CENTRALITIES. Carlo PICCARDI. DEIB - Department of Electronics, Information and Bioengineering Politecnico di Milano, Italy

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1 CENTRALITIES Carlo PICCARDI DEIB - Department of Electronics, Information and Bioengineering Politecnico di Milano, Italy carlo.piccardi@polimi.it Carlo Piccardi Politecnico di Milano ver. 09/11/2017 1

2 NODE CENTRALITY The centrality of a node is a measure of its importance in the network. Degree The importance of a node can trivially be captured by the number k i of its neighbors (i.e. interactions, communication channels, sources-destinations of information, etc.). The "hubs" are the most central nodes. In weighted networks, use the strength. Carlo Piccardi Politecnico di Milano ver. 09/11/2017 2

3 Betweenness The degree centrality may fail in some cases... The betweenness of node i is the number of shortest paths (connecting all the pairs of nodes of the network) that pass through i. Similar definition for link betweenness. Carlo Piccardi Politecnico di Milano ver. 09/11/2017 3

4 Anomalous nodes might emerge when comparing degree and betweenness. Example: the worldwide air transportation network (Guimerà et al., 2005) There are anomalous cities (=nodes) with very low degree but very high betweenness. Carlo Piccardi Politecnico di Milano ver. 09/11/2017 4

5 Closeness centrality A node is central if, on average, it is close (=short distance) to all other nodes: it has better access to information, more direct influence on other nodes, etc. The average distance from to all the other nodes is: The closeness centrality is defined as If the network is directed, we must distinguish between in- and out-closeness. If the network is weighted, several (non trivial) generalized definitions are available. Carlo Piccardi Politecnico di Milano ver. 09/11/2017 5

6 Eigenvector centrality The centrality i is (proportional to) the sum of the centralities of the neighbours (i.e., a node is important if it relates to many and/or important nodes). and 1/ N Letting T 1 2 i j a ij j, we obtain the eigenvector equation A If the network is connected (= A is irreducible), the centralities solution with 0, i 0 for all i (Frobenius-Perron theorem). i are given by the only Applications in social networks (who is the most influential individual?). Carlo Piccardi Politecnico di Milano ver. 09/11/2017 6

7 Authorities and Hubs In directed networks, we can take into account the different role of in- and outlinks. authority score x i : a node with large x i is pointed by highly ranked nodes hub score y i : a node with large xi a jiy j yi j y i points to highly ranked nodes j a ij x j For example, in the World Trade Network: authorities (= nodes with large x i ) are countries with large import flows ("consumers") hubs (= nodes with large y i ) are countries with large export flows ("producers") Carlo Piccardi Politecnico di Milano ver. 09/11/2017 7

8 RANDOM WALKS ON NETWORKS A random walk is a path formed by a sequence of random steps. The term is first attributed to Karl Pearson [Nature, 1905]. Applications in ecology, economics, psychology, computer science, physics, chemistry, biology, etc. Many variants: discrete vs continuous time uniform vs non-uniform step Markovian vs non-markovian process etc. Carlo Piccardi Politecnico di Milano ver. 09/11/2017 8

9 Random walks on networks In a binary (unweighed) network, the random walker in node chooses an out-link with uniform probability: In a weighted network, the out-link is chosen with probability proportional to its weight: is the transition matrix. Carlo Piccardi Politecnico di Milano ver. 09/11/2017 9

10 Random walks and Markov chains = state probability = probability of being in node at time ( ) evolves according to the Markov chain equation If the network is strongly connected the transition matrix is irreducible there exists a unique stationary state probability distribution positive ( for all )., which is strictly = fraction of time spent on node = centrality of node If undirected networks, is the (rescaled) node strength In directed networks, the in-strength turns out to be mostly correlated to (example: WWW). Carlo Piccardi Politecnico di Milano ver. 09/11/

11 Example: the World Trade Network (2008) The trading system can be modelled as a directed, weighted network: (million US dollars) from country to country is the export flow The strongly connected component includes countries (94% of the total). The network is extremely dense ( ) and very heterogeneous (multi-scale) in node degrees, node strengths, and link weights. Carlo Piccardi Politecnico di Milano ver. 09/11/

12 In the WTN, centrality strongly correlates with the in-strength......but it also fairly correlates with the out-strength (because the latter correlates with ). Carlo Piccardi Politecnico di Milano ver. 09/11/

13 from wikipedia.org PageRank ("Google") centrality The solution to Most directed networks are not connected. is not unique or non positive, or the Markov chain might be not even well defined. Teleportation: at each time step, the random walker has probability to jump to a randomly selected node. The network becomes complete (all-to-all) thus connected there exists a unique strictly positive solution to. = PageRank of node A proper value for γ? Not too large (the network would be heavily modified) nor too small ( too sensitive to ). The standard (Google) value is. Carlo Piccardi Politecnico di Milano ver. 09/11/

14 The WWW is an extremely heterogeneous network with self-organized structure. Google ranking exploits the network structure for retrieving information. Examples of PageRank values (in logarithmic scale Technical problem: (re)computing in a network with trillions of nodes. Carlo Piccardi Politecnico di Milano ver. 09/11/

15 Other random-walk-based centralities Closeness and betweenness centrality assume that information flows along the shortest path, but The definition of shortest path itself is questionable in weighted network. In dense networks, alternative (non-shortest) paths may significantly contribute. Mean First Passage Time (MFPT): random-walk-based measure of the distance = probability that a random walker started in takes exactly steps to arrive in for the first time It is measured in number of hops, as usual distances on graphs. It accounts for all paths, weighted by their probability. It is non-symmetrical:, even for undirected networks. All s can effectively be computed (via linear algebra) given the transition matrix. Carlo Piccardi Politecnico di Milano ver. 09/11/

16 Random walk closeness centrality [Noh and Rieger 2004] World Trade Network Random walk betweenness centrality [Newman 2005; Blochl et al 2011] =expected number of visits to, starting from and terminating the walk at the first passage in Carlo Piccardi Politecnico di Milano ver. 09/11/