penalty Modularity (Q): the modularity of the community detection result Split penalty (SP): the fraction of edges that connect nodes of different communities Q s = Q – SP: solving the problem, favoring small communities, of Modularity

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Q s with Community Density Resolution limit: Modularity optimization may fail to detect communities smaller than a scale Intuitively, put density into Modularity and Split Penalty to solve the resolution limit problem Equivalent definition

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Example of Two Well-Separated Communities Modularity (Q)Split Penalty (SP)Q s = Q – SPQ ds 2 communities0.50 1 community0000.245

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Example of Two Weakly Connected Communities Modularity (Q)Split Penalty (SP)Q s = Q – SPQ ds 2 communities0.3570.1430.2140.339 1 community0000.25

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Ambiguity between One and Two Communities Modularity (Q)Split Penalty (SP)Q s = Q – SPQ ds 2 communities0.30.20.10.263 1 community0000.249

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Ambiguity between One and Two Communities Modularity (Q)Split Penalty (SP)Q s = Q – SPQ ds 2 communities0.25 00.188 1 community0000.245

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Example of One Well Connected Community Modularity (Q)Split Penalty (SP)Q s = Q – SPQ ds 2 communities0.1670.333-0.1670.0417 1 community0000.23

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Example of One Very Well Connected Community Modularity (Q)Split Penalty (SP)Q s = Q – SPQ ds 2 communities0.04550.455-0.409-0.239 1 community0000.168

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Example of One Complete Graph Community Quality on a complete graph with 8 nodes Modularity (Q)Split Penalty (SP)Q s = Q – SPQ ds 2 communities-0.07140.571-0.643 1 community0000

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Modularity Has Nothing to Do with #Nodes

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5-clique Example Modularity (Q)Split Penalty (SP)Q s = Q – SPQ ds 30 communities0.87580.090910.78480.8721 15 communities0.88790.045450.84240.4305 ∆Q s =(0.8424-0.7848)=0.0576 > ∆Q=(0.8879-0.8758)=0.0121

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Thanks! Q & A

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Example of Two Weakly Connected Communities Modularity (Q)Split Penalty (SP)Q s = Q – SPQ ds 2 communities0.3090.250.05860.264 1 community-0.005860.125-0.1310.202

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