# SCAN: A Structural Clustering Algorithm for Networks

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1 SCAN: A Structural Clustering Algorithm for Networks Xiaowei Xu, Nurcan Yuruk, Zhidan Feng (University of Arkansas at Little Rock) Thomas A. J. Schweiger (Acxiom Corporation)

2 Networks scaling: #edges connected to respective vertex

3 Network basics Network graph G = {V, E} V is set of vertices, E is set of edges connecting the vertices Hub Hubs: vertices that bridge (many) clusters Outliers: vertices marginally connected to clusters featuring a weak association Outlier network clustering/graph partitioning: division of graph into set of disjoint sub-graphs in order to find hidden structures existing approaches aim at finding clusters based on the number of edges between vertices or clusters

4 SCAN A Structural Clustering Algorithm for Networks based on definitions of DBSCAN detects clusters, hubs and outliers using structure and connectivity of the vertices as a clustering criterion vertices sharing a certain quantity of neighbors should be grouped into one cluster, hubs and outliers should be isolated runtime complexity wrt. n vertices and m edges in a network: O(m) focusses on simple, undirected and unweighted graphs

5 Definitions for structure-connected clusters (I) VERTEX STRUCTURE, the structure of v is defined by its neighborhood STRUCTURAL SIMILARITY two objects sharing a similar structure in terms of their neighborbood, will have a large structural similarity

6 Definitions for structure-connected clusters (II) ε NEIGHBORHOOD apply threshold to structural similarity in order to assign cluster memberships ε = 0.65 CORE VERTEX core vertex shares structural similiarity of at least ε with at least μ neighbors

7 Definitions for structure-connected clusters (III) DIRECT STRUCTURE REACHABILITY if vertex is in ε Neighborhood of a core vertex, they should be in the same cluster w v ε = 0.65, μ = 2 STRUCTURE REACHABILITY a vertex is structure-reachable from, if there is a chain of vertices such that is directly structure-reachable from w ε = 0.65, μ = 2 v

8 Definitions for structure-connected clusters (IV) STRUCTURE CONNECTIVITY two non-core vertices belonging to the same cluster may not be structure-reachable, as the core-condition does not hold for them. However, they still belong to the same cluster, if they are structure-reachable from the same core vertex. This is called structure-connectivity. w u v ε = 0.7, μ = 3

9 Definitions for structure-connected clusters (V) STRUCTURE-CONNECTED CLUSTER a non-empty subset is called a structure-connected cluster, if all vertices in C are structure-connected and C is maximal wrt. structure reachability 1. Connectivity: 2. Maximality: CLUSTERING a clustering P consists of all structure-connected clusters

10 Definitions for structure-connected clusters (VI) If a vertex is not a member of any cluster, it is either a hub or an outlier, depending on its neighborhood. HUB if an isolated vertex a hub has neighbors belonging to two or more different clusters, it is 1. v is not a member of any cluster: 2. v bridges different clusters: OUTLIER an isolated vertex is an outlier, if and only if all its neighbors either belong to only one cluster or do not belong to any cluster 1. v is not a member of any cluster: 2. v does not bridge different clusters:

11 Algorithm SCAN for each vertex v not yet classified, it is checked whether vertex is a core if so, a new cluster is expanded from this vertex: 1. the algorithm generates a new cluster id, looks for all unclassified vertices in the neighborhood of the core and inserts them into a queue vertices which are directly structure-reachable from core vertex v 2. it traverses the queue until it is empty and establishes all vertices which are directly structure-reachable from the respective vertex w vertices which are structure-reachable from core vertex v (via vertex w) 3. the same cluster id is assigned to all those vertices and if they are not labeled as a non-member yet, they are also inserted into the queue if not, it is labeled as a non-member non-member vertices which have edges to two or more clusters, are classified as hubs. Otherwise, they are classified as outliers. ε-value between 0.5 and 0.8, μ = 2

12 Experimental Evaluation (I) Comparison to FastModularity algorithm: hierarchical network clustering algorithm optimizing the modularity modularity measures whether division of a network into communities is a good one in terms of many edges within communities and preferably little edges between communities 1. initally, each vertex is the only member of a community 2. iteratively, the algorithm greedily merges the two communities causing the largest increase of modularity, until all vertices are members of the same community

13 Experimental Evaluation (II) Customer Segmentation dataset

14 Experimental Evaluation (III) FastModularity clustering result SCAN clustering result

15 Experimental Evaluation (IV) Books about US politics dataset

16 Experimental Evaluation (V) FastModularity clustering result SCAN clustering result

17 Discussion + identifies not only clusters, but also outliers and hubs + linear run-time complexity wrt. # edges, each vertex is visited only once - performance highly depends on sensitive input parameters - ignores domain knowledge in clustering attributes - assumes that network is homogeneous and adjacency matrix is already defined

18 Q & A

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