15-382 C OLLECTIVE I NTELLIGENCE – S19 L ECTURE 35: N ETWORKS 2 T EACHER : G IANNI A. D I C ARO
I MPORTANCE / P OWER IN NETWORKS Certain positions within the network give nodes more impor portanc nce / / pow power o Directly affect/influence others This and fol ollow owing g o Control the flow of information slides are adapted from om Kristina Le Kri Lerman’s sl slides s o Avoid control of others Influencers in social networks, vehicles for disease spreading, hubs in road networks, key infrastructures on the Internet, leaders in animal societies, … 2
I MPORTANCE OF A NODE : N ETWORK C ENTRALITY "What characterizes an important node?" § Networ ork Centrality indices provide answers to the question by defining a mapping that attributes a real-valued number to each node § These number can be used to determine a ranking among the nodes of a network 3
C ENTRALITY IN N ETWORKS § Ce Centrality encodes the relationship between structure (topology) and importance/power (flows of information and control) in interconnected systems à Certain positions within the network give nodes more power or importance § How do we measure importance? Who can directly affect/influence others? [direct information transfer] o • Highest degree nodes are “in the thick of it” o Who controls information flows ? [relaying information transfer] • Nodes that fall on shortest paths between others can disrupt the flow of information between them o Who can quickly inform most others? [multicasting, few hops dissemination] • Nodes who are close to other nodes can quickly get/give information to them 4
D EGREE CENTRALITY ntrality of node 𝑜 is the number of other nodes 𝑜 is § Degr gree Cent connected to o A node with high degree has high potential communication activity 4 5 node In- Out- Total degree degree degree 1 1 0 1 1 2 3 2 3 2 5 3 1 3 4 4 5 4 2 1 3 1 5 2 1 3 2 3 5
M ATHEMATICAL REPRESENTATION USING ADJACENCY MATRIX Adjacency matrix A 1 2 3 4 5 4 5 1 0 1 0 0 0 2 1 0 0 1 1 0 3 0 1 0 1 1 2 3 4 0 0 0 0 1 5 0 1 0 0 0 node In- Out- Total degree degree degree out = å Out-degree : row sum d i A ij 1 0 1 1 j 2 3 2 5 3 1 3 4 4 2 1 3 in = å In-degree : column sum d i A ji 5 2 1 3 j 6
B ETWEENNESS CENTRALITY ntrality of node 𝑜 is the number of shortest paths § Betweenne nness Cent ( geodesics ) connecting all pairs of other nodes that pass through 𝑜 o Node with highest betweenness can potentially control or distort communication between a large number of nodes Paths from 1 4 5 1 à 2 1 à 2 à 3 1 à 2 à 4 1 à 2 à 4 à 5 1 Paths from 2 2 à 3 2 à 4 2 à 4 à 5 2 3 Paths from 3 3 à 2 3 à 4 3 à 5 4 5 1 Paths from 4 4 à 5 à 2 4 à 5 à 2 à 3 4 à 5 2 3 Paths from 5 5 à 2 5 à 2 à 3 5 à 2 à 4 7
C LOSENESS CENTRALITY § Node that is closer to all other nodes can reach all other nodes in shortest amount of time (on average) o This node should best avoid being controlled by others! § Clos osene ness cent ntrality is the reciprocal of the sum of geodesic distances from a node to all other nodes 𝑂 1 𝐷 𝑜 = 𝐷 𝑜 = ∑ & 𝑒(𝑛, 𝑜) ∑ & 𝑒(𝑛, 𝑜) In not strongly connected graphs, the distance between two nodes that are not connected is set to 1/ ∞ =0 8
S ELF - CONSISTENT MEASURES OF CENTRALITY § Katz (1953): Katz scor ore o “not only on how many others a person is connected to, but who he connects to” o One’s status is determined by the status of the people s/he is connected to § Bonacich (1972): Eige genvector or centrality o Node’s centrality is the sum of the centralities of its connections o Relative centrality can be computed as an average over the centralities of the directly connected nodes o We can use the degree centrality as a reference centrality for the 𝑑 / 𝑑 0 = 1 2 𝑑 / 𝜇 /∈𝒪(0) 9
E IGENVECTOR CENTRALITY Bonacich (1972): Eige genvector or centrality § o Node’s centrality is the sum of the centralities of its connections 𝑑 8 = 1 2 𝑑 / Using the adjacency matrix 𝐵 for degree centrality: 𝜇 /∈𝒪(0) 𝑑 8 = 1 Where 𝐻 is the entire network graph, 𝜇 2 𝐵 8/ 𝑑 / 𝒪 is 𝑜 ’s neighborhood /∈: 𝜇 𝑑 8 = 2 𝐵 8/ 𝑑 / This is an eige genvector or equation on ! /∈: 𝜇𝒅 = 𝐵𝒅 c is the eigenvector of A associated to the largest eigenvalue 𝜇 (only 𝒅 ≥ 0 are of interest + Perron-Frobenius theorem) 10
E IGENVECTOR CENTRALITY 𝜇𝒅 = 𝐵𝒅 𝜇 𝑑 8 = 2 𝐵 8/ 𝑑 / /∈: o The value of the 𝑜 -th component of the eigenvector provides the value of the relative centrality of node 𝑜 o There’s a scale factor, such that only the relative values are meaningful (for ranking the nodes) o Computational aspects: find the largest eigenvalue, compute the eigenvector (for very large matrices) 11
E IGENVECTOR CENTRALITY o Me Message ge passing g computation by estimate bootstrapping: o Start with an initial guess for the centrality of a node (e.g., number of neighbors), then iterate for each node 𝑗 : o Get centrality values from neighbors and update 𝑑 8 according to 𝑑 8 ← > ? ∑ /∈𝒪 𝐵 8/ 𝑑 / o Over time, estimates become more and more accurate, converging to the correct estimates in the limit o Convergence can be assessed by any L1, L2, L ∞ metric of choice for detecting that the vector of centralities undergo no significant changes o Any synchronous or asynchronous scheme can be adopted for selecting the nodes performing the update, examples from algorithms for asynchronous value iteration (dynamic programming) can be useful to devise computationally-effective schemes 12
M ESSAGE P ASSING BY P USH -P ULL G OSSIP The message passing algorithm algorithm can be implemented in a fully distributed o and decentralized way, including running it on the network itself Me Message ge passing g computation or estimation of network-level node-local properties o can be realized according to a general scheme based on pus push-pull gos gossip of information data, which is based on a model of epidemic spreading Active thread Passive thread Each node has two running threads • Active: runs periodically, selects a neighbor 𝑞 (peer), sends ( pus push ) its state value 𝑡 to 𝑞 , receives ( pull pull ) 𝑞 ’s state value, use it to update its local state 𝑡 • Passive: listen to requests from neighbors, when contacted receives state value from requester, sends its state value, updates its local state Ø 𝑡 is the state value held by a node, that can be the centrality value, as well as any other state value of interest related to the network 13
E IGENVECTOR CENTRALITY : P OWER METHOD o Pow ower method od (Pow ower iteration on) : form of centralized numeric implementation of the message passing approach for computing centrality values, that iterates the following equation: 𝐵𝒅 B 𝒅 BC> = 𝐵𝒅 B Ø This method is a general approach for the numeric calculation on of of the large gest genvalue of a diagonalizable matrix 𝐵 and, accordingly, of the eige eige genvector or assoc ociated to o the large gest (dom ominant) eige genvalue o Assumptions (for convergence): matrix 𝐵 is diagonalizable, has a dominant eigenvalue, and the starting vector 𝒅 D has a non-zero component in the direction of an eigenvector associated to the dominant eigenvalue o It’s a simple and possibly slow to converge algorithm o Appropriate for large and sparse matrices 14
A LPHA - CENTRALITY (B ONACICH , 1987) § Alp Alpha Centralit lity : Similar to eigenvector centrality, but the degree to which a node centrality contributes to the centralities of other nodes depends on a parameter 𝛽 § Mathematical interpretation: o 𝑑 8 (𝛽) is the expected number of paths activated directly or indirectly by node 𝑗 å c i ( a ) = (1 + a c j ( a )) A ij j A + a A 2 + a 2 A 3 + ... ( ) å c i ( a ) = j 15
A CLOSER LOOK AT A LPHA -C ENTRALITY 4 5 § Alpha-Centrality matrix 1 2 3 § 1 st term: number of paths of length 1 (edges) between i and j § Contribution of this term to c i ( a ) is S j A ij 0 1 0 0 0 0 0 1 1 0 c ( a ) = A + a A 2 + a 2 A 3 + ... 0 1 0 1 1 0 0 0 0 1 0 1 0 0 0 16
A CLOSER LOOK AT A LPHA -C ENTRALITY 4 5 § Alpha-Centrality matrix 1 c ( a ) = A + a A 2 + a 2 A 3 + ... 2 3 § 2 nd term: number of paths of length 2 between i and j 0 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 1 1 0 0 1 0 1 2 x 0 1 0 1 1 0 1 0 1 1 = 0 1 1 1 1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 1 0 17
A CLOSER LOOK AT A LPHA -C ENTRALITY 4 5 § Alpha-Centrality matrix 1 c ( a ) = A + a A 2 + a 2 A 3 + ... 2 3 § 3 rd term: number of paths of length 3 between i and j 0 0 1 1 0 0 1 0 0 0 0 1 0 1 2 0 1 0 1 2 0 0 1 1 0 0 2 1 1 1 0 1 1 1 1 x 0 1 0 1 1 0 2 1 2 2 = 0 1 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 1 1 0 0 1 0 0 0 0 1 0 1 2 18
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