Hierarchical Organization in Complex Networks by Ravasz and Barabasi İlhan Kaya Boğaziçi University

Abstract Many real neworks have two properties: Scale-free A high degree of clustering These properties are consequence of a Hierarchical Organization, implying Small groups of nodes organize in a hierarchical manner into increasingly large groups, while maintaining the scale-free topology

Abstract-Cont’d In hierarchical networks, the degree of clustering follows a scaling law, which is used to identify presence of a hierarchical organization in real networks WWW, actor network, internet at domain level, semantic web complex networks are shown to be hierarchical by this scaling law Hierarchy is a fundamental characteristic of many complex systems

What do we know so far? Random graph theory by Erdos and Renyi, used to describe complex networks for almost four decades But we learnt that Complex Networks like WWW, sexual network, actor collaboration network are far from being random, but governed by strict organizing principles.

Principle 1: Most networks display a high degree of clustering There exists some groups of nodes highly interconnected to each other, but have less connections to outside of the group Clustering Coefficient: probability that two neighbors of a selected node, i, {having k i links to first neighbors} is also connected, given as: Actual links between neighbor s of i Total number of links, If neighbor s form a clique

Clustering-cont’d Measurements indicate that most real networks have a higher clustering coefficient than same size random networks, which is In real networks, clustering coefficient is independent of the number of nodes in the network, N. Probability that two nodes connected

Principle 2: they are scale-free It means that probability that a randomly selected node has k links, follows a power law

Proposed Models have problems: Random Network Model neither captures scale-free nor clustering nature of real networks Predicts an exponential degree distribution. C i decreases as N -1, with number of nodes Scale free model account for power-law degree distribution, but predicts clustering coefficient as a function of nodes in network.

A Disregarded but fundamental feature… Hierarchical Topology Inconsistency between models and empirical results stems from a hierarchical topology: Easily identifiable, modular groups of nodes: highly interconnected with each other, a few links to nodes outside the group Examples of such groups are: In society, friends or co-workers; in Internet, communities with shared interest; Models producing power-law {Scale-free models} only look at degree distribution, blind to see modular topology

Solution? Hierarchical Network Model Brings modularity, high degree of clustering, scale-free topology under a single roof. Assumption: modules combine into each other in a hierarchical manner, generating a hierarchical network. Produces a power law degree distribution and a clustering coefficient as a function of node degree, which is called scaling law, independent of network size. This scaling law is used to determine presence of hierarchy in real networks

Hierarchical Network Generation Start with a small cluster of densely linked five nodes, N=5 Generate four replicas of this module and connect 4 peripheral nodes of each cluster to the central node of original cluster, N=25 Generate 4 identical replicas of this 25- node cluster and connect 16 peripheral nodes of each cluster to the central node of the original cluster.

Hierarchical Network Generation

Statistical Properties of HNM HNM is scale-free Has a power –law degree distribution with Number of nodes, N=5 7

Statistical Properties of HNM Has a clustering coefficient independent of number of nodes in network

Statistical Properties of HNM “In Deterministic Hierarchical Networks, Clustering coefficient of a node with k links follows the scaling law”, Dorogovtsev et. al. which is

Statistical Properties of HNM As figure depicts, Scale-free model does not follow a clustering coefficient scaling law as well as random model. Instead, in SF networks clustering coefficient is independent of node degree, k. Therefore scaling law can be used to identify hierarchy in a complex network.

Statistical Properties of HNM In our 125- node HNM network center nodes of clusters follows a scaling law Cluster size Clustering coefficient Node degree / /8384

Hierarchical Organization in Real Networks To investigate such a hierarchy exists in real networks C(k) function is measured, for several networks where topological maps are available: Actor collaboration Network Language Network WWW Internet At AS level At router level Power grid

Actor collaboration network Connect to actors if they acted in same movie. N= nodes, links C(k)~k -1, thus network has a hierarchical topology. Majority of actors appear in only 1 movie, small k, and neighbors are part of same cast, thus have links to each other; Which produces a high C(k), actually 1.

Actor collaboration network High-k nodes includes many actors who acted in several movies, thus their neighbors are not necessarily linked to each other Which produces a smaller C(k).

Language Network Connect two words if they appear as synonyms in Webster dictionary It is scale-free C (k) versus degree curve follows a scaling law as C(k)~k -1 Thus network has a hierarchical topology

Internet at AS level Connect two nodes {Autonomous Systems, domains}, if there is at least a router connecting them. N=65520, links It has a hierarchical topology as C(k)~k -1

Internet at router level Connect two routers if they have link btwn them. It is scale free However its C(k) graph does not show a scaling law, instead it is independent of the degree k. It is non-hierarchical

Power Grid Network Nodes are generators, transformers Links are high power transmission lines 4941 nodes, links C(k) is independent of k Possibly because of geographical organization and constraints like high- distance link costs

Stochastic Hierarchical Model Is this scaling law is universal or could there be different scaling exponents? We show that these scaling exponent changes as we create a stochastic version of deterministic Network model.

Stochastic Hierarchical Network generation Start with a five-node small cluster of densely linked nodes Make 4 copies of the cluster Pick randomly p faction of the nodes of new clusters, and connect them independently to the original cluster nodes. To select whom to connect use preferential attachment In the next step do this operation for p 2 faction of the nodes of newly added 4 clusters of 25-nodes.

Properties of Stochastic Hierarchical Network Model : Degree Distribution

Properties of Stochastic Hierarchical Network Model : Clustering Coefficient

Conclusion It is clear that most networks have a nodular topology quantified by the large clustering coefficient that they display. Many small densely linked clusters are combined to form larger but less cohesive groups, which in turn combine again to form less interconnected and larger groups

Conclusion – Cont’d Hierarchical nature of scale free networks are captured by a simple scaling law, offering us a straight forward method to identify hierarchy in real networks.