Centrality measure computation
Betweenness and closeness

• Graph \(G=(V,E)\)
  • Undirected
  • Unweighted
  • Connected
• Distance \(d(u, v)\)
  • # of edges in shortest path from \(u\) to \(v\)
• Eccentricity in subgraph​ \(S\)
  • \(e_S(v)=\max_{w\in S} d( v, w)\)
• Central nodes in \(S\)
  • Min \(e_S\)
• Peripheral nodes in \(S\)
  • Max distance from central

• Hodology
  • Study of pathways

• Closeness centrality in the case of directed weakly connected graphs
  • ​Very realistic case
• Two approaches
  • Generalise "classical" definition\[\kappa(v) =\frac{\rho(v) - 1}{\varphi(v)}\frac{\rho(v)-1}{n-1}=\frac{(\rho(v) - 1)^2}{(n-1)\varphi( v)}\]where \(\rho(v) = |\Rho(v)|\) and \(\Rho(v)\) is set of nodes reachable from \(v\)
  • Observe that infinite distance should not contribute\[\eta(v)=\sum_{w \in V-\{v\}}\frac{1}{d(v,w)}\](called harmonic centrality)

Interactive version available at http://schochastics.net/sna/periodic.html

• Correlation between centrality measures
  • Lot of papers
    • Not always consistent
  • An example (Florentine families)
    • Pearson's correlation coefficient
      • 1 totally correlated, -1 totally anti-correlated

• Density axiom
  • \(k\)-clique and directed \(k\)-cycle connected by edge \(\{x,y\}\)
  • Centrality of \(x\) is strictly larger than centrality of \(y\)

• A "jungle" of centrality measures

  • Sometimes strongly correlated, sometimes not

  • Different computational complexity

  • Different interpretation

• Difficult to "validate" them

  • Not so many benchmarks

  • Why Bacon should be more important than Clooney?

• Application dependent notion

  • Bioinformatics seems to be an interesting application field

  • Importance of nodes is sometimes clear

• Naive algorithm

  • For every node \(v\), set \(\beta(v)=0\)

  • For every node \(s\), perform BFS to find all shortest paths from \(s\) to all other nodes \(t\): let \(\Sigma_{s,t}\) be the set of these paths

  • For every node \(v\) and for each pair \(s,t\), count number of times \(v\) appears in a path in \(\Sigma_{s,t}\), divide by \(|\Sigma_{s,t}|\) and add to \(\beta(v)\)

  • For every node \(v\), return \(\beta(v)\)

• Naive algorithm very expensive in terms of space

  • Storing all shortest paths between all pairs of nodes

  • Little improvement

    • Storing only paths from one source

      • For every node \(v\), set \(\beta(v)=0\)

      • For every node \(s\), perform BFS to find all shortest paths from \(s\) to all other nodes \(t\): let \(\Sigma_{s,t}\) be the set of these paths

        • For every node \(v\) and for every target \(t\), count number of times \(v\) appears in a path in \(\Sigma_{s,t}\), divide by \(|\Sigma_{s,t}|\) and add to \(\beta(v)\)

      • For every node \(v\), return \(\beta(v)\)

  • ​​Still unfeasible in the worst case

    • ​Why?

• Brandes' algorithm

  • Integrating the computation of the betweenness centrality in the backward phase (construction of the shortest paths)

    • Without constructing the shortest paths

  • For every node \(v\), set \(\beta(v)=0\)
  • For every node \(s\)
    • For every node \(v\), \(\delta_s(v)=0\)
    • Perform (as before) BFS from \(s\) to all other nodes \(t\) and compute number of shortest paths \(\sigma_{s,t}\) from \(s\) to \(t\)

    • Going back to \(s\), increment \(\delta_s(v)\) as appropriate when node \(v\) is reached by using \(\sigma_{s,\bullet}\)-values

    • Add \(\delta_s(v)\) to \(\beta(v)\)

  • ​​Return \(\beta(v)\)

• Forward phase: computing number of shortest paths
  • When a new parent is discovered, add its number of shortest paths from the source

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• Backward (accumulation) phase: computing contributions to betweenness centrality
  • Pair-wise dependency\[\delta_{s,t}(v)=\frac{\sigma_{s,t}(v)}{\sigma_{s,t}}\]
    • Hence, \(\beta(v)=\sum_{s,t}\delta_{s,t}(v)\)
  • One-sided dependency\[\delta_s(v)=\sum_t\delta_{s,t}(v)\]
    • Hence, \(\beta(v)=\sum_{s}\delta_{s}(v)\)
  • Basic idea
    • \(\delta_s(v)\) can be computed recursively by using \(\delta_s(w)\) for \(w\) immediately following \(v\) on some shortest path from \(s\)
      • ​\(\mathrm{PH}_s(v)\): set of these nodes \(w\)

• Fact 1. If \(\sigma_{s,t}(v)\neq0\) then \(\sigma_{s,t}(v) = \sigma_{s,v}\sigma_{v,t}\)

• Fact 2. If the arc \((u,v)\) belongs to a shortest path from \(s\) to \(t\), then \(\sigma_{s,t}(u,v) = \sigma_{s,u}\sigma_{v,t}\)
  • \(\sigma_{s,t}(u,v)\): number of shortest paths from \(s\) to \(t\) passing through edge \((u,v)\)

• Fact 3. If \(\sigma_{s,t}(v)\neq0\) then \(\sigma_{v,t} = \sum_{w\in\mathrm{PH}_s(v)}\sigma_{v,t}(v,w)\)

• Brandes' lemma. For every \(s\neq v\)\[\delta_s(v)=1+\sigma_{s,v}\sum_{u\in\mathrm{PH}_s(v)}\frac{\delta_s(u)}{\sigma_{s,u}}\]
• Proof
  • If \(t=v\), then contribution of \(t\) to \(\delta_s(v)=1\)
  • Let \(T_u\) be set of \(t\neq v\) such that \(\sigma_{s,t}(v,u)\neq 0\) and let \[T=\bigcup_{u\in\mathrm{PH}_s(v)}T_u\]
  • Then\[\delta_s(v)=1+\sum_{t\in T}\frac{\sigma_{s,t}(v)}{\sigma_{s,t}} = 1+\sum_{t\in T}\frac{\sigma_{s,v}\sigma_{v,t}}{\sigma_{s,t}} = 1+\sigma_{s,v}\sum_{t\in T}\sum_{u\in\mathrm{PH}_s(v)}\frac{\sigma_{v,t}(v,u)}{\sigma_{s,t}}\\=1+\sigma_{s,v}\sum_{u\in\mathrm{PH}_s(v)}\sum_{t\in T_u}\frac{\sigma_{v,t}(v,u)}{\sigma_{s,t}}=1+\sigma_{s,v}\sum_{u\in\mathrm{PH}_s(v)}\sum_{t\in T_u}\frac{\sigma_{u,t}}{\sigma_{s,t}}\\=1+\sigma_{s,v}\sum_{u\in\mathrm{PH}_s(v)}\frac{1}{\sigma_{s,u}}\sum_{t\in T_u}\frac{\sigma_{s,u}\sigma_{u,t}}{\sigma_{s,t}}=1+\sigma_{s,v}\sum_{u\in\mathrm{PH}_s(v)}\frac{\delta_s(u)}{\sigma_{s,u}}\]

• Accumulation phase: computing contributions to betweenness centrality

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\[\delta_s(v)=1+\sigma_{s,v}\sum_{u\in\mathrm{PH}_s(v)}\frac{\delta_s(u)}{\sigma_{s,u}}\]

• Complexity of Brandes' algorithm
  • Time \(O(nm)\)
  • Space \(O(n+m)\)
• Unlikely that we can do better
• Other results
  • Approximation
    • Through sampling
  • Dynamic version
  • Distributed version

• Textbook algorithm
  • Time \(O(nm)\)
  • Space \(O(n+m)\)
• Unlikely that we can do better
• Approximation
  • Through sampling
• What about looking for the top-\(k\) nodes?

• \(k\)-Hitting Set (\(k\)-HS) problem
  • Domain \(X\)
  • Collection \(C\) of subsets of \(X\) with \(|X|\leq k\log|C|\)
  • Is there \(c\in C\) such that, for any \(y\in C\), \(c\cap y\neq \emptyset\)?
• Conjecture: there is no \(\epsilon\) such that, for all \(k\geq 1\), there is algorithm solving \(k\)-HS in time \(O(n^{2-\epsilon})\) 
  • \(n\): size of input

• From HS to farness
  • Example: \(X = \{a,b,c,d\}\) and \(C = \{\{a,b\}, \{b,c\}, \{c,d\}\}\)

• Computing closeness top node
  • iFUB: perform complete BFS starting from a (small) set of (cleverly) selected nodes
  • "Orthogonal" idea
    • Perform a partial BFS starting from all nodes \(v\)
      • In non-increasing order of degree
    • Each pBFS
      • Receives the starting node \(v\) and a lower bound \(x\) on the closeness centrality of top node
      • Returns \(0\) if \(v\) is not the top node, \(\kappa(v)\) otherwise
• Generalises to
  • Top \(k\) nodes
  • Directed weakly connected graphs

• pBFS(\(v\),\(x\))

• The algorithm
  • \(\kappa(v) = 0\) for each \(v\)
  • \(\kappa_\mathrm{top} = 0\)
  • For each \(v \in V\) in decreasing order of degree
    • \(\kappa(v) = \mathrm{pBFS}(v,x)\)
    • If \(\kappa(v) \neq 0\), then \(\kappa_\mathrm{top} = \kappa(v)\)
  • Return \(\kappa_\mathrm{top}\)

• The IMDB network
  • Nodes: actors
  • Edges: at least one co-presence

• The case of directed weakly connected graphs
  • ​Closeness definition\[\kappa(v) =\frac{\rho(v) - 1}{\varphi(v)}\frac{\rho(v)-1}{n-1}=\frac{(\rho(v) - 1)^2}{(n-1)\varphi( v)}\]where \(\rho(v) = |\Rho(v)|\) and \(\Rho(v)\) is set of nodes reachable from \(v\)
  • Lower bound on farness\[\varphi(v)\geq\tilde{\varphi}_d(v) = \varphi_d(v) - \tilde{\gamma}_ {d+1}(v)+(d+2)(\rho(v)-n_d(v))\]
• ​We don't know \(\rho(v)\)
  • ​We need to do a BFS from each node
• Idea
  • We can compute upper and lower bounds on \(\rho(v)\) and use them to compute lower bound for closeness centrality

• Using the bounds
  • If \(\alpha(v)\leq\rho(v)\leq\omega(v)\), then\[\frac{1}{\kappa(v)} \geq(n-1)\min\left(\frac{\tilde{\varphi}(v,\alpha(v))}{(\alpha(v)-1)^2},\frac{\tilde{\varphi}(v,\omega(v))}{(\omega(v)-1)^2}\right)\]where\[\tilde{\varphi}_d(v,x) = \varphi_d(v) - \tilde{\gamma}_ {d+1}(v)+(d+2)(x-n_d(v))\]
    • Let \(a=d+2\) and \(b=\tilde{\gamma}_{d+1}(v)+a(n_d(v)-1)-\varphi_d(v)\)
      • ​Note that \(a>0\) and \(b>0\) (since \(\varphi_d(v)<a(n_d(v)-1)\))
    • We know that\[\varphi(v)\geq\varphi_d(v) - \tilde{\gamma}_ {d+1}(v)+a(\rho(v)-n_d(v))\\=a(\rho(v)-1)+\varphi_d(v)-\tilde{\gamma}_{d+1}(v)-a(n_d(v)-1)=a(\rho(v)-1)-b\]
    • Hence \[\frac{1}{\kappa(v)}=\frac{(n-1)\varphi( v)}{(\rho(v)-1)^2}\geq(n-1)\frac{a(\rho(v)-1)-b}{(\rho(v)-1)^2}\]
    • Function \(g(x)=\frac{a(x-1)-b}{x^2}\) in interval \([x_1,x_2]\) with \(x_1,x_2>0\) has minimum in \(x_1\) or \(x_2\): bound follows

• Computing the bounds \(\alpha(v)\) and \(\omega(v)\)
  • Compute graph of strongly connected components (DAG)
  • If \(u,v\in C\), then \(\rho(u)=\rho(v)=\sum_{D\in R(C)}|D|\)
    • \(R(C)\): set of components reachable from \(C\)
  • Hence, we need lower and upper bound \(\alpha(C)\) and \(\omega(C)\) on \(\sum_{D\in R(C)}|D|\)
  • We process components in reverse topological order
    • \(\alpha(C)=|C|+\max_{D\in N(C)}\alpha(D)\)
    • \(\omega(C)=|C|+\sum_{D\in N(C)}\omega(D)\)

• \(\alpha(C_6)=4=\omega(C_6)\)
• \(\alpha(C_2)=6=\omega(C_2)\)
• \(\alpha(C_5)=7=\omega(C_5)\)
• \(\alpha(C_4)=9=\omega(C_4)\)
• \(\alpha(C_3)=10<26=\omega(C_3)\)
• \(\alpha(C_1)=11<34=\omega(C_1)\)

• The importance of being important
  • Fast spreading of information
  • Control of the information flow among vertices
  • Fast collection  of information from other nodes
• Increasing the value of centrality by adding links can increase the importance within the network

• \(\varphi(x_1)=1+1+3+2+3=10\)

• \((x_1,x_6)\): \(\varphi(x_1)=1+1+3+2+1=8\)

• \((x_1,x_5)\): \(\varphi(x_1)=1+1+2+1+2=7\)

• \((x_1,x_4)\): \(\varphi(x_1)=1+1+1+2+3=8\)

• Maximum Closeness Improvement Problem (MCI)
  • ​Input: a graph \(G=(V,E)\), a vertex \(u \in V\) and an integer \(k \in \mathbb{N}\)
  • A set \(S=\{v| v \in V\setminus N(u)\}\), such that \(|S|\leq k\)
  • Maximize \(\kappa(v)\)in the graph obtained from \(G\) by adding the edges \((u,v)\) with \(v\in S\)
• For each \(\gamma \geq 1-\frac{1}{15e}\), there is no \(\gamma-\)approximation algorithm for the MCI problem, unless \(P=NP\)
  • Reduction from the dominating set problem
• A greedy algorithm approximates MCI within a factor \(1-\frac{1}{e}\)
  • A function \(f\) is monotone submodular if for any pair of sets \(S\subseteq T \subseteq X\) and for any element \(e\in X\setminus T\), \(f(S\cup\{e\}) - f(S) \geq f(T\cup \{e\}) - f(T)\)
  • For each vertex \(u\), \(\kappa(u)\) is monotone and submodular with respect to any feasible solution for MCI

• The DBLP network
  • Nodes: researchers
  • Edges: at least one co-authorship