Inferring Coarse Views of Connectivity in Very Large Graphs
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1 Inferring Coarse Views of Connectivity in Very Large Graphs Reza Motamedi, Reza Rejaie, Walter Willinger, Daniel Lowd, Roberto Gonzalez 10/8/14 1
2 Introduction! Large-scale networked systems (e.g. OSNs) are often represented as graphs! Characterizing the connectivity structure of such a graph provides deeper insights about the system! Coarse view of a graph allows a top-down analysis Identify a few tightly connected regions along with their inter-, and intra-region connectivity If needed/desired, zoom in on individual region and recurse! How can one capture a coarse view of large graphs? 10/8/14 2
3 Obtaining coarse view of a graph! Community detection techniques optimize an objective function Detects communities with 100s of nodes in real-world graphs Some techniques have limited scalability! Graph partitioning techniques divide the graph into strongly connected partitions May produce balanced partitions May require seeds for each partition or the number of partitions as input 10/8/14 3
4 This paper presents! The design of a scalable technique (WalkAbout) to infer coarse (regional) views of a graph! An illustration of WalkAbout in action for inferring the regional connectivity of Flickr, Twitter, Google+! A study of the relationship between regional- and community-level views of a large graph! An initial attempt at answering the question Are (Flickr) regions meaningful? 10/8/14 4
5 Random Walks (RW)! Consider an undirected, connected, non-bipartite graph G = [V, E]! The probability that a very long RW visits node x converges to deg(x) 2 E T G (ε)! The mixing time is the walk length at which the probability of being at node x is within ε of the stationary distribution We use mixing time rather informally, not specifying ε 10/8/14 5
6 Behavior of Many RWs! Starting V RWs in parallel (one from each node)! V(x,wl): the expected number of RWs that are at node x after wl steps! As wl reaches the mixing time, the number of walkers at node x converges to V(x, wl) V deg(x) 2* E deg(x) => V(x, wl) 2 E V! degree/visit ratio (dvr) converges to average node degree 10/8/14 6
7 Validation through Simulations! Use simulation over synthetic graphs to explore the dependency of dvr on different parameters More results in the paper wl=10 wl=20 wl=50 wl= Avg. degree=24.74 Avg. degree=33.94 Avg. degree=44.30 Avg. degree= PDF 0.1 PDF 0.1 dvr dvr dvr 40 deg<50 deg> wl /8/14 7
8 Detecting Regions Key Idea! Suppose a graph consists of a few weakly connected regions! Starting RWs from randomly selected nodes on graph G = [V, E] that has multiple regions Region i is G i = [V i, E i ]! If wl is close to the mixing time of regions, a majority of RWs remain in their starting region the graph can be viewed as disconnected regions deg(x) dvr i (x) = E[V(x, wl)] = 2 E i V i! dvr i (x) converges to average node degree of region i 10/8/14 8
9 Key Idea (cont d)! Regions with different average degree form separate peaks in the dvr histogram Region: a non-overlapping range of dvr values! Formation of peaks is a transient phenomenon As wl increases beyond the mixing time of regions, dvr for all nodes converges to a single value Ø The similarity of dvr implies tighter connectivity among nodes in a region Ø dvr signal is indirect and efficient => scalable 10/8/14 9
10 Validation on Synthetic Graphs! A graph with two regions (average degree of 70, 60) connected with b bridge edges.! Only changing a single region or the bridge PDF pdf PDF Avg. Degree dvr Region size vdr x Bridge Size dvr /8/14
11 WalkAbout! Using many short RWs to infer/explore regional connectivity of large graphs The number of regions, nodes per region, and determining inter-, intra-region connectivity! Basic challenges The variation and rate of convergence of dvr is inversely proportional with node degree (i.e. noise of low degree nodes) Regions having similar average deg. & different mixing times! Identifying regions in two steps: Detecting the core (high degree) nodes of each region Mapping low degree nodes to the detected cores per region 10/8/14 11
12 WalkAbout Main Steps! Emulating RWs and generating the dvr histogram Removing low degree nodes ( D min ) to reduce noise! Identifying core of each region Search for the walk length that leads to pronounced peaks Detect a peak & its associated dvr range => nodes per region! Mapping low degree nodes to cores Based on the relative reachability (using multiple RWs)! Producing the regional view 10/8/14 12
13 Inferring vs Exploring! WalkAbout provides a few parameters that affect the resulting regional view (, wl) D min Parameters can be set based on the domain knowledge! Sensitivity to these parameters offers insight about the graph structure! Developing WalkAbout as an interactive tool with GUI Publicly available at 10/8/14 13
14 WalkAbout in Action! Inferring regional view of connectivity of the LCC for Flickr, Twitter and Google+! To contrast: Apply Louvain Communitiy detection method! Default setting D min = 500 See the tech report for results on the sensitivity to D min Flickr Twitter GPlus Nodes 1.6M 41.6M 51.7M Edges 31.M 1,468M 869.4M Communities 28K 39K 24K 10/8/14 14
15 Regional View of Flickr PDF PDF wl = 30, D min = 500 R0 R R3 R wl dvr R dvr Cores Regions Regions Regions Size %Nodes %Edges Avg.Deg Mod. R R R R R /8/14 15
16 Lessons Learned! Regions with closer dvr tend to have stronger interregion connectivity Incorrectly placed high degree nodes Regions with different sizes and mixing times! The number of peaks changes with walk length The number/selection of peaks affect the regional view! Identified regions could be very imbalanced in size Detecting possible sub-regions in a hierarchical manner 10/8/14 16
17 Regions & Communities! Comparing/relating the regional and community views Typical community is much smaller and more modular Largest communities have sizes comparable to regions Ø Orders of magnitude more communities! The highest degree nodes per region are placed in a few communities with size & modularity comparable to regions! Modularity Average Degree Size Louvain Large Louv 10/8/14 WA 17 0 TW G+ FL OR TW G+ FL OR TW G+ FL OR
18 Mapping Communities to Regions! Community c is mapped to region R that contains most of its nodes Mapping confidence: fraction of c s nodes located in R! Across regions of all OSNs For 75% of communities, the confidence is 100% For 90% of communities, the confidence is more than 80% Ø Regions can be viewed as a collection of communities Ø A coarser view of the graph 10/8/14 18
19 Per-Region Analysis of Communities! Are the characteristics of communities generally reveal the features of their region? No strong relation between the modularity of communities in a region and the modularity of the region! The inter-connectivity among communities is critical to determine features of each region Modulairty FL TW G Average Degree FL TW 2 G /8/ R0 R1 R2 R3 R R0 R1 R2 R3 R4 R5 0.2 FL R0 R1 R2 R3 R4 R5 R0 R1 R2 R3 R4 R0 R1 R2 R3 R4 R5 100 R0 R1 R2 R3 R4 R R0R1R2R3R4 R0 R1 R2 R3 R4 R5 100 R0 R1 R2 R3 R4 R5 Size TW G
20 Run-time! Comparing the run times of WalkAbout and the Louvain community detection technique On Intel X5650 (2.66GHz) computer with 72GB RAM! Splitting WalkAbout run time to dvr calculations to detect core, and Mapping of low degree nodes to those cores! WalkAbout exhibits a shorter run time for large graphs FL Louvain WA: Map to Core WA: dvr TW G /8/14 Second 20
21 A New Kind of Validation! Do users in a region exhibit a similar social attributes Need social context for users! 99K social groups in Flickr: group name, users/group Group name provides info about group interest or context Map each group to a region where most users are located Mapping confidence for R1-R4 is high even for large groups e.g. group names in R1 related to male nudity. Ø Social forces appear to derive the formation of regions Group Mapping Confidence /8/ R0 R1 R2 R3 R4
22 Conclusion & Outlook! WalkAbout, a new technique to infer/explore coarse views of large graphs! Applying WalkAbout to three major OSNs! Are regions meaningful? Relating the regional- and community-level views Showing social cohesion of regions in Flickr! Future plans Exploring the recursive application of WalkAbout Multi-scale characterization of graph connectivity and its application to examine graph evolution 10/8/14 22
23 Inferring Coarse Views of Connectivity in Very Large Graphs Reza Motamedi, Reza Rejaie, Walter Willinger, Daniel Lowd, Roberto Gonzalez 10/8/14 23
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