Weighted Networks IST402 – Network Science Acknowledgement: Roberta Sinatra Laszlo Barabasi

How do people find a new job? interviewed 100 people who had changed jobs in the Boston area. More than half found job through personal contacts (at odds with standard economics). Those who found a job, found it more often through “weak ties”. HYPOTHESIS: The strong ties are within groups, weak ties are between groups. Network Science: Weighted Networks Granovetter, 1973 WHY WEIGHTS MATTER: THE STRENGTH OF WEAK TIES

Examples of weighted networks Extended metrics to weighted networks: Weighted degree and other definitions (C, knn). Correlations between weights and degrees Scaling of weights Betweenness centrality Granovetter : The strength of weak ties OUTLINE Network Science: Weighted Networks

Nodes: airports Links: direct flights Weights: Number of seats  2002 IATA database  V = 3880 airports  E = direct flights  = 10 k max = 318  = 4 l max = 16  = 10 5 w max = 10 7  N min = 10 3 N max = 10 7 Network Science: Weighted Networks EXAMPLE 1: AIRLINE TRAFFIC

Nodes: scientists Links: joint publications Weights: number of joint pubs. i, j: authors k: paper n k : number of authors δ i k =1 if author i contributed to paper k Network Science: Weighted Networks Newman and Girvan, cond-mat/ (2003) EXAMPLE 2: SCIENCE COLLABORATION NETWORK

EXAMPLE 3: INTERNET domain2 domain1 domain3 router Nodes: routers Links: physical lines Link Weights: bandwidth or traffic Network Science: Weighted Networks

Nodes: metabolites Links: reactions Link Weights: reaction flux Network Science: Weighted Networks E. Almaas, B. Kovács, T. Vicsek, Z. N. Oltvai, A.-L. B. Nature, 2004; Goh et al, PRL EXAMPLE 4: METABOLIC NETWORK

weak links strong links Nodes: mobile phones Links: calls Weights: number of calls or time on the phone Network Science: Weighted Networks Onella et al, PNAS 2007 EXAMPLE 5: MOBILE CALL GRAPH

DEFINITIONS: Weighted Degree Network Science: Degree Correlations

GRAPHOLOGY 2 Unweighted (undirected) Weighted (undirected) protein-protein interactions, wwwCall Graph, metabolic networks

a ij and w ij In the literature we often use a double notation: A ij and w ij (somewhat redundant) Adjacency Matrix (A ij ) Weight Matrix (W ij ) Node Strength (weighted degree) s:

Strength distribution P(s): probability that a randomly chosen node has strength s Weight distribution P(w): probability that a randomly chosen link has weight w In most real systems P(s) and P(w) are fat tailed. Network Science: Weighted Networks EMPIRICAL FINDING 1: P(s) AND P(w) ARE FAT TAILED

If there are no correlations between k and s, we can approximate w ij with : Science collaboration network: Weight appear to be assigned randomly/ Network Science: Weighted Networks Barrat, Barthelemy, Pastor-Satorras, Vespignani, PNAS 2004 EMPIRICAL 2: RELATIONSHIP BETWEEN STRENGTH AND DEGREE

If there are no correlations between k and s, we can approximate w ij with : Science collaboration network: Weight appear to be assigned randomly/ Barrat, Barthelemy, Pastor-Satorras, Vespignani, PNAS 2004 EMPIRICAL 2: RELATIONSHIP BETWEEN STRENGTH AND DEGREE

Airport network: β=1.5 Randomized weights: s= k: β=1 Correlations between topology and dynamics: β>1: strength of vertices grows faster than their degree  the weight of edges belonging to highly connected vertices have a value higher than the one corresponding to a random assignment of weights.  the larger is an airport’s degree (k), disproportionally more traffic it can handle (s). In most systems we observe: EMPIRICAL 2: NODE STRENGHT AND DEGREE Barrat, Barthelemy, Pastor-Satorras, Vespignani, PNAS 2004

 (assuming no degree correlations) Weight correlation: Scaling relationship: WEIGHT CORRELATIONS(2)

Weight correlation: Scaling relationship: Science collaboration: β=1  Θ=0 (YES!) (confirming the lack of correlations between k and w)  The weight between two authors does not depend on their degree. Airline network: β=1.5  Θ=0.5 (YES!)  The traffic between two airports depends on their individual degrees. Network Science: Weighted Networks Barrat, Barthelemy, Pastor-Satorras, Vespignani, PNAS 2004 WEIGHT CORRELATIONS(2)

Weight correlation: Scaling relationship: E. coli Network Science: Weighted Networks Macdonald, Almaas, Barabasi, Europhys Lett 2005 WEIGHT CORRELATIONS(2)

Weight correlation: Scaling relationship: Weight-degree: AirlineCollaboration β=1.5β=1 Θ=0.5Θ=0 Network Science: Weighted Networks Macdonald, Almaas, Barabasi, Europhys Lett 2005 SUMMARY

The end Network Science: Weighted Networks

WEIGHTED CORRELATIONS (Clustering, Assortativity) Network Science: Degree Correlations

If c i w /c i >1: Weights localized on cliques If c i w /c i <1: Important links don’t form cliques If w and k uncorrelated: c i w =c i Un-weighted Weighted If w ij =  c i =c i w a ij a ih a jh =1 only for closed triangles Barrat, Barthelemy, Pastor-Satorras, Vespignani, PNAS 2004 WEIGHTED CLUSTERING COEFFICIENT

Weighted Collaboration network: C(k) and C w (k) are comparable Airline network: C(k) < C w (k): accumulation of traffic on high degree nodes Network Science: Weighted Networks EXAMPLES: WEIGHTED CLUSTERING COEFFICIENT

If k nn w (i)/k nn (i) >1: Edges with larger weights point to nodes with larger k Un-weighted Weighted If w ij =  k nn,i =k nn,i w Measures the affinity to connect with high- or low-degree neighbors according to the magnitude of the interactions. Network Science: Weighted Networks Barrat, Barthelemy, Pastor-Satorras, Vespignani, PNAS 2004 WEIGHTED ASSORTATIVITY

Weighted Assortative behavior in agreement with evidence that social networks are usually have a strong assortative character Assortative only for small degrees. For k>10, k nn (k) approaches a constant  uncorrelated structure in which vertices with very different degrees have similar k nn Network Science: Weighted Networks EXAMPLES: WEIGHTED ASSORTATIVITY

If Y 2 (i) » 1/k i 1: No dominant link weights If Y 2 (i) ~ 1/k i : A few dominant link weights Y 2 (k) » 1/k  No dominant connection Air Traffic: Network Science: Weighted Networks DISPARITY

~ k Mass flows along linear pathways E. Almaas, B. Kovács, T. Vicsek, Z. N. Oltvai, A.-L. Barabasi. Nature, 2004 Network Science: Weighted Networks APPLICATION: DISPARITY IN METABOLIC FLUXES

Inhomogeneity in the local flux distribution ~ k Mass flows along linear pathways E. Almaas, B. Kovács, T. Vicsek, Z. N. Oltvai, A.-L. B. Nature, 2004; Goh et al, PRL Network Science: Weighted Networks E. Almaas, B. Kovács, T. Vicsek, Z. N. Oltvai, A.-L. Barabasi. Nature, 2004

BETWEENNESS CENTRALITY Network Science: Degree Correlations

Load: # of packets that a node should handle during the shortest path-based transport from all nodes to all others. 1 1/2 1 1 History: [Freeman ‘77]  Measure of “centrality” of a person in a social network.  How “influential” is a person, assuming that influence is determined by the mutual communication among the population via shortest paths? Network Science: Weighted Networks Goh et al, PRL 2001 BETWEENESS CENTRALITY OR LOAD IN NETWORKS

1 1/2 1 1 σ st : total number of shortest paths from node s to node t and σ st (v) is the number of shortest paths from s to t going through v. Betweenness centrality (BC) of a node v: BC scales as the number of pairs of nodes (s ≠ t ≠v) so we rescale it by (N-1)(N-2)/2 (N: number of nodes in the giant component) Network Science: Weighted Networks BETWEENNESS CENTRALITY OR LOAD IN NETWORKS

N 1 (N 2 ) is the number of nodes in region C 1 (C 2 ). First equality: the term for which s and t are in the same region does not contribute since in this case σ st (v) = 0.  although v has a small degree, its BC is large as intuitively expected. C1C1 C2C2 v N1N1 N2N2 Network Science: Weighted Networks M. Barthelemy, Eur. Phys. Journal 2004 BETWEENNESS CENTRALITY OR LOAD IN NETWORKS

The load distribution follows a power law for scale-free networks. ER graph static model g=2.2 (  ) g=2.5 (  ) g=3.0 (  ) g=4.0 (  ) g=  (  ) P(g) g δ: load exponent. Network Science: Weighted Networks Goh et al, PRL, 2001 BC DISTRIBUTION ON NETWORK MODELS

The load distributions of many real-world SF networks also follow power laws. Collaboration network, protein interaction network, metabolic network of eukaryotes, etc. Internet, metabolic network of archaea, WWW, etc. P(g) g g Network Science: Weighted Networks Goh et al, PNAS 2002 LOAD DISTRIBUTION OF REAL NETWORKS

Assuming k i and g i are ordered in similar way, power-law load distribution implies the scaling relation g(k) ~ k h Internetconfig. modelcollaboration g(k): average load of the vertices of degree k g Network Science: Weighted Networks RELATIONSHIP BETWEEN BC AND DEGREE Gokk Pre 03

Both δ (load exponent) and η depend on the degree exponent γ. We need to understand this behavior. Network Science: Weighted Networks M. Barthelemy, Eur. Phys. Journal 2004 SUMMARY OF THE EMPIRICAL SCALING LAWS:

Tree (or tree-like sparse network): Arbitrary scale-free networks: The power law scaling is not proved for an arbitrary network. Network Science: Weighted Networks M. Barthelemy, Eur. Phys. Journal 2004 SUMMARY

ER graph P(g) g ER: does NOT hold HOLDS? Network Science: Weighted Networks M. Barthelemy, Eur. Phys. Journal 2004 Goh et al, PRL, 2001 SUMMARY: ERDOS-RENYI NETWORK

Links the δ (load exponent), η and the degree exponent γ. If we calculate one of them (e.g. η) then we get the other one (e.g. δ) For large g (and large k):  Network Science: Weighted Networks M. Barthelemy, Eur. Phys. Journal 2004 SCALING RELATIONS

For a tree (or a network that has no loops): η=2 This was shown by several groups: Goh et al, PNAS 2002, using a rate equation method for the BA model with m=1 (i.e. a tree). Szabo, Alava, Kertesz, PRE 2002, for a tree. Barthelemy EPJ 2004, heuristic argument.  For trees the load exponents is uniquely determined by the degree exponent. Network Science: Weighted Networks M. Barthelemy, Eur. Phys. Journal 2004 SCALING RELATIONS

MODELING WEIGHTED NETWORKS Network Science: Degree Correlations

MODELING GOAL: Generate a model that can explain the observed scaling in real networks: Help us calculate analytically the scaling exponents. Network Science: Weighted Networks A. Barrat, M. Barthalemy, A. Vespignani, PRL 92, (2004); PRE 70, (2004)

Growth: add a node with m links, connecting to a node i with probability: Weights: each new node brings a weight w 0, added to s i, updating the weights on the links of i: Network Science: Weighted Networks CMODEL Each link of node i will increase its weight proportional to the weight it already has.

Weight Rearrangements:  =0: BA model (all weights are equal to w 0 )  << w 0 : Negligible effect (   =0: BA model) Scientific collaborations: the birth of a new collaboration will not strenghten earlier collaborations  ~ w 0 : Traffic: the new traffic is transferred to the neighboring nodes  >> w 0 : a new edge generates a multiplicative effect, boosting the weight of the other links Network Science: Weighted Networks WEIGHTED NETWORKS: MODEL

Rate equation approach yields separate equations for s i and k i, whose solution is: Network Science: Weighted Networks A. Barrat, M. Barthalemy, A. Vespignani, PRL 92, (2004); PRE 70, (2004) ANALYTICAL RESULTS

Models of weighted networks SUMMARY Network Science: Weighted Networks Empirical results on weighted networks: Empirical Finding 1: P(s) and P(w) are fat tailed Weighted degree correlations and clustering coefficient Generating Weights: Universality and scaling in betweeness centrality